Lecture notes for STAT3006 / STATG017
Stochastic Methods in Finance 1
Julian Herbert
Department of Statistical Science, UCL
2011-2012
Contents
1 Financial Markets and Products 8
1.1 Financial markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Equities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Fixed income (FI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Time value of money 12
2.1 Compound interest and present value . . . . . . . . . . . . . . . . . . . 12
2.2 Compound interest with non-annual payments . . . . . . . . . . . . . . 13
2.3 Present value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Government Bond valuation . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Introduction to Derivatives 18
3.1 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Option Payos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Speculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Combining derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Arbitrage and the pricing of forward contracts 28
4.1 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Example - Arbitrage opportunities in a forward contract . . . . . . . . 28
4.3 Pricing forward contracts for securities that provides no income . . . . 30
4.4 Value of a forward contract . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Forward contracts on a security that provides a known cash income . . 32
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Stochastic Methods in Finance 1
4.6 Known dividend yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7 Forward foreign exchange contracts . . . . . . . . . . . . . . . . . . . . 34
5 Pricing Options under the Binomial Model 35
5.1 Modelling the uncertainty of the underlying asset price . . . . . . . . . 35
5.2 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2.1 Arbitrage opportunity . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.2 No-arbitrage pricing . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 One-step binomial tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 A replicating portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 Risk-neutral valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.6 Appendix: A riskless portfolio . . . . . . . . . . . . . . . . . . . . . . . 42
5.6.1 Example revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.6.2 No-arbitrage pricing . . . . . . . . . . . . . . . . . . . . . . . . 43
5.6.3 Riskless portfolio - general result . . . . . . . . . . . . . . . . . 44
5.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Applications of the Binomial Model 46
6.1 The value of a forward contract . . . . . . . . . . . . . . . . . . . . . . 46
6.1.1 A replicating strategy . . . . . . . . . . . . . . . . . . . . . . . 46
6.1.2 Risk-neutral valuation . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 A European put option . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Two-step binomial trees . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.4 General method for n-step trees . . . . . . . . . . . . . . . . . . . . . . 51
6.5 Pricing of American options . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Calculus refreshers 55
7.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Chain rule dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Partial dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4 Linear ordinary dierential equations . . . . . . . . . . . . . . . . . . . 57
8 Continuous-time stochastic processes for stock prices 58
8.1 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.2 Other processes and Markov property . . . . . . . . . . . . . . . . . . . 59
8.3 Taking limits of the random walk . . . . . . . . . . . . . . . . . . . . . 60
8.4 Brownian motion (Wiener process) . . . . . . . . . . . . . . . . . . . . 61
8.5 Denition of Brownian motion . . . . . . . . . . . . . . . . . . . . . . . 62
8.6 Generalised Brownian Motion process . . . . . . . . . . . . . . . . . . . 64
8.7 Ito^ process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.8 A process for stock prices: the geometric Brownian motion . . . . . . . 66
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Stochastic Methods in Finance 1
8.9 Appendix: Brownian Motion as a limit of a discrete time random walk 68
9 Introduction to stochastic calculus and Ito^'s lemma 70
9.1 Ordinary and stochastic calculus . . . . . . . . . . . . . . . . . . . . . . 70
9.2 Ito^'s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.3.1 Derivation of the SDE of a generalised Brownian motion . . . . 73
9.3.2 Solution of the SDE of a geometric (exponential) Brownian motion 74
9.3.3 Logarithm of stock prices . . . . . . . . . . . . . . . . . . . . . 75
9.3.4 Generic transformation of stock prices . . . . . . . . . . . . . . 75
9.3.5 Forward price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.4 Appendix: second order eects in the limit . . . . . . . . . . . . . . . . 76
9.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 The Black-Scholes model 78
10.1 Lognormal property of stock prices . . . . . . . . . . . . . . . . . . . . 78
10.2 The Black-Scholes-Merton dierential equation . . . . . . . . . . . . . . 79
10.2.1 Assumptions of the Black-Scholes model . . . . . . . . . . . . . 79
10.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 82
10.3 Black-Scholes formulas for the pricing of vanilla options . . . . . . . . . 83
10.4 BSM PDE using a riskless portfolio approach . . . . . . . . . . . . . . 85
10.5 Simple extensions to the Black-Scholes model . . . . . . . . . . . . . . 86
10.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
11 Hedging and the Greeks 88
11.1 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
11.2 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
11.3 Gamma and gamma neutral portfolios . . . . . . . . . . . . . . . . . . 90
11.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
12 Volatility 93
12.1 Estimating volatility from historical data . . . . . . . . . . . . . . . . . 93
12.2 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
12.3 Volatility smiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
12.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
13 Risk-neutral pricing in continuous time 99
13.1 The risk-neutral process . . . . . . . . . . . . . . . . . . . . . . . . . . 99
13.2 Risk-neutral pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
13.2.1 Example: European call option . . . . . . . . . . . . . . . . . . 100
13.3 A useful log-normal distribution result . . . . . . . . . . . . . . . . . . 102
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Stochastic Methods in Finance 1
13.4 The risk-neutral Monte Carlo approach to derivative pricing . . . . . . 102
13.4.1 Simulating Geometric Brownian motions . . . . . . . . . . . . . 103
13.4.2 Generating random variables . . . . . . . . . . . . . . . . . . . . 103
13.5 Appendix - Risk neutral pricing and the Black-Scholes equation . . . . 104
iv J Herbert UCL 2011-12
STOCHASTIC METHODS IN FINANCE 1
2011{12
Julian Herbert
email: julian@stats.ucl.ac.uk
Lectures: Term 1, Tue 5{7pm, starting 4 October. No lecture in reading week.
In Course Assessment: In class assessment on Week 5; Tuesday 1 Novem-
ber 5pm
Course guidance notes
This introductory note provides some guidance around the course.
Aims of course
To introduce mathematical and statistical concepts and tools used in the nance in-
dustry, in particular stochastic models and techniques used for derivative pricing.
Prerequisites
No prior knowledge of nance is assumed, though an interest in the subject is a distinct
advantage.
A course covering probability and distribution theory, and a foundation in calculus
and dierential equations is required. For example you should be comfortable with
ordinary dierential equations (in particular be familiar with linear and exponential
growth equations), Taylor expansions, partial dierentiation and standard integration
techniques such as integration by parts and substitution. You should also be comfort-
able with conditional probabilities, Markov processes, probability spaces, probability
density functions and normal distributions.
Stochastic Methods in Finance 1
Course content
1. Intro to nancial products, markets and derivatives
2. Time value of money
3. Arbitrage pricing
4. The Binomial pricing model
5. Brownian motion and continuous time modelling of assets
6. Stochastic calculus
7. The Black-Scholes framework
8. Risk-neutral pricing
Lecture notes
The printed notes should be used as a guide to some of the key topics in the course,
and are aimed at providing written copies of some of the working in the lectures so
that students can concentrate on the subject without needing to copy too many lines
of algebra. At the end of each set of printed lecture notes a selection of further reading
is provided, which will give a guide to the sections of the course-related texts in which
the lecture topic can be found, as well as a selection of more advanced reading for the
interested student.
Note that the printed lecture notes do not contain everything that is outlined in
the lectures, and so should not be used as a substitute for lecture attendance. The full
set of course material consists of material presented in the lectures, combined with the
handouts and the ideas and techniques developed through exercise sheets and the two
workshops.
Exercise sheets
The exercise sheets are an integral part of the learning on the course. These should
be attempted on a weekly basis. As well as providing opportunities for consolidating
learning from the lectures, these exercises will also be used as a learning tool themselves
to explore the concepts in the course. Therefore a student that does not complete all
the exercises will not have achieved the full set of learning outcomes for the course.
These exercise sheets will not be marked and so do not need to be handed in.
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Stochastic Methods in Finance 1
The course builds quickly on material learnt previously. You should therefore plan
each week to review your lecture notes, review the previous week's exercise solutions,
as well as attempt the current week's exercises before the next lecture. It is strongly
recommended that you attempt the exercises each week as subsequent lectures usually
build on these exercises.
Workshops
The two workshops also form an integral part of the learning for the course. They will
provide an opportunity to further develop your understanding of concepts introduced
in the lectures, and also introduce some new techniques. They are usually timetabled
for Friday afternoons - see the Statistics timetable for the specic dates and times for
this course.
In course assessments
The in course assessment (ICA) will be a closed book classroom test during the lecture
of the 5th week of term.
Oce hours
Oce hours for students requiring help with course material will be after each lecture,
arranged by prior appointment only. Please indicate the area of the course and prob-
lem that you require help on when arranging this.
Reading List
The course is self sucient, but this is a list of both related texts and some wider
reading. There is now a vast range of books on mathematical nance and derivative
pricing, so this is necessarily just a small selection with the aim of providing a good
range of approaches taken.
Additional books are also referenced at the end of some of the lecture notes in the
reading list sections where they are particularly relevant to the lecture topic.
Main Related Texts
- John C. Hull (2005) Options Futures and Other Derivative Securities. Now in the
8th edition, Prentice Hall.
- Martin Baxter & Andrew Rennie (1996) Financial Calculus. Cambridge University
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Stochastic Methods in Finance 1
Press.
- Paul Wilmott (2001) Paul Wilmott Introduces Quantitative Finance Wiley
Further introductory texts in the area
These books provides good introduction to the main topics covered in the course, each
presenting the material with a dierent perspective.
- Salih Neftci (2000) An Introduction to the Mathematics of Financial Derivatives.
Academic Press.
Excellent coverage of the mathematics needed for pricing derivatives, explained in an
easy to follow way. Includes a good introductory presentation of stochastic calculus.
- Alison Etheridge (2002), A course in Financial Calculus. Cambridge University
Press
A nice introduction to the principals underlying derivative pricing, with the emphasis
on laying foundations of understanding. Takes a similar approach to the Baxter and
Rennie book, but at a slightly simpler level.
- Wilmott, Howison, and Dewynne (1995), The Mathematics of Financial Deriva-
tives. Cambridge University Press.
This book presents derivative pricing from a physicist's perspective, and so focuses on
solving partial dierential equations, a less common perspective in more recent books.
Worth a read for a good explanation of the p.d.e.s relevant to nance, and particularly
for those with a background in applied maths, but not essential for this course.
- Tomas Bjork (1998) Arbitrage Theory in Continuous Time. Oxford University
Press.
Good coverage of the key topics. Could be used as an excellent introductory text if you
are comfortable with a more mathematical presentation. Provides some good intuition.
- Desmond Higham (2004), An Introduction to Financial Option Valuation. Cam-
bridge University Press
A simple presentation of the basics of options pricing, this book also provides a guide
to computational implementation of the results, with guides to Matlab code at the end
of each chapter. Useful for those who want to work on their own to implement and
extend the results we see in the course.
- Sean Dineen - Probability theory in nance - a mathematical guide to the Black-
Scholes formula. American Mathematical Society, Graduate Series in Mathematics.
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Stochastic Methods in Finance 1
A good introduction to the the use of martingales in no-arbitrage pricing. Provides
a more mathematically rigorous treatment of topics covered in this course, but also
takes the time to introduce the motivation for and intuition of the results presented,
and proceeds at a slow pace. Chatty style of writing.
More Advanced Texts
- Ricardo Rebonato (2004) Volatility and Correlation; Second Edition; Wiley.
- Steven Shreve (2004) - Stochastic Calculus for Finance - 1 Discrete-Time models;
2 Continuous-Time models. Springer.
A rigorous, mathematical approach, and goes into much more depth of the underlying
probability theory results used in martingale pricing. Two volumes.
- Marek Musiela and Marek Rutkowski - Martingale Methods in Financial Mod-
elling (Springer)
In-depth coverage of a wide range of models and products, using a rigorous, probability
based approach.
- John Cochrane - Asset Pricing - Excellent book that presents general frameworks
for the range of pricing techniques, with an aim of unifying asset pricing techniques and
"clarifying, relating and simplifying the set of tools we have all learned in a hogdepodge
manner". Hence goes beyond no-arbitrage derivative pricing covered in this course to
include other types of pricing models (including CAPM etc), and on the way develops
an awareness of the links between theoretical nance and macro-economics. Also in-
cludes interesting chapters on the theory of statistical estimation. Aimed at graduate
level, so I suggest reading after completing some introductory courses in asset pricing.
- Stanley Pliska, Introduction to Mathematical Finance: Discrete Time Models
(Blackwell Publishing)
More of an introductory text, but I have put in this section as it looks more widely
at models beyond derivative pricing. Provides a rigorous study of use of risk-neutral
probability measures, takes a mathematical approach, but is still accessible if you are
comfortable with calculus and elementary probability theory, and prepared to put some
work in.
- Glenn Shafer & Vladimir Vovk (2001) Probability and Finance: It's Only a Game!
Wiley
Interesting book that presents a new framework for looking at probability and nance.
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Stochastic Methods in Finance 1
Tough going without a strong background in probability theory.
Background and references in nance
These books provide guides to nancial concepts and jargon, and may be useful as
references as the course progresses, or for students with no prior nance knowledge.
- Michael Brett (1995) How to read the nancial pages. 4th edition, Century Busi-
ness.
- Brian Butler, David Butler & Alan Isaacs (1997) Dictionary of nance and bank-
ing. Oxford University Press.
- John Downes, Jordan Goodman (2003) Dictionary of Finance and Investment
Terms 6th edition, Barron's.
A selection of other nance related books
- Richard Lindsey and Barry Schachter (2007), How I became a quant - insights from
25 of Wall Street's elite (Wiley)
Twenty ve dierent industry professionals contribute a chapter each, and therefore
this book provides an eclectic mix of views on a range of areas and industries in which
quantitative techniques are used. Also highlights some of the challenges of developing
modelling techniques that are eective in a business environment, as well as a number
of personal views on career steps and challenges. Worth bearing in mind that you
don't have to empathise with all of the authors (or even any) to have a rewarding
and successful career in nance - there is a huge range of dierent working cultures in
industry.
An excellent read for those interested in using their stats, maths, economic or pro-
gramming skills in the nance industry. An even better read for those of you who
have these skills and are not currently interested. And probably essential reading if
there is anyone who thinks they know they only want to research hybrid quantos stoch.
vol. models for one of 4 select IBs/develop equity stat-arb trading algorithms on their
favourite platform... etc.
- Roger Lowenstein (2000) - When genius failed - the risk and fall of Long Term
Capital Management
Well told story of the potential nancial crises resulting from hedge fund LTCM's po-
sitions in 1998, and the response of regulators and investment banks. A good case
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Stochastic Methods in Finance 1
study in the importance of stress testing underlying model assumptions.
- Michael Lewis - Liars Poker
Classic book telling the story of the world of bond trading and investment banking in
the eighties. Still relevant today.
- Andrew Ross Sorkin (2009) Too big to fail - Inside the battle to save Wall Street
Another well put together telling of a nancial crisis, this time the story of the fall of
Lehmans in 2008, based on interviews with many of the key players in banking and
regulation at the time.
7 J Herbert UCL 2011-12
Chapter 1
Financial Markets and Products
In this lecture we provide a basic introduction to the nancial markets that will provide
the background to the problems in nance that we will study in the course. We also
look at the types of risks that nancial institutions operating in these nancial markets
are exposed to.
1.1 Financial markets
The nancial markets are institutions and procedures that facilitate transactions in all
types of nancial securities (claim on future income).
We can have organised security or stock exchanges, or over-the-counter markets
(for very specic requests).
Why do we have nancial markets?
In order to transfer eciently funds from economic units who have them to people
who can use them
To reallocate risk and to manage risk
1.2 Equities
Equity (or stock or share) is the ownership of a small piece of a company (claim
on the earnings and on the assets of the company).
The shareholders are the people who own the company, and have a say in the
running of the business by the directors.
Most companies give out lump sums every 6 months or a year, which are called
dividends.
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Stochastic Methods in Finance 1
The shares of large companies are traded in regulated stock exchanges.
Later in the course what we will want to study (and model) is the stock price. The
price of a share is determined by the market, and depends on the demand and supply
of shares in the market.
1.3 Fixed income (FI)
FI securities are nancial contracts between two counterparties where a xed exchange
of cash ows is agreed (which depends on the interest rate).
Interest rate is the cost of borrowing or the price paid for the rental of funds and is
usually expressed as % per year.
Broadly, there are two types of interest:
xed interest rate: locked for a certain period of time
oating interest rate: changes from time to time
Bond: a debt security that promises to make payments periodically (are issued by
government, companies, local authorities, etc.)
zero-coupon bond: pays only a known xed amount (the principal) at some given
date in the future (the maturity date).
coupon-bearing bond: similar to zero-coupon bond, except that it also pays
smaller quantities (the coupons) at specic intervals up to and including the ma-
turity date.
Example. A bond that pays 1,000 GBP in 3 years with a coupon of 10% per year
paid semi-annually. Income from the bond:
time from now $
6 months 50
12 months 50
18 months 50
24 months 50
30 months 50
36 months 1,050
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Stochastic Methods in Finance 1
) In the US:
Treasury bills: maturity 1 year (normally zero-coupon)
Notes: maturity between 2 and 10 years (coupon-bearing bonds)
Bonds: maturity 10 years (long bond = 30 years)
) In the UK: bonds that are issued by the government are called gilts.
An important question is: what is the value of the bond? For instance, if a T-bill
pays 1,000 USD in one year, how much should I pay for it now? (linked to the time
value of money)
Issuance of bonds and equities are the two main sources of funding for companies.
In the event of liquidation (or bankruptcy) of a company the debt holders (i.e. bond
holders) are paid out of the remaining assets of the company before the equity holders.
Only when all debt holders have received all promised cash ows will equity holders
start to receive any compensation, if there is anything left by then. On the other hand,
if a company increases its value considerably, then bond holders will still only receive
the xed payments agreed in the bond contract, whereas the value of the equities does
not have a ceiling. This means that equities are generally more risky than bonds, in
the sense that there is more variance around their expected returns.
1.4 Currencies
Dierent countries use dierent currencies. If you want to exchange one currency for
another you have to use the spot exchange rate.
Example 1,000 GBP are worth 1,480 USD if the spot exchange rate GBP/USD is
1.48. This is important if you have to consider the exchange rate in transactions.
The spot exchange rate may be regulated by the government of the country (in
order to control the growth and the investment of foreign capital) or it can uctuate
freely (determined by the market).
A currency is strong when its value is rising relative to other currencies (it appre-
ciates) and weak when it is falling (depreciates).
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1.5 Commodities
These are raw products such as precious oil, metals, food products, etc. Some com-
modities have organised exchanges for their trading. Prices uctuate according to
demand and supply.
1.6 Indices
An index is a weighted sum of a collection of assets, which is designed to represent the
whole market. These are often held by investors to gain exposure to investment in a
whole market.
Example FT-SE 100 is designed to represent the equity market in the UK.
1.7 Further reading
{Paul Wilmott Introduces quantitative nance { sections 1.1 - 1.5
{Robert Kolb and Ricardo Rodriguez (1996) Financial Markets, Blackwell
{Michael Brett (1995) How to read the nancial pages. 4th edition, Century Business.
11 J Herbert UCL 2011-12
Chapter 2
Time value of money
One of the most important concepts in nance is that $1 today is worth more than
$1 received in the future. This is referred to as the time value of money. One way
of looking at this this concept is that $1 today can generate income without any risk
through interest.
Note: To simplify things in this section we make the assumption that the interest
rate is constant and the same for all maturity dates 1.
2.1 Compound interest and present value
The compound interest is the interest that occurs when the interest paid on an invest-
ment during the rst period is added to the principal during the second period. In
other words, interest is paid on the interest received in the previous period.
Example. We invest $100 for one year at 6%. The future value after one year (FV1)
equals the present value PV (the principal) plus the interest:
FV1 = PV + PV 0:06 = PV (1 + 0:06)
Similarly, the future value after two years (FV2) equals the value after one year FV1
plus the interest computed on the future value itself:
FV2 = FV1 + FV1 0:06 = FV1(1 + 0:06) = PV (1 + 0:06)2
In general we obtain:
FVn = PV (1 + 0:06)
n
1In reality the risk-free interest rate depends on the length of the time period over which the
money is held. For more details of this see for example Hull chapter 4.
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Stochastic Methods in Finance 1
If we denote the interest rate by r, the value of the investment after n years is:
FVn = PV (1 + r)
n
2.2 Compound interest with non-annual payments
Let us study the previous example, but under the assumption that the interest rate is
paid every 6 months, so that 6 / 2 = 3% is paid every 6 months.
0 months ) 100
6 months ) 100 + 100

0:006
2

= 100

1 + 0:006
2

12 months ) 100

1 + 0:006
2



1 + 0:006
2

= 100

1 + 0:006
2
2
Note that: 100

1 + 0:006
2
2
> 100 (1 + 0:006)
18 months ) 100

1 + 0:006
2
3
24 months ) 100

1 + 0:006
2
4
So if compounding occurs m times during a year, then the future value after n years
is:
FVn = PV

1 +
r
m
nm
where nm is the total number of interest payments.
Example. Let us consider what is the value of $100 after one year compounded at
15% if we compound in dierent ways:
(a) annually (m = 1): $115.00
(b) semi-annually (m = 2): $115.56
(c) quarterly (m = 4): $115.87
(d) monthly (m = 12): $116.08
(e) weekly (m = 52): $116.16
(f) daily (m = 365): $116.18
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Stochastic Methods in Finance 1
There seems to be a limit as m!1. If n = 1:
FV1 = PV

1 +
r
m
m
lim
m!1

1 +
r
m
m
= er
Therefore we obtain:
FV1 = PV e
r
When we use the limit as m!1 we say that we use continuous compounding.
Example. Find the FV of $100 after 1 year and 3 years continuously compounded
at 10%.
FV1 = 100 e0:101
FV3 = 100 e0:103
This happens because
lim
m!1

1 +
r
m
mn
= (er)n = ern
and therefore
FVn = PV e
rn
Similarly, if we want to compute the value, say, after 6 or 18 months, we convert the
months into years (0.5 and 1.5 respectively) and then compute:
FV6months = 100 e0:100:5
FV18months = 100 e0:101:5
Let us consider now the following problem: we can either compound continuously
at rate rc or compound m times a year at rate rm. What value should rc take with
respect to rm in order to obtain the same FV after n years (and vice-versa)?
Let us start from the usual formulas:
If we use continuous compounding we have: FVn = PV ercn
If we use m period compounding we have: FVn = PV

1 + rm
m
nm
From here, if we want the two future values to be the same, we have to equate them
and solve:
ercn =

1 +
rm
m
nm
Therefore we obtain:
14 J Herbert UCL 2011-12
Stochastic Methods in Finance 1
rm = m

erc=m 1

rc = m log

1 + rm
m

2.3 Present value
Using the formulae in the previous sections, we can calculate the present value of a
future cash amount. This is essentially how much should be invested today to achieve
a certain amount in the future if all interest payments are re-invested at the same
interest rate.
For m = 1 we have: FVn = PV (1 + r)n ) PV = FVn(1 + r)n
For general m: FVn = PV

1 + r
m
nm ) PV = FVn 1 + rmnm
For continuous compounding: FVn = PV ern ) PV = FVnern
2.4 Government Bond valuation
The value of a bond is the present value of its expected cash ow. The best way to
understand how this works is by looking at some examples.
Example. A T-bill 2 pays $1,000 after one year and the interest rate is 6% with
continuous compounding. What is the value of the bond today?
Value = PV ($1; 000) = $1; 000e0:061
If the interest rate increases, the value of the bond goes down.
Example. A coupon-bearing, default-free bond pays 6% coupons semi-annually and
gives back a $1,000 principal after 3 years. The rate that we use for discounting is 5%
with continuous compounding. What is the value of the bond?
Future cash ow:
2A T-bill is issued by the US Government and therefore assumed riskless i.e. that there will be no
default and so the promised payments are guaranteed.
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Stochastic Methods in Finance 1
6 months $30
12 months $30
18 months $30
24 months $30
30 months $30
36 months $1,030
How do we compute the value? We said that it equals the present value of its
expected cash ow, so we have to compute the present values of the cash ows above
and sum them:
Present values:
6 months $30 e0:050:5
12 months $30 e0:051
18 months $30 e0:051:5
24 months $30 e0:052
30 months $30 e0:052:5
36 months $1; 030 e0:053
Total
Example. What is the value of a bond that pays $1,000 annually forever, when the
discount rate is 10% and is applied using annual compounding? (perpetual bond)
1st payment after 1 year: PV = 1;000
1+r
2nd payment after 2 years: PV = 1;000
(1+r)2
: : :
Once again, the value of the bond is the sum of these present values, that is:
Value =
1X
i=1
1; 000
(1 + r)i
= 1; 000

1
1 1
1+r
1
!
= 1; 000

1 + r
r
1

= 1; 000

1 +
1
r
1

=
1; 000
r
In this case, therefore, the value of the bond is $1;000
0:1
= $10; 000.
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Stochastic Methods in Finance 1
2.5 Further reading
Paul Wilmott Introduces quantitative nance { section 1.6 covers ground sim-
ilar to this lecture, and also provides a dierential equation based presentation of
continuously compounded interest rates, which will be useful for later lectures in the
course.
17 J Herbert UCL 2011-12
Chapter 3
Introduction to Derivatives
A derivative is an instrument whose value depends on the values of other more basic
underlying securities.
Some examples are:
Forward contracts
Future contracts
where you have the obligation to buy or sell a security at some time in the future. We
also have:
Options, i.e. the option to buy or sell something in the future, and
Swaps, which involve the exchange of future cash ows.
Where can we buy a derivative?
(a) Organised exchanges (standard products)
(b) Over the counter (from nancial institutions, can be non-standard products)
Why are derivatives useful?
(a) We can use them to protect ourselves from future uncertainty (hedge the risk)
(b) We can use them to speculate on the future direction of the market
(c) Or to change the nature of an asset or liability, often its risk prole (e.g. to swap
a xed rate loan with a oating rate loan) or its tax status
18
Stochastic Methods in Finance 1
(d) Or to change the nature of an investment without selling one portfolio and buying
another
Who uses derivatives?
(a) Hedgers (aim to reduce risk)
(b) Speculators (\gamble" on the future direction of the markets)
(c) Arbitrageurs (aim to make money without any risk)
3.1 Forwards
A forward contract is an agreement to sell or buy an asset at a certain time in the
future for a certain price (delivery price).
If the buyer knows that he will need to buy the asset in the future, then enter-
ing into a forward contract provides the benet of eliminating the uncertainty of the
price the buyer will have to pay in the future. Thus the buyer has protection against
rises in the price of the asset. Similarly, if a seller has an asset that he knows he
will want to sell in the future, then entering into a forward contract guarantees the
amount he will receive for it, eliminating uncertainty in the price received for the asset.
The main features of this contract are:
It can be contrasted with a spot contract (buy or sell immediately)
It is an over-the-counter product
Usually no money changes hands until maturity
If you agree to buy, you have a long position
If you agree to sell, you have a short position
Example. A forward contract can be used, for instance, to hedge foreign currency
risk. Suppose an American trader expects to receive $1m in 6 months and wants to
hedge against exchange rate movements. He enters into an agreement to sell dollars
and buy pounds after 6 months at a rate 1.60. He is long in pounds and short in
dollars. His net payo per forward contract is shown in Figure 3.1.
If after 6 months the rate is 1.7, then without the forward contract he would need
$1.7m to buy $1m, but he can actually buy them with only $1.6m, due to his forward
contract,) he makes a prot compared to his situation without the forward contract.
On the other hand, if the rate falls to 1.5, then without the forward contract he
would need only $1.5m to buy $1m, but under the forward contract he has to buy it
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Stochastic Methods in Finance 1
-2
-1.5
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5
$/£ rate at maturity
Pa
yof
f
Figure 3.1: Payo for foreign exchange forward, long position
at 1.6 ) he has a loss compared to his situation without the forward contract. The
crucial point is that entering into the forward contract means that he has xed the
dollar amount he will receive for the $1m in 6 months time. He has eliminated the
uncertainty - we say he has hedged his position.
Say, instead, that a trader has chosen to be long in dollars and short in pounds.
His net payo for this position is shown in Figure 2.
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5
$/£ rate at maturity
Pa
yof
f
Figure 3.2: Payo for foreign exchange forward, short position
So forward contracts are designed to neutralise risk by xing the price that the
hedger will pay or receive for the underlying asset.
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3.2 Futures
A future contract is similar to a forward contract but has some dierences:
(a) It is traded on an exchange
(b) Everything about the contract and the underlying asset is well specied
(c) The value of the contract is calculated daily and any prot or losses are adjusted
in an account that you have with a broker (margin account)
(d) Closing out a futures position means entering into osetting (trade)
(e) Most contracts are closed out before maturity
3.3 Options
A call option is an option (but not an obligation) to buy a certain asset by a
certain date for a certain price (strike price).
A put option is an option to sell a certain asset by a certain date for a certain
price.
An American option can be exercised at any time up to expiration date.
A European option can be exercised only on the expiration date itself.
There are two sides to every option contract:
(a) The party that has bought the option (long position)
(b) The party that has sold (or written) the option (short position)
3.4 Option Payos
A European call option will be exercised if the asset price is above the strike price
at the option expiration date, and then the buyer of the option will receive a payo
worth ST X, where X is the strike price and ST is the value of the asset at the option
expiration date. If the asset price is below the strike price at the option expiration date,
then the option will not be exercised, and will be worthless. Therefore the European
call option has a payo function max[ST X; 0]. Similarly, the payo function for a
put option is max[X ST ; 0]:
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The payo from a derivative can be represented in a payo diagram. Payo di-
agrams are useful tools for understanding options and combinations of options. The
payos from options are shown in the following diagrams, where X is the strike price
and ST is the asset price at time T (expiration date).
-20
-10
0
10
20
30
40
50
60
70
0 20 40 60 80 100 120
underlying asset price at T
Pa
yof
f
K
Figure 3.3: Payo for long position in European call, strike price K
-20
-10
0
10
20
30
40
50
60
0 20 40 60 80 100 120
underlying asset price at T
Pa
yof
f
K
K
Figure 3.4: Payo for long position in European put, strike price K
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Stochastic Methods in Finance 1
-70
-60
-50
-40
-30
-20
-10
0
10
20
0 20 40 60 80 100 120
underlying asset price at T
Pa
yof
f
K
Figure 3.5: Payo for short position in European call, strike price K
-60
-50
-40
-30
-20
-10
0
10
20
0 20 40 60 80 100 120
underlying asset price at T
Pa
yof
f
K
K
Figure 3.6: Payo for short position in European put, strike price K
Derivatives can be used for a number of reasons, in particular, options can be used
to;
(a) Hedge: We can use them to protect ourselves from future uncertainty (hedge
the risk)
(b) Speculation: We can use them to speculate on the future direction of the
market. As we shall see, options provide leaverage for speculation.
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3.5 Hedging
We have seen in an earlier lecture how forward contracts can be used for hedging.
Other types of hedging can be based on options contracts, where the objective is not
to x a price, but to provide insurance. Consider, for instance, the following example.
Example. An investor owns 1,000 shares at $102 each. He wants to hedge the risk
of a fall in share price and buys six-month European put options with a strike price of
$100 for $4 each. Therefore he pays $4,000 to buy the options, which guarantee that
the shares can be sold for at least $100 each in six months.
If the share price goes below $100, he can exercise the options and obtain a payo
of $100,000, with a prot of $96,000. If the share price stays above $100, he keeps
(or sells) the shares. Now the value of the holding is above $100,000 (prot above
$96,000). Whatever happens, his total asset value, originally $102,000, can not fall
below $96,000.
3.6 Speculation
An example of speculation using forward contracts could be the following.
Example. An agent believes that the $/$ exchange rate will increase, and wants
to exploit this change in the rate. He takes the risk and enters a three-month long
forward contract, where he agrees to buy $100,000 at a $/$ rate of 1.65. If the rate
actually goes up, say to 1.7, he can buy for $1.65 an asset worth $1.7, and so he makes
a prot of (1:7 1:65) 100; 000 = $5; 000.
The way speculator use options is more complex, and is illustrated in the following
example.
Example. Suppose a speculator believes a certain stock price will increase, and there-
fore wants to gain by buying now $6,400 worth of stock. Suppose the current price is
$64 and that a three-month call option with $68 strike price is selling for $5. Two
strategies are possible: buy 100 shares or buy 1,280 options.
If the price goes up to $75, the rst strategy gives a prot of 100 (75 64) =
$1; 100. The second strategy is more protable: he can exercise the options and receive
a payo of 1; 280 (75 68) = $8; 940; subtracting the cost paid for the options we
have a prot of 8; 940 6; 400 = $2; 540.
However, if the price goes down to $55, then the rst strategy gives a loss of
100 (64 55) = $900, when the second strategy gives a loss of $6,400.
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Stochastic Methods in Finance 1
This example illustrates what is sometimes known as leaveraging. For a given
amount invested ($6,400 in the above example), the use of options allows greater
exposure to the movements of the underlying asset than investing in the asset alone.
3.7 Combining derivatives
The standard call and put options are termed \plain vanilla" derivatives. Financial
institutions can design derivatives which are sold to customers or are combined with
bond and stock issues in order to make them more attractive to the situation and
needs of the customer. These derivatives are loosely called \exotic options".
The standard vanilla options can be combined to provide a wide range of risk pro-
les that may suit a number of customer requirements. Some simple examples are
given here.
(a) Straddle: 1 call and 1 put at the same strike price
(b) Strangle: 1 call at strike price X2 and one put at strike price X1 < X2 (used,
for instance, if there is a lot of volatility in the stock)
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Stochastic Methods in Finance 1
(c) Spread: two options of the same type (i.e. two calls or two puts), one long and
one short. We can have bull or bear spreads, and these can be obtained from both
calls and puts.
BULL SPREAD from calls
BULL SPREAD from puts
The holder will benet if the stock price increases.
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Stochastic Methods in Finance 1
BEAR SPREAD from calls
BEAR SPREAD from puts
The holder will benet if the stock price decreases.
27 J Herbert UCL 2011-12
Chapter 4
Arbitrage and the pricing of
forward contracts
4.1 Arbitrage
Inherent in most investments is a level of risk. You do not know for sure the return
you are going to make on your investment, or sometimes even if you are going to make
your initial investment back. The exception we have discussed earlier is an investment
in Government bonds, the so-called risk-free investment. We assume that there will
be no default on these bonds, hence they are \risk free".
Arbitrage is an important concept in nance. Arbitrage is a situation where by
combining two or more nancial products we make an investment that is guaranteed
to yield a prot with no investment or cost, with absolute certainty. In other words, a
successful arbitrage involves making money without any risk or investment.
We have discussed how a central question in nance is the valuation or pricing of
nancial products. A key concept in pricing or valuing products is the assumption
that there are NO arbitrage opportunities in the nancial markets.
This essentially relies on the fact that if there were arbitrage opportunities, then im-
mediately someone would capitalize on these, thus aecting the market prices through
demand and supply and eliminating the opportunity. In this course, and indeed in the
majority of theoretical nance, it is assumed that this happens instantaneously. In
other words, we assume that there are NO arbitrage opportunities in the markets.
4.2 Example - Arbitrage opportunities in a forward
contract
Assume that a forward contract on a non-dividend paying stock matures in 3 months.
i.e. the contract involves delivery of the stock in 3 months time. The stock price is
28
Stochastic Methods in Finance 1
$100 now, the three-month risk-free interest is 12% p.a. Suppose the forward contract
price is $105, i.e. the stock will be delivered for payment of $105 in 3 months time.
Questions to consider:
Is this the correct price for the forward contract today?
What do we mean by \correct price?"
Can I make money without any risk?
Consider the following trading strategy:
Time 0:
{ Borrow $100 at 12%
{ Buy a share at $100
{ Sell a forward at $105
3 months:
{ Get $105 for the share
{ Pay back $103 (including $3 interest payment at the risk free rate of 12%)
{ ) prot $2
This has yielded a riskless prot of $2 with no initial investment. In other words,
an arbitrage opportunity.
What about the case where the forward price is $102? Consider an investor who
has a portfolio with one share of the stock. Can he make money without any risk?
Time 0:
{ Sell the share at $100
{ Invest the $100 at 12%
{ Buy a forward at $102
3 months:
{ Get $103
{ Pay $102 for the share
{ ) prot $1
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This has yielded a riskless prot of $1 with no initial investment, so that an arbitrage
opportunity exists if $102 is the forward price. Clearly neither of the two forward prices
considered, $105 or $102, are sustainable in a market without arbitrage opportunities.
They cannot be the correct price for the forward contract in such a market.
In fact, whenever the forward price in this example is higher or lower that $103
there is an arbitrage opportunity.
What happens in the market? The rst strategy is protable when the forward
price is greater than $103. This will lead to an increased demand for short forward
contracts, and therefore the three-month forward price of the stock will fall. On the
other hand, the second strategy is protable when the price is smaller than $103,
therefore we will see an increase in the demand for long forward contracts and in turn
an increase in the three-month forward price of the stock. These activities of traders
will cause the three-month forward price to be exactly $103.
Therefore, from now on the basic assumption in derivative pricing is that
there are no arbitrage opportunities in the financial markets
Question: How did we calculate the gure $103 in this example?
4.3 Pricing forward contracts for securities that pro-
vides no income
The principal in the above example can be extended to price forward contracts in
general.
Assumptions: 1) No transaction costs 2) The market participants can borrow or
lend money at the same continously compounded risk-free rate r.
We also further assume that the underlying provide no income. Extensions to the
argument are needed if the underlying provides an income, such as a share yielding
dividends. We do not consider this here.
The example in the previous section indicates that the delivery price of a forward
contract should be the Future Value (FV) of the underlying security price. If S is the
spot price and F is the forward price, then
F = SerT
where T is the time to maturity and r is the corresponding risk-free interest rate. If
F < SerT or F > SerT , then there are arbitrage opportunities. See exercises for a
proof of this.
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Stochastic Methods in Finance 1
Example. If the stock price is $40 with no dividends, and the interest rate is 5%,
then a forward contract after 3 months should have delivery price equal to
F = 40e0:050:25 = 40:50
This is the correct price so that the initial value of the contract is zero, and the contract
is therefore a fair one.
4.4 Value of a forward contract
As we have seen, the no-arbitrage principle requires that the value of a forward contract
at the time it is rst entered into is zero, i.e. the delivery price equals the forward
price. The value of the contract, however, can change afterwards and become positive
or negative, because the \fair" forward price, if recalculated, can change as the price
of the underlying changes over time (while the delivery price of the contract remains
the same).
At any time between the beginning of the contract and the delivery date, the value
f of a long forward contract with delivery price K can be found by considering the
following two portfolios:
Port. A: One long forward contract + cash KerT
Port. B: One security
Here T now denotes the remaining time until delivery, and r the associated risk-free
interest rate. We consider only non-wasting securities, which can be kept indenitely
with no storage costs. Thus the argument is not directly applicable to e.g . commodity
futures.
After time T , KerT will become K and I will use this money to buy one unit of the
security. Therefore at time T the two portfolios have the same value (independent of
the security price), which means that they must have the same value today (otherwise
there is arbitrage).
The value of portfolio A today is: f +KerT , where f is the value of the forward
contract.
The value of portfolio B today is S.
Therefore, since the value of the two portfolios is the same we have f +KerT = S,
and so the value of the contract is:
f = S KerT
This value is zero if and only if K = SerT .
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Notice that S = FerT , where F is the current forward price, i.e. the delivery price
that would apply if the contract were entered in today. Therefore, we can rewrite the
above as f = (F K)erT . This equation shows that we can value a long forward
contract on an asset by assuming that the cash value of the asset at the maturity of the
forward contract is the forward price F . In fact, under this assumption, the contract
will give a payo of F K, which is worth (F K)erT today.
Example. Let us consider a six-month forward contract on a one-year T-bill with
principal of $1,000. The delivery price is $950, and the six-month interest rate is 6%
(with continuous compounding). The current bond price is $930. Then the value of
the contract is:
f = S KerT = 930 950 e0:06 612 = 8:08
Example. Consider now a forward contract on a non-dividend paying stock that
matures in six months. The spot price is $1 and the risk-free interest rate is 10%.
Therefore the forward price is F = SerT = 1 e0:1 612 = 1:05127, and the value f is
zero.
After three months the spot price is $1.05, and the interest rate remains the same.
What is the value of the contract now? We obtain f = 1:05 1:05127e0:1 312 =
0:02469.
4.5 Forward contracts on a security that provides
a known cash income
Let us consider now a (non-wasting) security that provides known cash incomes of ci
at time ti for a number of time points in the future.
The spot price S already re ects the value of all future known incomes, and in
particular up until T . However these incomes will not be received if we enter a forward
contract with maturity at T as they will go to the holder of the security at the times
ti before T . Therefore, to compute the future price we have rst to subtract from the
spot price the present value of the missed future incomes, I = PV (income until time
T ). S I is how much I would be prepared to pay now for this security if I would
receive no income until time T . The forward price is then:
F = (S I)erT
Work through the argument of Section 4.2 to show that this must be the no-arbitrage
price in this known-dividend case.
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Example. Consider a ten-month forward contract on a stock with spot price $50.
The interest rate is 8% (with continuous compounding) per annum. We assume that
dividends of $0.75 per share are expected after 3, 6 and 9 months. What is the forward
price?
The present value of future incomes is given by:
I = 0:75

e0:08
3
12 + e0:08
6
12 + e0:08
9
12

= 2:16
Therefore the price of the contract is F = (50 2:16)e0:08 1012 = 51:14.
The value of the contract is
f = S I KerT
for a delivery price of K.
4.6 Known dividend yield
Consider now the case where the underlying asset provides a known dividend yield
which is paid continuously at an annual rate q. Then the forward price is
F =
h
SeqT
i
erT = Se(rq)T
Example. Let us consider a forward contract with maturity at 18 months, where
the underlying asset provides a continuous dividend yield at 5%. The interest rate
is 8% and the spot price is $1.20. The price of the contract is therefore F =
1:20e(0:080:05)
18
12 = 1:25523.
If the delivery price is K, then the value of the contract is
f = SeqT KerT
Note. The formula given above can be explained by a simple no-arbitrage argument.
Consider a strategy as follows: at time 0
Buy spot eqT of the asset at price S per unit and reinvest income from the asset
in the asset. You spend SeqT .
Short one forward contract on one unit of the asset
Since the holding of the asset grows at rate q, at time T I have eqT eqT = 1 unit of
the asset, and I sell it for F (forward price).
The present value of the cash in ow FerT must equal what I spend to enter the
strategy, otherwise there is arbitrage, and therefore FerT = SeqT i.e. our formula
33 J Herbert UCL 2011-12
Stochastic Methods in Finance 1
above.
(What would an arbitrageur do if F < Se(rq)T ? And what if F > Se(rq)T ?)
4.7 Forward foreign exchange contracts
We consider now the case where we want to buy or sell foreign currencies in the future.
For example, if we want to buy $1m after six months, then we can buy a forward
contract for $1m at the appropriate exchange rate. Observe that the holder of the
foreign currency earns interest at the risk-free rate prevailing in the foreign country.
Denote that by rf ; if r is the domestic rate, then
F = Se(rrf )T
Note that this equation is identical to the one in the previous section with q re-
placed by rf . This is because a foreign currency can be regarded as an investment
asset paying a known dividend yield, which in this case is the risk-free rate of interest
in the foreign currency.
If the delivery price is K, then the value of a foreign exchange contract is given by:
f = SerfT KerT
Example. The six-month interest rate in the US and the UK are 5% and 6% re-
spectively. The current exchange rate is $1.6/$1. The forward rate for a six-month
contract is then:
F = 1:6e(0:050:06)
6
12 = 1:592
34 J Herbert UCL 2011-12
Chapter 5
Pricing Options under the Binomial
Model
We have seen in the last lecture that there is a fair contract price for forward contracts
that does not allow arbitrage. Any other price will allow arbitrage. We now look at
how we can determine this price in for other more general derivatives.
5.1 Modelling the uncertainty of the underlying as-
set price
In order to model the value of a variable that changes over time we will develop models
based on stochastic processes.
We can use discrete time, where the variable changes only at certain xed points
in time, or we can use continuous time, where the variable changes at any time.
Also, the variable can be continuous (can take any value within a range), or it can
be discrete (takes only certain values).
We will start now with discrete time and discrete variables.
In order to introduce the basic logic behind option pricing we start from an ex-
tremely simple model, the one-step binomial tree.
5.2 A simple example
Assume that the price of a stock is currently $20, and after three months it will, with
equal probabilities, either be $22 or $18 (discrete time, discrete variable). We want
to nd the value of a European call option with strike price $21.
The payo of the call option will be as follows:
35
Stochastic Methods in Finance 1




Discrete Continuous
Discrete X
Continuous
TIME
VARIABLE

If after three months the stock price goes up to $22, the option will be worth
max(22-21,0) = $1
If after three months the stock price goes down to $18, the option will be worth
max(18-21,0)=0
We can see this as a game where with 50% probability you get $1 and with 50%
probability you get $0.

0.5
0.5
Time: 0 3 months
£18
£22 Option payoff=£1
Option payoff=£0
£20
The classical decision theory says that the game is \fair" when the expected payo
of the game is $0. This means that in this case:
E[Payo at T Call option value] = 0
36 J Herbert UCL 2011-12
Stochastic Methods in Finance 1
and therefore we should compute the option value as
Call option value = E[Payo at T]
= 0:5 1 + 0:5 0 = 0:5
In this case we would conclude that a \fair" price for the option would be $0.5.
But is this the \correct" price? We have learnt that the \correct" price is such that
there are no arbitrage opportunities in the market, so we can check if a price of $0.5
allows for arbitrage.
5.2.1 Arbitrage opportunity
Suppose that the 3-month risk-free continuous interest rate is 4.4% p.a. Consider for
instance the following portfolio:
Short (sold) 0.25 shares
Long (bought) 1 call option
An investment in the riskless bond of $4.45 (at the risk-free rate)
If the option is sold for $0.5, then I could adopt the following strategy (setting up four
portfolios like the one above):
Time 0
{ Buy 4 options (-$2)
{ Sell 1 share at $20
{ Invest the $17.80 at 4.4%
This provides a net gain of 20 - 17.80 - 2 = $0.20.
Time 3 months - two possible situations
{ Stock price is $22
Get 4 $1 = $4 payo from options
Get 17:80 e0:0443=12 = $18 from cash risk-free investment
Buy back share for $22
) Prot = 4 + 18 - 22 = 0
{ Stock price is $18
Get nothing from options
Get $18 from cash risk-free investment
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Buy back share for $18
) Prot = 18 - 18 = 0
Notice that, regardless of what happens in the future (whether the stock goes up
or down), my position after 3 months is neutral, and I cannot lose any money.
The initial prot of $0.20 at time zero means that regardless of the stock price
in the future, I can make this prot of $0.20 today, with no chance of a loss
in the future. This is an arbitrage opportunity. Hence we conclude that $0.50
cannot be the \correct" price for the option.
5.2.2 No-arbitrage pricing
So how can we obtain a price for the option such that no arbitrage opportunities can
arise? Let us consider again the riskless and share assets in the portfolio we just
introduced, but taking the reverse positions (i.e. long the stock and short the riskless
asset); the value of the risk-free investment combined with the stock position, after
three months, will be:
If the stock price goes up to $22: 0:25 22 4:45e0:0443=12 = 1
If the stock price goes down to $18: 0:25 18 4:45e0:0443=12 = 0
Compare this with the payo of the option in each case. We can see that the value of the
portfolio always equals the value of the call option: it is therefore called a replicating
portfolio - it replicates the value of the option in all future states of the world (which
in the case of the binomial model, is the the two possible stock movements, up and
down). In order to avoid arbitrage, the value of this portfolio must equal the value of
the derivative at all times (prove this to yourself).
The value of this portfolio at time zero is
PV = 4:45 + 20 0:25 = 0:55
and so the no-arbitrage price for the option at time zero must also be $0.55.
Note that, unlike the earlier argument, this approach does not depend on the
probabilities of the dierent possible outcomes; instead it depends on the risk-free
interest rate.
We have found that if we can nd a replicating strategy for a derivative, we are
then able to price it. However the replicating portfolio we set up was no co-incidence.
The amounts of stock and riskless bond were deliberately chosen to make the portfolio
replicating. We now turn to the question of how we can determine what the amounts
should be for a general derivative pricing problem.
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5.3 One-step binomial tree
To look now at a general framework for the pricing of options we need to introduce
some notation.
Let S be the price of the stock at time 0, and let assume that at time T the stock
can either go up to u S or down to d S (u > 1 and d < 1); u 1 and 1 d
equal, respectively, the proportional increase or decrease in the stock price.
We want to price a European call option that gives the right to buy at time T
the stock at a strike price of X.
Let us call fu and fd the values of the option at T when the price goes up or
down, respectively, and let f be the value of the option at time 0 (i.e. its price).
We have: fu = maxfSuX; 0g and fd = maxfSdX; 0g.
Let p be the probability of the stock price going up at T (market probability).
All the above is summarised in the diagram in Figure 1.

1-p
p
Time: 0 T
dS
uS Derivative payoff = f_u
Derivative payoff =f_d

S
Derivative value = f
Figure 5.1: One-step binomial tree
This diagram will help us derive a no-arbitrage price for the call option. There are
two ways of doing this: the rst is similar to the example we saw earlier, i.e. it involves
the construction of a replicating portfolio i.e. a portfolio of stock and cash that takes
up the same values as the option at time T on both branches, while the second is based
on the construction of a riskless portfolio, which we shall look at further in the next
section.
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5.4 A replicating portfolio
Let us consider a portfolio of x worth of riskless zero-coupon bonds and y stocks.
A replicating portfolio has the property that its value tracks the target value (in
this case the derivative value) exactly over time. It is constructed to have the same
terminal value as the derivative. By no-arbitrage, it therefore has the same value as
the derivative at all times prior to maturity, including time 0.
According to our notation, f is the value of the option at time 0, while fu and fd
are the option values at T if the stock price has gone up or down, respectively (see the
diagram above). How can we construct our replicating portfolio? Let us look at the
value of the portfolio:
At 0: 0 = x+ yS
At T:
T =
(
u = xe
rT + ySu ifST = Su
d = xe
rT + ySd ifST = Sd
For the portfolio to be replicating, we must have u = fu and d = fd, i.e. choose x
and y such that
xerT + ySu = fu
xerT + ySd = fd
The solution is given by
x =
ufd dfu
u d e
rT y =
fu fd
Su Sd
With this choice, the portfolio has the same value at T as the option. By no-arbitrage,
the values at time zero must be the same, and we obtain:
f = x+ yS =
ufd dfu
u d e
rT +
fu fd
Su SdS
as the price of the derivative today. This can be rearranged as follows:
f = erT
(
erT d
u d fu +
u erT
u d fd
)
= erT fp^fu + (1 p^)fdg
where we have written
p^ :=
erT d
u d
Notice that d < erT < u, otherwise there would be arbitrage opportunities, and
therefore p^ is a probability, since the above implies 0 < p^ < 1.
40 J Herbert UCL 2011-12
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5.5 Risk-neutral valuation
Notice that the formula for f has a very nice interpretation: if we consider the following
diagram:

1-p^
p^
Time: 0 T
dS
uS Derivative payoff = f_u
Derivative payoff =f_d

S
Derivative value = f
we see that f = erT E^[fT ], where
fT =
(
fu ifST = Su
fd ifST = Sd
and E^ denotes the expectation with respect to p^, i.e. E^[fT ] = p^fu+(1p^)fd. Therefore
f is the discounted, expected payo under the probability p^, which does not depend
on the market probability p.
What is the interpretation of p^? If we calculate the expected stock price at time T
with respect to this probability we obtain
E^[ST ] = p^Su+ (1 p^)Sd
= p^S(u d) + Sd
=
erT d
u d S(u d) + Sd = e
rTS
Therefore
E^[ST ] = e
rTS
which is the forward price.
In a risk-neutral world, all individuals are indierent to risk and require no com-
pensation for risk, and the expected return on all securities is the same (the risk-free
rate). The result above implies that setting the probability of the stock price going
41 J Herbert UCL 2011-12
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up to Su equal to p^ is equivalent to assuming that the return on the stock equals the
risk-free rate. Therefore, p^ is the probability of an \up"-step in a risk-neutral world,
and the option value is the discounted expected payo in that risk-neutral world. This
method of deriving option value is called risk-neutral valuation.
Example. Let us apply this to our numerical example: as before, r = 0:044, S = 20,
u = 1:1, d = 0:9 and T = 0:25. We computed earlier p^ = 0:5553.
The expected payo form the option is then 0:5553 1+ 0:4447 0 = 0:5553, and
if we discount back to time 0 we obtain the call option value: 0:5553e0:0440:25 = 0:55
as before.
Risk-neutral valuation is also related to what is sometimes called the change in
measure approach to option valuation. This refers to the fact that a probability is a
measure. When we use p^ to calculate our discounted expected payos rather than the
actual market probabilities p, we are \changing the probability measure".
We will explore this technique further in a continuous time framework later in the
course.
5.6 Appendix: A riskless portfolio
5.6.1 Example revisited
Consider again the example outlined in section 5.2 above.
We will now look at the pricing problem in a slightly dierent way. Consider the
following portfolio:
Long (bought) 0.25 shares
Short (sold) 1 call option
If the option is sold for $0.5 (which we already know is a price that oers arbitrage
opportunities), then I could adopt the following strategy (selling four portfolios like
the one above):
Time 0
{ Buy 4 options (-$2)
{ Sell 1 share at $20
{ Invest the $18 at 4.4%
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Time 3 months Note that this time this portfolio costs nothing to set up and
gains nothing. After three months:
{ Stock price is $22
Get $4 from options
Get $18.20 from investment
Buy back share for $22
) Prot = 4 + 18.20 - 22 = 0.20
{ Stock price is $18
Get nothing from options
Get $18.5482 from investment
Buy back share for $18
) Prot = 18.20 - 18 =0.20
Regardless of the stock price I make a prot of $0.20 - so we have again shown
(using a slightly dierent approach) that a price of $0.50 for the option allows
an arbitrage opportunity.
Notice that this diers from the replicating portfolio approach in that it costs
nothing to set up, but guarantees a prot in the future, The replicating portfolio
made a prot at set-up at time 0, but was guaranteed to produce a neutral
position in the future.
5.6.2 No-arbitrage pricing
As with the replicating approach above, we can use this construction to nd a price
for the option such that no arbitrage opportunities can arise. Let us consider again
the portfolio we just introduced; its value after three months will be:
If the stock price is $22: 0:25 22 1 = 4:5
If the stock price is $18: 0:25 18 0 = 4:5
We can see that the value of the portfolio does not change with the value of the stock:
it is therefore called a riskless portfolio. In order to avoid arbitrage, its value at time 0
should be simply the present value of $4.5 computed using the risk-free discount rate
(4.4%), since it is a riskless portfolio. Hence:
PV = 4:5 e0:044 312 = 4:45
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We know that the value of the portfolio at time 0 is 0.25 times the value of the share at
time 0 minus the value of the call option f (I am in a short position), and this equals
the PV above. Therefore:
0:25 20 f = 4:45
and so we obtain:
f = 5 4:45 = 0:55
the no-arbitrage price for the option is $0.55 as we found previously. We know need
to look at how to establish a riskless portfolio in a general derivative pricing situation.
5.6.3 Riskless portfolio - general result
Consider the notation introduced in section 5.3. In order to construct a riskless port-
folio we can start with shares long and 1 call option short. We will then nd what
needs to be. The value of this portfolio is as follows:
At 0: 0 = S f
At T:
T =
(
Su fu ifST = Su
Sd fd ifST = Sd
The portfolio is riskless i the value at T is independent of the price of the stock, i.e.
i Su fu = Sd fd, which gives:
=
fu fd
Su Sd
With this the portfolio is riskless and its value at T equals Su fu.
In order to avoid arbitrage, the value of the portfolio at time 0 must equal the value
at T discounted at the risk-free rate r:
0 = T e
rT
This is equivalent to:
S f = [Su fu]erT
which gives the following expression for the value of the option at time 0:
f = erT [p^fu + (1 p^)fd]
where
p^ =
erT d
u d
(Check!)
This is exactly the same result as before under the replicating portfolio pricing
approach.
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5.7 Further reading
Martin Baxter & Andrew Rennie, Financial Calculus. { Chapter 1 and 2
Marek Capinski & Tomasz Zastawniak, Mathematics for Finance: An Introduction to
Financial Engineering (Springer) { Chapter 1, 8.1-8.2.
John C. Hull, Options Futures and Other Derivative Securities { Section 9.1 (in the
5th edition)
45 J Herbert UCL 2011-12
Chapter 6
Applications of the Binomial Model
We are going to look now at some applications of the binomial model for the pricing of
derivatives. We will show that the model can be used not only for option pricing, but
also for other derivatives, giving the example of a forward contract. We then examine
the case of a European put option.
We will then move on to the application of the binomial model for the evaluation of
two-step and three-step trees, and look at the pricing of American options, addressing
the issue of optimal early exercise.
6.1 The value of a forward contract
The binomial model can be used to compute the value of any derivative (not only
options), provided that we can express its features using a binomial tree. In this
section we look at an example considering a forward contract, but a similar analysis
can be undertaken for other derivatives.
A forward contract with delivery price K is pictured in the diagram, and can be
thought of as the payo fT given by
fT =
(
fu = SuK
fd = SdK
6.1.1 A replicating strategy
We can now determine the no-arbitrage price for the forward in the binomial model
by replicating the forward contract using an amount of stock and an amount of the
riskless investment (investment in risk-free bonds). Consider the portfolio that consists
of long one unit of stock, and short KerT units of the riskless investment (i.e. you
have sold the riskless, zero-coupon government bonds to \borrow" money, and so will
have to pay the risk-free rate of interest).
46
Stochastic Methods in Finance 1

Time: 0 T
dS
uS
payoff = fd = dS-K

S
payoff = fu = uS-K
Derivative value = f
At time T this will be worth
ST KerT erT = ST K
regardless of the future price of the stock (we have written ST for the price of the stock
at time T ).
This is exactly the payo of the forward derivative at time T , and so we have found
a replicating portfolio for the forward. Therefore the price of the derivative at time
zero is the value of this portfolio at this time, which is S0 KerT . Note that this is
the usual value for a forward contract we saw in a previous lecture. For this to be a
fair contract at construction, we need this price to be zero, and so we need
S0 KerT = 0
which implies
K = SerT ;
which is consistent with our previous result.
6.1.2 Risk-neutral valuation
We can now use the risk-neutral approach in the binomial model and verify that the
formula for the value f at time zero gives the correct delivery price SerT (forward
price).
We have seen that the value of the contract at zero is given by
f = erT [p^ fu + (1 p^)fd]
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This, in our case, gives the following expression once we substitute in the appropriate
values for fu and fd:
f = erT [p^ (SuK) + (1 p^)(SdK)]
which simplies to
f = erT [p^ Su+ (1 p^)SdK]
Replace now the formula for the risk-neutral probability p^
p^ =
erT d
u d
and obtain
f = erT
"
erT d
u d Su+
u erT
u d SdK
#
= erT
"
erTSu Sud+ Sud erTSd
u d K
#
= erT
"
erTS(u d)
u d K
#
= S KerT
Again this is the usual value for a forward contract, and therefore we see that the
binomial model gives consistent results also for other derivatives (not only options).
Once again, we see that the only delivery price which gives zero initial value to the
forward contract is K = SerT .
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6.2 A European put option
We consider again the example presented in the last lecture, but this time we look at
a put option. Once computed the price of the put option, we will compare it with
the call price obtained previously, and check whether a result called put-call parity is
satised.
Assume that the price of a stock is currently $20, and after three months it will
either be $22 or $18 (discrete time, discrete variable). We want to nd the value of a
European put option with strike price $21.
If after three months the stock price is $22, the put option will be worth $0,
otherwise it will be worth $3 (see diagram below).

Time: 0 3 months
£18
£22 Put option payoff
f_u = £0
Put option payoff
f_d = £3
£20
Therefore u = 1:1, d = 0:9, r = 0:044, T = 0:25, fu = 0 and fd = 3. The
risk-neutral probability of an up-step is
p^ =
erT d
u d =
e0:0440:25 0:9
1:1 0:9 = 0:5553
as before.
The price of the option is given by the usual formula
f = erT [p^ fu + (1 p^)fd]
which gives in this case
f = e0:0440:25[0:5553 0 + 0:4447 3] = 1:32
So the price of a put option is $1.334, while the call option was priced $0.55 (from
previous lecture).
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We can know look at the put-call parity relationship for the price of put and call
options, given by the following:
Call Put = S KerT
with the usual notation (K is the strike price). In our case, S = 20, K = 21 and r and
T are the same as above. It follows that
S KerT = 20 21e0:0440:25 = 0:77
This equals the dierence between the call and put price that we found, since 0:55
1:32 = 0:77, and therefore the put-call parity result is satised.
6.3 Two-step binomial trees
We can now extend the discussion to two-step binomial trees where the asset price will
change twice before maturity.
Example. Consider, for instance, a case similar to the previous one, but where the
time to maturity is now six months, and the prices change twice, at three and at six
months. The up- and down-steps are again of 10% at both times (u = 1:1 and d = 0:9).
The stock price at zero and the strike price remain the same of the past example, i.e.
S = 20, X = 21, but we now use r = 12% for the risk-free rate. We want to nd the
price of a European call option at $21.

C
A
fC=£0
fB=£2.026
fD=£3.2
£20
£18
£22
£24.2
£19.8
£16.2
B
D
E
F
fE=£0
fF=£0
fA=£1.280
3 months 6 months 0 Time:
To do so, we need to compute the value of the option at each node (denote the
nodes with the letters A - F as in the diagram). The values at maturity are easy to
50 J Herbert UCL 2011-12
Stochastic Methods in Finance 1
compute at each of the nal nodes. To obtain the values at three months we need
to compute the risk-neutral probability p^ and then apply the one-step formula. From
there we can then obtain the initial value applying again the one-step formula with
the three-months values just obtained. Notice that the risk-neutral probability is the
same for all steps, since we have constant u and d, and are given by
p^ =
e0:120:25 0:9
1:1 0:9 = 0:6523 ! 1 p^ = 0:3477
Let us compute the values. We know that at the nal nodes the option is worth
fD = 3:2
fE = fF = 0
We can obtain the values after three months as:
fB = e0:120:25[0:6523 3:2 + 0:3477 0] = 2:0257
fC = 0
It follows that the price of the option is
f = fA = e
0:120:25[0:6523 2:0257 + 0:3477 0] = 1:2823
6.4 General method for n-step trees
The following general approach can be used for the valuation of European options for
two-step, three-step, : : :, n-step binomial trees:
(a) Compute the risk-neutral probability for every one-step binomial tree (in the
large tree)
(b) Compute the option values at the terminal nodes (the payo function)
(c) Work backwards and compute the option values at each intermediate node using
risk-neutral valuation
Example. We give here an example of the general method using a three-step three for
a European option, where each step is long t, and the up and down jumps are related
as follows:
u =
1
d
The three-step tree is shown in the diagram.
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C
A
S
dS
uS
u2S
udS
d2S
B
D
E
F
t 2t 0 Time:
G
H
I
J
u3S
u2dS
d2uS
d3S
3t
The method above gives:
(a) Compute p^ = e
rtd
ud
(b) Find fG, fH , fI and fJ
(c) Then:
{ fD = e
rt[p^ fG + (1 p^)fH ]
{ fE = e
rt[p^ fH + (1 p^)fI ]
{ fF = e
rt[p^ fI + (1 p^)fJ ]
and:
{ fB = e
rt[p^ fD + (1 p^)fE]
{ fC = e
rt[p^ fE + (1 p^)fF ]
which give a value for the option equal to
f = fA = e
rt[p^ fB + (1 p^)fC ]
6.5 Pricing of American options
Let us start with an example of an American put option. Consider the following two-
step tree with u = 1:2, d = 0:8, r = 0:05, and each step is one-year long. The initial
price of the stock is $50, and the strike price is $52.
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If you use risk-neutral valuation for European put option with strike price of $52
you obtain the values on the left tree. Let's see if the values are correct for an American
option.

C
A
fC=9.4656
fB=1.4147
fD=0
50
40
60
72
48
32
B
D
E
F
fE=4
fF=20
fA=4.1925

C
A
fC=12=max{12,
9.4656}
fB=1.4147=
max{1.4147,0)
fD=0
50
40
60
72
48
32
B
D
E
F
fE=4
fF=20
fA=5.0894
At the terminal nodes D, E and F you can exercise in both cases (European or
American option), so the values are the same. At node B if you exercise (American)
you get nothing, and therefore the value for an American option is the same as the
European option value (1.4147). At node C, if you exercise you get $12, and the value
if you do not exercise is 9.42636. Therefore you should exercise and the value for the
American option at C is $12.
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Now in order to nd the value at A you use risk-neutral valuation
f = fA = e
0:051[p^ 1:4147 + (1 p^)12] = 5:0894
So the general valuation method for an American option is
(a) Compute the risk-neutral probability for every one-step binomial tree (in the
large tree)
(b) Compute the option values at the terminal nodes using the payo function
(c) Work backwards and compute the option values at each intermediate node us-
ing risk-neutral valuation. Test if early exercise at each node is optimal. If it
is, replace the value from the risk-neutral valuation with the payo from early
exercise.
(d) Continue with the nodes one step earlier
54 J Herbert UCL 2011-12
Chapter 7
Calculus refreshers
These notes are intended to give you reminders of various important classical calculus
results that we will use. The result (10.3) and the solution to equation (7.4) are
particularly relevant. For more details see an introductory calculus book.
7.1 Taylor series
A function f(x) can be expanded as a Taylor series around the point x0 as follows
f(x) = f(x0) + f
0(x)(x x0) + f
00(x0)
2!
(x x0)2 + f
000(x0)
3!
(x x0)3 + :::
=
1X
i=0
f (i)(x0)(x x0)i
i!
where f (i) is the ith derivative of f , and we need to assume that the derivatives at
x0 all exist. The special case of this expansion in which x0 = 0 is sometimes called
Maclaurin's series.
If f() is a function of two variables, say f(x; y); then a Taylor series expansion can
be made about a point (x0; y0) as follows
f(x; y) = f(x0; y0) +
@f(x0; y0)
@x
(x x0) + @f(x0; y0)
@y
(y y0) +
1
2!
@2f(x0; y0)
@x2
(x x0)2 + 1
2!
@2f(x0; y0)
@y2
(y y0)2 +
1
2!
2
@2f(x0; y0)
@x@y
(x x0)(y y0) + ::: (7.1)
This can be generalised to the case where f is a function of n variables.
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Stochastic Methods in Finance 1
7.2 Chain rule dierentiation
The classical chain rule for dierentiation says that for dierentiable functions f and
g we have
[f(g(s))]
0
= f
0
(g(s)) g
0
(s)
or equivalently
d
ds
f(g(s)) =
df
dg
(g(s))
dg
ds
(s)
or also equivalently
f(g(t)) f(g(0)) =
Z t
0
f 0(g(s)) g0(s)ds
=
Z t
0
f 0(g(s))dg(s)
where we have written h0(t) for the ordinary derivative of a function h at t.
7.3 Partial dierentiation
For f a function of x and y, with x and y both functions of a single independent
variable t, the extension of the chain rule gives us
df
dt
=
@f(x; y)
@x
dx
dt
+
@f(x; y)
@y
dy
dt
We can also write this as
df =
@f
@x
dx+
@f
@y
dy (7.2)
when y and x depend on one or more other variables.
For our course it will be useful to consider this result heuristically as an application
of the Taylor series. If we take the limit as x0 tends to x and y0 tends to y and we
write dx x x0; dy y y0 and df(x; y) f(x; y) f(x0; y0), then we can express
result (7.1) as
df =
@f
@x
dx+
@f
@y
dy +
1
2!

@2f
@x2
dx2 +
@2f
@y2
dy2 + 2
@2f
@x@y
dxdy
!
+ ::: (7.3)
In ordinary calculus the second order (and higher order) terms in (10.3) tend to
zero so that we obtain result (7.2).
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7.4 Linear ordinary dierential equations
A vast topic - here we only present one simple ordinary dierential equation (ODE) to
remind the reader of one basic technique, and of the important example of exponential
growth. The ODE
dx
dt
= rx; (7.4)
(also sometimes written as dx = rx dt), where r is a constant, can be solved using
the variable separable technique, or in other words by writing the ODE as the integral
equation Z 1
rx
dx =
Z
1 dt
Integrating this then gives
1
r
ln(x) = t+ k1
) ln(x) = rt+ rk1
) x = ert+rk1
) x = Kert (7.5)
where k1 and K are constants. This is the equation for exponential growth at rate r.
To obtain particular solutions, dierential equations are usually solved together
with boundary conditions. In the above simple example, we can determine the
value of the arbitrary constant K with the added information that say
x(0) = x0: (7.6)
Then substituting this into (7.5) we can nd K as
x(0) = x0 = Ke
r0 = K
so that the particular solution to the ODE (7.4) together with the boundary condition
(7.6) is
x = x0e
rt:
When we move into the world of partial dierential equations an initial boundary
condition can specify the value of the function across a curve. However boundary
conditions do not always have to be initial conditions, they can, for example, be
terminal conditions, or other more complicated types of boundary conditions.
57 J Herbert UCL 2011-12
Chapter 8
Continuous-time stochastic
processes for stock prices
We now explore further suitable stochastic processes to describe the behaviour of stock
prices. A typical example of stock price movements is given in the graph below, which
shows the price of the UK FTA index from 1963-1992.
The binomial model we saw in the past lectures is one approach to modelling the
movement of real prices. In order to implement it on realistic time scales we will
need to have a large number of time steps and hence use a large number of nodes.
Although there are sometimes benets to the discrete time approach to modelling the
stock prices, solving for derivative prices can be computationally expensive as we are
often not able to obtain explicit solutions over many nodes. One alternative approach
that can lead to the derivation of some explicit theoretical results for option pricing is
to consider a continuous time limit.
58
Stochastic Methods in Finance 1
A continuous time limit of the binomial model can lead us to a model based on
the important stochastic process Brownian motion. This increases our exibility for
asset price modelling and allows the use of mathematical techniques such as stochastic
calculus.
First of all we look again brie y at a special case of the binomial model, and how
it can be extended over time to model the stock prices. We will then consider the
appropriate limits to use to move into continuous time.
8.1 Random walk
We will start now from this simple discrete-time, discrete-value (discrete-state space)
stochastic process that has natural similarities with the binomial model we just looked
at.
Here fxtg is a random variable that begins at a known value x0, and at time
t = i jumps by a random variable i from xi1, for i = 1; 2; 3; : : : So for example
x1 = x0+ 1, and x2 = x1+ 2. Assume that i are independent, identically distributed
random variables with
i = with probability 12
i = + with probability 12
for i = 1; 2; : : :, so that xt takes a jump at each time step of size , either up or down,
each with probability 1
2
. The jumps are independent, and therefore we can describe
the dynamics of xt with the following equation
xt = xt1 + t = x0 +
tX
i=1
i
where t is a random variable that takes only values or with equal probability 12 .
Therefore the random variable xt = x0 +
Pt
i=1 i is the position after t steps of a
random walker who started from x0, where each step is equally likely to be up or down
by . The random process fxt; t = 0; 1; : : : ; Ng is the path followed by the walker over
time.
8.2 Other processes and Markov property
Because the probability of an up or down jump is 1
2
, at time t = 0 the expected value
of xt is x0 for all t. One way to generalise this process is by changing the probabilities
for an up- or down-jump to be greater than 0.5 (random walk with drift).
Another way to generalise this process is to let the size of the jump at each time
t be a continuous random variable. For example, we could let the size of the jump
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Stochastic Methods in Finance 1
be normally distributed with mean zero and variance 2. In this case we will call xt
a discrete-time, continuous-state stochastic process. Another example of a discrete-
time continuous-state process is the rst-order autoregressive process, abbreviated as
AR(1). It is given by the equation
xt = + xt1 + t
where and are constants, with 1 < < 1, and t is normally distributed with
zero mean.
Both the random walk (with discrete and continuous states, with or without drift)
and the AR(1) process satisfy the Markov property, and are therefore called Markov
processes. This property is that the probability distribution for xt+1 depends only on
xt, and not additionally on what happened before time t. For example, in the case of
the simple random walk, if xt = 6 and = 1, then xt+1 can equal 5 or 7, each with
probability 1
2
. The values before xt are irrelevant once we know xt.
The random walk (with discrete and continuous states, with or without drift) pro-
cess satises the Markov property, and are therefore called Markov processes. This
property is that the probability distribution for xt+1 depends only on xt, and not ad-
ditionally on what happened before time t. For example, in the case of the simple
random walk, if xt = 6 and = 1, then xt+1 can equal 5 or 7, each with probability
1
2
.
The values before xt are irrelevant once we know xt.
8.3 Taking limits of the random walk
If our N steps take time T to be carried out, each one takes t = T=N time units.
As we move from discrete to continuous time, both the time between jumps, t, and
the size of the jumps, , will need to tend to zero. However the rate at which each
tends to zero, and hence the relationship between the time gap t and the jump size
, will also be important. We will need to x this relationship in some way when we
take limits.
We might initially think that as we take limits we should set = kt for some
positive constant k, so that the ratio of =t is always constant. However it turns
out that this choice will not work and we will need to nd another approach (see
appendix).
Remember that we are going to take limits as both t! 0 and ! 0, whereas t
is xed as the length of the overall interval we are looking at. It turns out that for the
variance of the process to \behave well" and remain nite, it is the ratio 2=t that
we need to keep constant while t ! 0 and ! 0, rather than the ratio of =t. If
we call this constant
2 := 2=t;
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Stochastic Methods in Finance 1
then we can see that the variance of an incremental change in the process over time
t is 2t, and hence is proportional to t, the size of the time length we are looking at.
The standard deviation is therefore proportional to the square root of t.
This property, that the variance of the incremental change in our process is pro-
portional to the length of time we are considering, is one of the key properties of the
continuous time stochastic process that we will use, Brownian motion.
8.4 Brownian motion (Wiener process)
Brownian motion or a Wiener process is a continuous-time stochastic process with the
following important properties1:
The Brownian motion process has independent increments. This means that the
probability distribution for the change in the process over any time interval is
independent of any other (non-overlapping) time interval (a property that is also
true for the discrete time random walk).
It is a Markov process. As explained earlier, this means that the probability
distribution for all future values of the process depends only on its current value,
and is unaected by past values of the process or by any other current infor-
mation. As a result, the current value of the process is all one needs to make
a best forecast of its future value. This property follows from the independent
time interval property.
Changes in the process over a nite interval of time are normally distributed,
with a variance that increases linearly with the time interval.
The Markov property is particularly important in the context of modelling stock mar-
kets, since it implies that only current information is useful for forecasting the future
path of the process. Stock prices are often modelled as Markov processes, on the
grounds that public information is quickly incorporated in the current price of the
stock, so that the past pattern of the prices has no forecasting value. This is called the
weak form of market eciency. If it did not hold, investors could in principle \beat
the market" through technical analysis, by using the past pattern of prices to forecast
the future. The fact that a Wiener process has independent increments means that we
can think of it as a continuous-time version of a random walk.
1In 1827, the botanist Robert Brown rst observed and described the motion of small particles
suspended in a liquid, resulting from the apparent successive and random impacts of neighbouring
particles; hence the term Brownian motion. In 1905, Albert Einstein proposed a mathematical theory
of Brownian motion, which was developed further and made more rigorous by Norbert Wiener in 1923.
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Stochastic Methods in Finance 1
8.5 Denition of Brownian motion
Brownian motion can be formally dened by the following properties. The process zt
is Brownian motion if and only if
1. All non-overlapping increments of the process are independent (so that zs2 zs1
is independent of zt2 zt1 for all 0 s1 < s2 t1 < t2).
2. For 0 s < t the increment zt zs is normally distributed with mean 0 and
variance t s.
3. zt is continuous and z0 = 0.
We can think of the second property as saying that a process z = fztgt>0 following
Brownian motion has increments that can be expressed as
z = zt+t zt =
p
t t
where t N(0; 1).
Let us examine what the conditions above imply for the change of z over some
nite interval of time T . If we want to study the dierence zT z0, we can use N
intervals with each interval step being t = T=N , and obtain
zT z0 = zT zTT=N + zTT=N zT2T=N + + zT=N z0
=
NX
i=1
i
p
t
where the i N(0; 1) and they are independent. Each \small" dierence has mean
0 and variance equal to the interval length t, which for N intervals will be T=N . It
follows that zT z0 is normally distributed with
E[zT z0] = 0
and
Var[zT z0] = N T
N
= T
i.e.
(zT z0) N(0; T )
An example of a sample path of the Brownian motion process is shown below.
Example. Suppose zt follows a Brownian motion process where z1 = 25 and t = 1
year. What are the distributions of z2 and z6? From above, we know that (zT2zT1)
N(0; (T2 T1)). Therefore for T2 = 2 and T1 = 1 we have
(z2 25) N(0; 1) ) z1 N(25; 1)
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Stochastic Methods in Finance 1
Figure 8.1: Example of a Brownian motion sample path
Similarly, when T = 6 we have z6 N(25; 5).
Notice that in the case where the process starts at zero, i.e. z0 = 0, the generic
variable zt follows the normal distribution N(0; t).
Example. If Z N(0; 1), then the process xt =
p
tZ is continuous and is marginally
distributed as a N(0; t). Is xt a Brownian motion process?
The answer is no, since the increments do not respect the conditions for a Wiener
process. In fact, we have xt+txt =
p
t+tpt

Z. It follows that the increment
follows a normal distribution with zero mean and variance given by
p
t+tpt
2 1 = t 2t
24s1 + t
t
1
35
which is not t.
Moreover, the increments are not independent. For instance, if we consider the
two increments xt+t xt and xt x0 we have that their correlation is given by the
following (where we use the fact that x0 = 0 and Z
2 is a 21):
E(xt+t xt)(xt x0) = E(xt+t xt)xt
=
p
t+t
p
tE(Z2) E(x2t )
= t
24s1 + t
t
1
35 6= 0
It follows that xt is not a Brownian motion process.
Example. If zt and wt are two independent Brownian motion processes starting at
zero, and is a constant between 1 and 1, then the process xt = zt +
p
1 2wt is
continuous and has marginal distributions N(0; t). Is xt a Brownian motion?
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Stochastic Methods in Finance 1
Here an increment is given by
xt+t xt = (zt+t zt) +
q
1 2(wt+t wt)
We know from the properties of Brownian motion that both the increments of zt and of
wt appearing above follow a N(0;t) distribution. It follows that the increment in xt
is the sum of a N(0; 2t) and a N(0; (1 2)t), i.e. a N(0;t), which is consistent
with the Brownian motion process properties.
Moreover, the increment in xt shown above will be independent of any other incre-
ment in the process xt over a non-overlapping time interval, since this is true for both
zt+t zt and wt+t wt. It follows that xt is indeed a Brownian motion.
8.6 Generalised Brownian Motion process
If z is a Brownian motion (or equivalently Wiener) process , we have seen that E[z] =
0 (drift) and Var[z] = t (variance).
We can construct a general class of processes x = fxtg such that for \small" time
periods t
x = xt+t xt = at+ bz
where a and b are constants. Then x N(at; b2t) since z is normally distributed
and:
E[x] = at (new drift)
and
Var[x] = b2t (new variance)
With arguments similar to those in Section 1, we obtain that the behaviour of the
change over a time interval T is given by
(xT x0) N(aT; b2T )
Example. Consider a generalised Wiener process with x0 = 50, a = 20 and b
2 = 900.
At the end of six months (T = 0:5) we have (x0:5x0) N(0:5a; 0:5b2), and therefore
the value of the variable after six months has distribution N(x0 + 0:5a; 0:5b
2), i.e. a
N(60; 450).
We say that x satises the stochastic dierential equation
dx = adt+ bdz
where a is the drift rate, b2 is the variance rate and dz is the \error" term. If b = 0
(no variability) then dx = adt, i.e. x = x0 + at: the process grows linearly with time.
When b 6= 0, the term bz adds variability around the line x = x0 + at.
Sample paths for a generalised Brownian motion process and Brownian motion are
shown below.
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Stochastic Methods in Finance 1
8.7 Ito^ process
A further type of stochastic processes can be dened where the drift and variance rate
are not constant anymore. A random process x = fxtg is an Ito^ process if for any t
and very small t
x = xt+t xt = a(x; t)t+ b(x; t)z
where a(x; t) is the drift rate and [b(x; t)]2 is the variance rate. We say that x satises
the following stochastic dierential equation
dx = a(x; t)dt+ b(x; t)dz
Ito^ processes are sometimes called diusion processes, as they can be used to model
the diusion of gas particles.
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Stochastic Methods in Finance 1
8.8 A process for stock prices: the geometric Brow-
nian motion
We are looking for an appropriate process to model stock prices. One model we may
consider using for non-dividend paying stock prices is the generalised Brownian motion
process
dS = dt+ dz
where is the drift of the stock price and the , is the square root of the variance
rate. The model implies that in a period of t
S = St+t St = t+
p
t
where N(0; 1) as usual.
Observe that the expected increase in the stock price in this time period is t,
which is independent of the stock price itself. This does not seem appropriate because
growth in a stock price is usually related to the size of the stock price itself. It is
the return from a stock that we are interested in, which is measured in terms of a
percentage change in price.
This model will also allow the possibility of negative values for the stock price S
which is clearly not appropriate.
A more appropriate model is to look at the percentage change as following gener-
alised Brownian motion, i.e.
S
S
=
St+t St
St
= t+ z
This gives a model for the actual stock price S as
S = Stt+ Stz
i.e. S=S N(t; 2t). We call this process a geometric Brownian motion, and
we say that it satises the following stochastic dierential equation
dS = Sdt+ Sdz ()
If = 0 (no variance) then S is a risk-free asset and dS = Sdt, which is equivalent
to dS=dt = S, i.e. S 0 = S. The solution to this dierential equation is
ST = S0e
T
For = 0 the price grows at a continuously compounded rate of per unit.
The geometric Brownian motion has two parameters: the rate of return and the
volatility (risk).
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Stochastic Methods in Finance 1
It turns out that when 6= 0 the process that solves the stochastic dierential
equation () is given by
St = S0 exp
(

2
2
!
t+ z
)
(8.1)
i.e. () corresponds to the exponential of a generalised Brownian motion. We shall
look into this further in lectures to come. Following the formula above, the geometric
Brownian motion is also called exponential Brownian motion.
Example. Consider a stock that pays no dividends, has a volatility of 30% per annum
and provides an expected return of 15% per annum. Then = 0:15 and = 0:30:
dS
S
= 0:15dt+ 0:30dz
This means that as an approximation we can write
S
S
= 0:15t+ 0:30
p
t
If the price is now $100 what is the distribution of the price after one week?
One week is 1=52 years, i.e. t = 0:0192; St = 100. We obtain:
St+(1week) 100
100
= 0:15 0:0192 + 0:30
p
0:0192
where N(0; 1). It follows:
St+(1week) = 100:288 + 4:16
Therefore our approximation gives:
St+(1week) N(100:288; 4:162)
An approximate 95% probability interval is then given by
100:288 1:96 4:16 St+(1week) 100:288 + 1:96 4:16;
which gives
92:14 St+(1week) 108:44:
With the more precise lognormal formula for a 95% probability interval, derived from
(8.1) above,
Se(
2
2
)T1:96pT ST Se(
2
2
)T+1:96
p
T
we obtain
92:17 St+(1week) 108:49:
They are almost identical, since t here is very small (0.0192). If t is larger the rst
procedure is no longer accurate, and we need to use the lognormal approach.
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Stochastic Methods in Finance 1
8.9 Appendix: Brownian Motion as a limit of a
discrete time random walk
In later lectures we will be working with Brownian motion (also called a Wiener pro-
cess), which is a continuous-time stochastic process. Here we provide an outline of
how to move from a version of the discrete-time binomial model (the random walk) to
continuous-time Brownian motion, by taking appropriate limits.
Consider a process xn that follows the random walk outlined in section 8.1. After
n steps the process can be expressed as
xn = x0 +
tX
i=1
i:
We are interested in the incremental change xN x0 (i.e. the distribution of the
increase or decrease in the process after N time steps) when N is large.
Suppose now that the time between the jumps up or down of the random walk
is small and is t, so that jumps take place at times t; 2t; 3t : : : The number of
jumps up to a time point t will be n = t=t. As we move from discrete to continuous
time, both the time between jumps, t, and the size of the jumps, , will need to
tend to zero. However the rate at which each tends to zero, and hence the relationship
between the time gap t and the jump size , will also be important. We will need to
x this relationship in some way when we take limits.
We might initially think that as we take limits we should set = kt for some
positive constant k, so that the ratio of =t is always constant. However as we shall
see, this choice will not work and we will need to nd another approach.
As we know that n, the number of jumps between time 0 and time t, is going
to become large in the limit, we can start by using the central limit theorem. One
version of the central limit theorem tells us that if X1; X2; : : : ; Xn are a sequence of
independent identically distributed random variables with nite means and nite
non-zero variances 2, and Sn =
Pn
i=1Xi, then
Sn n

p
n
! N(0; 1)
in distribution as n!1: In our case the mean of each jump i is zero ( = 12 12 = 0),
and we can calculate the variance as
E[2i ] (E[i])2 = 2( 12 + 12) 0
= 2:
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Stochastic Methods in Finance 1
Therefore, the variance of
Pn
i=1 i is n
2 as all covariances are zero by independence of
the i. So for large n, we have that approximately
nX
i=1
i N(0; n2):
Substituting n = t=t as the number of jumps in our time interval for given t, the
variance of the incremental change in the process, xt x0, is
2
t
t:
Remember that we are going to take limits as both t ! 0 and ! 0, whereas
t is xed as the length of the overall interval we are looking at. We can see from
this variance result that, in order for the variance of the process to \behave well" and
remain nite, it is the ratio 2=t that we need to keep constant while t ! 0 and
! 0, rather than the ratio of =t. If we call this constant
2 := 2=t;
then we can see that the variance of an incremental change in the process over time
t is 2t, and hence is proportional to t, the size of the time length we are looking at.
The standard deviation is therefore proportional to the square root of t.
This property, that the variance of the incremental change in our process is pro-
portional to the length of time we are considering, is one of the key properties of the
continuous time stochastic process that we will use, Brownian motion.
69 J Herbert UCL 2011-12
Chapter 9
Introduction to stochastic calculus
and Ito^'s lemma
In the last few lectures we have tried to model the behaviour of stock prices using
some specic stochastic processes. But what about the behaviour of the price of a
derivative? We expect that in general the price of a derivative will be a function of
both time and the price of the underlying asset. If we specify a particular process
for the price of the underlying asset, can we obtain the dynamic of the price of the
derivative? The answer is yes, and this can be achieved using stochastic calculus.
9.1 Ordinary and stochastic calculus
In previous lectures we have seen that the various stochastic processes we analysed
satised certain stochastic dierential equations (SDE). The most general form of
SDE was of the type
dxt = a(xt; t)dt+ b(xt; t)dzt
where fztg is a standard Brownian motion, and a and b are some known (non-random)
functions of xt and time. In fact, when we write an SDE we are really just using
convenient notation to describe an integral equation. If we add an initial condition,
say that the process starts at a given point x0, then the SDE above can be interpreted
as a version of the integral equation
xt = x0 +
Z t
0
a(xs; s)ds+
Z t
0
b(xs; s)dzs (9.1)
Example. In the case of constant a(xt; t) = a and b(xt; t) = b we obtain the SDE of
the generalised Brownian motion
dxt = adt+ bdzt
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Stochastic Methods in Finance 1
and the associated integral equation simplies to
xt = x0 + at+ bzt
We are of course comfortable with the integral with respect to t (or equivalently with
respect to s in (9.1)), but we need to be careful with what it really means to talk about
the second integral in (9.1), which is with respect to Brownian motion (zt).
We have seen in the lectures on Brownian motion that in the case of no volatility
(b = 0) the process is just a deterministic function equal to the straight line xt = x0+at,
while in the case b 6= 0 we add noise around that line according to the Brownian motion
zt.
But what about the general case of variable a(xt; t) and b(xt; t)? We can try and
interpret our integral equation as follows: the ds-integral is an ordinary Riemann inte-
gral, and the dzs-integral perhaps could be interpreted as a Riemann-Stieltjes integral
for each trajectory. Unfortunately, this is not possible since one can show that the tra-
jectories are of locally unbounded variation1, and therefore the stochastic dzs-integral
cannot be dened in the traditional way.
To make the characteristics of the trajectories of a Brownian motion even clearer,
let us take a closer look at some graphs. First, consider a deterministic smooth (dier-
entiable) function with a quite jagged trajectory, which could be considered \similar"
to a Brownian motion trajectory. As we zoom in, we see that the function becomes
smoother and straighter, until eventually it becomes a straight line.
Dierentiable functions, however strange their global behaviour, are built from
straight line segments, and classic calculus is the formal acknowledgement of this.
Let us now take a look at a Brownian motion. As we zoom in we cannot obtain
a straight line. The process replicates itself (self-similarity property) even if rescaling
(zooming in).
Unfortunately this implies that the trajectory is not dierentiable at any t, and
therefore the tools of traditional calculus cannot be applied. What we need now is
stochastic calculus.
We are going to look here at stochastic dierentials and in particular at Ito^'s formula
for the dierential of a function of a stochastic process. Underlying the subject is the
careful denition of the integral with respect to Brownian motion in (9.1). We will not
look at this in detail in this introductory course, but instead will concentrate on some
intuition and some tools needed for manipulating stochastic dierentials.
1The variation of a process Xt over an interval [0; T ] is dened as sup
Pn
i=1 jXti Xti1 j where
: 0 = t+ 0 < t1 < ::: < tn = T; so that the supremum is taken over all possible partitions of [0; T ].
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9.2 Ito^'s formula
Ito^'s formula is a fundamental result of stochastic calculus, and gives us an explicit
understanding of the behaviour of functions of stochastic processes. The formula is
also known as Ito^'s lemma, and states the following:
Ito^'s formula. Consider a random variable x that follows an Ito^ process
dx = a(x; t)dt+ b(x; t)dz
where z is a Wiener process, a(x; t) is the drift rate and [b(x; t)]2 is the variance rate
(non-constant). Consider also a continuous function G = G(x; t) twice dierentiable
in x and once dierentiable in t. Then G is itself a random process and satises the
following stochastic dierential equation:
dG =
"
@G
@x
a(x; t) +
@G
@t
+
1
2
@2G
@x2
b2(x; t)
#
dt+
@G
@x
b(x; t)dz
Therefore G also follows an Ito^ process, with drift rate of
@G
@x
a(x; t) +
@G
@t
+
1
2
@2G
@x2
b2(x; t)
and variance rate of
@2G
@2x
b2(x; t)
The rigorous proof of the formula is outside the scope of this course.
However, when we are dealing with a continuous and dierentiable function of two
variables x and t, in ordinary calculus we usually express the dierential as
dG =
@G
@x
dx+
@G
@t
dt
This re ects the fact that we are making a Taylor expansion
dG =
@G
@x
dx+
@G
@t
dt+
1
2
@2G
@x2
(dx)2 +
1
2
@2G
@t2
(dt)2 +
@2G
@x@t
dxdt+
and then neglecting the terms of second or higher order. In our case, however, we have
to be careful because the variable x has special properties, since it satises the SDE
above. Therefore, after we make the Taylor expansion, we have to look carefully at the
second order terms to see if we can actually neglect them. It turns out that whilst the
terms in (dt)2 and in dxdt are indeed of higher order and hence can be dropped, the
term in (dx)2 is actually of order dt and so cannot be dropped (see Appendix for more
explanation of why this is the case). Hence if we go back now to our Taylor expansion,
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we see that we cannot neglect anymore all the terms that looked \second order", since
the term in (dx)2 is actually of order dt. The terms in (dt)2 and in dxdt, instead, are
of higher order, and can be dropped. Therefore we are left with
dG =
@G
@x
dx+
@G
@t
dt+
1
2
@2G
@x2
b2(x; t)dt ()
If we substitute in the denition of dx according to the SDE of the process we obtain
Ito^'s formula.
Note. Sometimes Ito^'s result is reported directly as above, i.e. stating the following:
Ito^'s formula (2). If x is an Ito^ process and G(x; t) is C2;1, then dG is given by (*).
This is absolutely equivalent to the formula we gave earlier (just substitute in dx from the SDE).
9.3 Examples
We are going to apply now Ito^'s formula to derive the SDE followed by some specic
processes. The rst two were given without proof in the last handout, and are derived
here using Ito^'s formula. The other examples are more general applications. In all
these examples we always start from an Ito^ process
dx = a(x; t)dt+ b(x; t)dz
and then specify a(x; t) and b(x; t) in order to obtain the processes of interest.
9.3.1 Derivation of the SDE of a generalised Brownian motion
Let us start from a standard Brownian motion. In this case our process x is simply
z, so the SDE will obviously be dx = dz, which is an Ito^ process with a(x; t) = 0 and
b(x; t) = 1. Let us now consider a transformation of the process with a shift, a change
of scale and the addition of a trend term:
G(x; t) = x0 + t+ x
where and are constants. We know that G follows a generalised Brownian motion,
since the dierences are given by
G = t+ x
which by denition is a generalised Brownian motion, since x in this case is a standard
Brownian motion.
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To apply Ito^'s formula we compute:
@G
@x
=
@G
@t
=
@2G
@x2
= 0
It follows that
dG = dt+ dz
is the SDE of a generalised Brownian motion with drift and volatility .
9.3.2 Solution of the SDE of a geometric (exponential) Brow-
nian motion
In the last handout we said that the solution of the SDE of the geometric Brownian
motion was the exponential of a generalised Brownian motion with certain parameters.
Now we have the show to prove it: we start from a generalised Brownian motion
satisfying the following SDE
dx =


2
2
!
dt+ dz
In this case we have an Ito^ process with a(x; t) = 2
2
and b(x; t) = .
From the previous example we know that the solution of the SDE is given by
x =


2
2
!
t+ z
Let us consider now the exponential transformation
G(x; t) = ex
We see that G is only function of x, so all the derivatives in t will be zero. Compute
now:
@G
@x
= ex = G
@2G
@x2
= ex = G
Then we have
dG =
"
G


2
2
!
+
1
2
G2
#
dt+Gdz
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which simplies to
dG = Gdt+ Gdz
i.e. the SDE of the geometric Brownian motion.
So we have proved that the solution of the SDE of a geometric Brownian motion is
G = exp
(

2
2
!
t+ z
)
i.e. the exponential of a generalised Brownian motion.
9.3.3 Logarithm of stock prices
In the last handout we established that an appropriate model for stock prices is the
geometric Brownian motion. From the previous example, then, we can write
S = S0 exp
(

2
2
!
t+ z
)
and therefore we have that the logarithm of stock prices follows a generalised (not
geometric!) Brownian motion. In fact:
Y = logS = Y0 +


2
2
!
t+ z
where Y0 = logS0, i.e. a generalised Brownian motion with drift rate 22 and
variance rate 2.
This result can also be obtained starting from the SDE for S
dS = Sdt+ Sdz
and applying Ito^'s formula to the function G(S; t) = logS.
9.3.4 Generic transformation of stock prices
We can derive a general formula valid for any function of stock prices. Consider the
usual geometric Brownian motion for S as above. We have
a(S; t) = S
b(S; t) = S
Applying Ito^'s formula, we obtain that a generic transformation of S and time, G =
G(S; t), follows a process with SDE
dG =
"
@G
@S
S +
@G
@t
+
1
2
@2G
@S2
2S2
#
dt+
@G
@S
Sdz
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9.3.5 Forward price
A direct application of the previous formula can be the determination of the SDE
satised by the forward price.
Consider a forward contract on a non-dividend paying stock. The forward price
is F = SerT , where T is the time to maturity. We can express this as a function of
current time t by setting T = tm t, where tm is the maturity date. We obtain
F (S; t) = Ser(tmt)
our function of interest. The process of stock prices is as usual a geometric Brownian
motion. We have
@F
@S
= er(tmt)
@F
@t
= rSer(tmt)
@2F
@S2
= 0
From the formula derived in the previous example we have
dF =
h
er(tmt)S rSer(tmt)
i
dt+ er(tmt)Sdz
Substituting F = Ser(tmt) this becomes
dF = ( r)Fdt+ Fdz
i.e. F is a geometric Brownian motion too, with expected growth rate r and
volatility . The growth rate of F is the excess return of S over the risk-free rate.
9.4 Appendix: second order eects in the limit
Here we look at the limit of dx2. Remember that x follows an Ito^ process, and hence
small changes in x are given by
x = a(x; t)t+ b(x; t)
p
t
where we used the Brownian motion property z =
p
t. Dropping the arguments
of functions a and b, the dierences squared are then given by
(x)2 = a2(t)2 + b2t2 + 2ab(t)3=2
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The rst and last terms are of order higher than t, but the central term has the same
order as t; we also know that 2 21 and therefore
E[t2] = t
Var[t2] = 2(t)2
As t ! 0, the variance of t2 will tend to zero faster than its expectation, so we
can treat the term as non-stochastic and equal to its expected value t. It follows
that (x)2 will tend to b2dt as t tends to zero, i.e.
(dx)2 b2dt
9.5 Further reading
Wilmott (ch. 7) and Hull (Section 10A in 4th edition) provide a heuristic discussion
of Ito's lemma similar to the one presented in these lectures, while Neftci (ch. 9 & 10)
and Bjork (ch. 3) go into slightly more detail.
See for example Stochastic dierential equations: An introduction with applications
by Bernt Oksendal (Springer) for a more advanced treatment of the Ito^ integral and
of stochastic dierential equations.
77 J Herbert UCL 2011-12
Chapter 10
The Black-Scholes model
Earlier in the course we have seen some examples of the pricing of options using the
binomial tree model. In that case the time was discrete, i.e. the price of the stock
underlying the option could only change at some specic moments, and the stock price
was discrete too, being able only to take two dierent values in the next time period.
The model we present here allows the stock price to change at any time, and to take
any value according to a continuous distribution. It is therefore a continuous time
continuous variable model.
We are going to outline the Black-Scholes method for pricing derivatives. In par-
ticular we are going to analyse the pricing of European call and put options on a
non-dividend-paying stock.
10.1 Lognormal property of stock prices
We assume as usual that the stock price follows a geometric Brownian motion:
dS = Sdt+ Sdz
where is the drift and is the volatility. As we have seen in earlier lectures, the
logarithm of stock prices follows a generalised Brownian motion with
d(logS) =


2
2
!
dt+ dz
Therefore we know that the dierences in log-prices have a Normal distribution as
follows:
(logST logS0) N
"

2
2
!
T; 2T
#
so that when S0 is given we obtain:
logST N
"
logS0 +


2
2
!
T; 2T
#
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Stochastic Methods in Finance 1
i.e. the logarithm of the stock prices has a Normal distribution. If a variable X is such
that logX Normal then we say that X has Lognormal distribution.
It follows that ST has a Lognormal distribution, and the moments are given by
E(ST ) = S0e
T
Var(ST ) = S
2
0e
2T
h
e
2T 1
i
10.2 The Black-Scholes-Merton dierential equa-
tion
We now look at how we can determine methods for calculating derivative prices for
the model we have introduced for the stock price process.
Earlier in the course we looked at simple discrete time, descrete state space models
for the stock price, and we saw that there were a number of techniques that we could
apply to price derivatives, all based on the assumption of no-arbitrage.
In this section we start to look at these techniques again, and apply them to our
continuous time model for the stock price process.
10.2.1 Assumptions of the Black-Scholes model
Assumptions:
1) The stock price process, St, follows a geometric Brownian motion
2) We can have any long or short position in the stock
3) No transaction costs
4) The stock pays no dividends (note: the model can be easily adapted to include
dividends)
5) No arbitrage
6) Trading is continuous in time
7) There is a risk-free interest rate that is constant and the same for all maturities
(this assumption can also be relaxed relatively easily to allow for varying but
still deterministic rate)
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Denote the option value process by f ; it is a function of the stock price S and of time
t1, and note that the above assumption number 1) means that we assume, as before,
that
dS = Sdt+ Sdz (10.1)
In our Black-Scholes world we also have a riskless asset with price process Bt say,
i.e. Bt a riskless zero coupon bond that is worth 1 at time zero, with
dB = rBdt; (10.2)
as we have seen in previous lectures.
We present here a heuristic outline of the derivation of a partial dierential equation
This will provide a avour of the important concepts underlying the approach.
We are going to If we can nd a portfolio that replicates the derivative then we
can use the absence of arbitrage to argue that this value of the derivative will be the
price process for this replicating portfolio. However, in order for us to be able to
demonstrate that any other price process allows arbitrage, and hence that this must
be the price of the derivative if there is no arbitrage, we need an additional one more
property for the replicating portfolio.
Consider a replicating portfolio consisting of t amount of the underlying stock,
and t of the riskless asset. At time t this portfolio will be worth
t = tSt + tBt (10.3)
where Bt is the value of the riskless asset.
Recall our replicating strategy for the binomial model. We argued that if we could
nd a portfolio strategy that was always worth the same as the derivative at the
maturity of the derivative in the future, then using no arbitrage we could argue that
the value of this portfolio today must be the value of the derivative. We therefore
need T = f(ST ; T ) = Payoff(ST ) where Payoff(ST ) is the payo function for the
particular derivative at maturity.
The dierence in our continuous time set up is that we now need a dynamic portfolio
strategy to replicate the derivative, as we have possible changes in the stock price over
continuous time. However, to apply our no-arbitrage argument throughout the lifetime
of the derivative, we also need the replicating portfolio strategy to be self-nancing.
Self-nancing portfolios
The replication argument we are going to use relies on the trading strategy we will
use to replicate the derivative not having an injection or withdrawal of money at any
1The fact that the derivative is a function of the stock and time is an assumption that we make.
It can be shown to be the case, but for this course we simply assume it.
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point prior to maturity - the no-arbitrage argument will break down if at any point
funds are added to the portfolio.
This means that for example any change in the amount of stock in the portfolio must
be funded entirely by changes in the amount of bond, and vice-versa.
We say that the portfolio must be self-nancing. In other words, whatever the
cost of setting up the portfolio, it must not use additional, exogenous money to sub-
sidise it at any point, nor have money taken out of it. Mathematically this requires
dVt = tdS(t) + dB(t); (10.4)
so that changes in the portfolio value are driven by changes to the stick and bond
prices only. Changes in the value of the portfolio are explainable in term of changes
in the value of the tradable constituent assets alone.
Any replicating portfolio we use to price must satisfy this result.
This means that changes in the value of the portfolio are solely due to changes in
the value of the two assets, so that we do not add or take out capital from the portfolio
after the initial construction. A self nancing strategy neither requires nor generates
funds between time 0 and time T . This requirements results in the equation
t =
Z t
0
sdSs +
Z t
0
sdBs
which gives the dierential equation for the value of the replicating portfolio:
dt = tdS + tdB
Substituting in for the dynamic processes for the stock and riskless asset price
processes from (10.1) and (10.2), we can now write
dt = tdS + tdB
= (tS + trBt)dt+ tSdZ (10.5)
We can also apply Ito^'s formula to the derivative value process, as it is a function
of the stock price and time. For to replicate the derivative it must also be a function
of the stock price and time, so that Ito^'s result gives
d =
"
@
@S
S +
@
@t
+
1
2
@2
@S2
2S2
#
dt+
@
@S
Sdz (10.6)
where the rst term is deterministic and the second is random.
Combining (10.6) and (10.5) we have
For the portfolio to be both replicating and self-nancing we need to equate both
the stochastic terms and the deterministic terms in the dynamics of these two equation.
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These two conditions are analogous to our simultaneous equations that we needed to
solve in the binomial model.
In other words, comparing (10.6) and (10.5) we nd for the portfolio to be repli-
cating we need
tS =
@
@S
S (10.7)
(tS + r tBt) =
@
@S
S +
@
@t
+
1
2
@2
@S2
2S2 (10.8)
Now recall that our aim is to nd parameters t and t to make this portfolio
replicating. We can already see what t must be - we should chose
t =
@
@S
for (10.7) to hold.
Eliminating t from (10.8) by using a re-arrangement of (10.3) ( tBt = ttSt),
we also have that
(tS + r(t tSt)) = @f
@S
S +
@f
@t
+
1
2
@2f
@S2
2S2 (10.9)
Substituting our result for t =
@f
@S
into this we have
(
@
@S
S + r(t @
@S
St)) =
@
@S
S +
@
@t
+
1
2
@2
@S2
2S2
which can be simplied as
rt = r
@
@S
St +
@
@t
+
1
2
@2
@S2
2S2
which is called the Black-Scholes-Merton partial dierential equation
The dierential equation (10.11) is an example of a class of partial dierential
equations called parabolic.
Exercise: Convince yourself of the need for the replicating portfolio to be
self-nancing by re-visiting the proof of this result, and showing where the
argument breaks down if the portfolio is not self-nancing.
10.2.2 Boundary conditions
In order to solve this dierential equation and obtain f we need to impose some
boundary conditions which will depend on which derivative (f) we want to price. This
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is common to all dierential equations. For example, say that we look at a function
g(x; y) such that
@g
@x
+
@g
@y
= 0
A solution is given by a(x y) + b, and without boundary conditions we cannot
nd specic values for the constants a and b. If we impose values for the function
g at specic points for x and y, say g(0; 0) = 0 and g(1; 0) = 2; we obtain a = 2
and b = 0. Similarly, we need to nd boundary conditions for our derivatives prices to
solve together with the partial dierential equation (10.11). See the handout "Calculus
Refresher" for more details.
The boundary conditions are determined by the properties of the particular deriva-
tive we are trying to value, usually depending on their payo function at the exercise
date. For a European call option with strike price X, the boundary condition is given
by the payo function at the exercise date
f = max(S X; 0) at time T
or equivalently
fT = max(ST X; 0):
Similarly for a European put option we use the boundary condition
f = max(X S; 0) at time T
10.3 Black-Scholes formulas for the pricing of vanilla
options
We will not solve the partial dierential equation (10.11) directly in this course, as this
requires techniques from a course in partial dierential equations. Here we just give
the solution, but we will determine this solution from the PDE through an indirect
risk-neutral pricing method in later lectures.
Using the above boundary conditions, the solutions of the Black-Scholes dierential
equation are given by the following formulas for the pricing of vanilla call and put
options.
1. European call option with strike price X and time to expiry T .
Call = S0N(d1)XerTN(d2) (10.10)
where N(:) denotes the cumulative distribution function of a standard Normal,
and
d1 =
log(S0
X
) +

r +
2
2

T

p
T
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d2 =
log(S0
X
) +

r 2
2

T

p
T
= d1
p
T
2. European put option with strike price X and time to expiry T .
Put = XerTN(d2) S0N(d1)
where the notation is the same as above.
Notice that both formulas do not depend on .
Put-call parity. Recall from workshop 1 the important result we determined be-
tween the price of a European call and a European put on the same underlying, with
equal strike prices and maturity times. We can verify that these Black-Scholes pricing
formulas satisfy this put-call parity result, i.e.
Call Put = S0 XerT
From the expressions above we have:h
S0N(d1)XerTN(d2)
i

h
XerTN(d2) S0N(d1)
i
=
= S0[N(d1) +N(d1)]XerT [N(d2) +N(d2)]
= S0 XerT
where the last result follows from the symmetry of the Normal distribution (i.e.
N(z) = 1N(z)).
Note - American options. When there are no dividends, it can be shown that early
exercise of an American call option is never optimal, and therefore the price is the
same as a European call option, and we can use the formula above. Unfortunately,
there is no analytic formula for the pricing of American put options, so in this case we
need to use numerical methods.
Example. The price of a non-dividend-paying stock is now $42, the risk-free interest
rate is 10% per annum, and the volatility is 20% per annum. What are the prices of a
European call and a European put option expiring in 6 months and with strike price
of $40?
We have S0 = 42, X = 40, r = 0:1, = 0:2 and T = 0:5 years. We rst compute
the constants d1 and d2 and obtain
d1 =
log(42
40
) +

0:1 + (0:2)
2
2

0:5
0:2
p
0:5
= 0:7693
d2 = d1 0:2
p
0:5 = 0:6278
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Therefore the prices of the options are given by
Call = 42N(0:7693) 40e0:10:5N(0:6278) = 4:76
Put = 40e0:10:5N(0:6278) 42N(0:7693) = 0:81
10.4 BSM PDE using a riskless portfolio approach
Here we demonstrate how a riskless portfolio argument can be used to give the B-S-M
pde. The approach is similar to the replicating approach used above. We rst use Ito^'s
formula for the derivative price as a function of the stock price and time. This gives
df =
"
@f
@S
S +
@f
@t
+
1
2
@2f
@S2
2S2
#
dt+
@f
@S
Sdz
where the rst term is deterministic and the second is random.
We can attempt to construct a riskless portfolio (i.e. get rid of the random term)
by rst constructing a portfolio consisting of:
1 : derivative
: share
This is the same as our approach under the binomial model. At any point in time the
value of is:
= f +S
and therefore the value of the portfolio changes according to
d = df +dS
But we know that df and dS are given by:
dS = Sdt+Sdz
df =
"
@f
@S
S +
@f
@t
+
1
2
@2f
@S2
2S2
#
dt+
@f
@S
Sdz
so together this gives
d =
"
@f
@S
S @f
@t
1
2
@2f
@S2
2S2 +S
#
dt+
"
@f
@S
S +S
#
dz
We can see that if we chose = @f
@S
then we can eliminate the random component of
this expression, and we have
d =
"
@f
@t
1
2
@2f
@S2
2S2
#
dt
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which is a riskless portfolio because there is no stochastic driver in the dynamics of the
portfolio price process, and hence no uncertainty about its future value. By denition,
a riskless portfolio grows at the risk-free rate r, so it satises
d = rdt
so from above we have
r = @f
@t
1
2
@2f
@S2
2S2
But from the denition of the portfolio r = rf + r @f
@S
S, and thus we obtain
rf = r
@f
@S
S +
@f
@t
+
1
2
@2f
@S2
2S2 (10.11)
which is the Black-Scholes-Merton dierential equation.
Compare this outlined approach with the arguments used for derivative pricing
under the binomial model in previous lectures.
Note. The portfolio we are using is not permanently riskless, but riskless only in an
innitesimally short period of time. As S and t change, @f=@S also changes. To keep
the portfolio riskless, we have to change frequently the quantity of stock.
10.5 Simple extensions to the Black-Scholes model
We have assumed that the risk-free rate is always constant, and crucially it is deter-
ministic (i.e. entirely predictable). As long as we keep it deterministic we can allow
the risk-free rate to vary over time, as the function r(t) say, and still follow through the
same Black-Scholes arguments outlined above. In this case the risk-free rate parameter
r in equation (10.11) should be replaced by
1
T t
Z T
t
r(u)du;
which can informally be seen as a form of averaging r over the remaining lifetime of
the option.
10.6 Further reading
The risk-free hedging approach to deriving the Black-Scholes formula is discussed in
many option pricing books, including Wilmott (ch. 8), Hull (ch. 11 of 4th edition).
Bjork (ch. 7) outlines the replicating strategy derivation of the BSM PDE.
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For more about solving PDEs both analytically and numerically see for example the
Wilmott, Howison and Dewynne book, referenced in the course guidance notes.
See also Rebonato - Volatility and Correlation for discussion of techniques for mod-
elling of volatility smiles.
See Bjork chapter 7 for further discussion of the replicating approach to showing the
B-S-M PDE result.
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Chapter 11
Hedging and the Greeks
11.1 The Greeks
A nancial institution that sells a derivative to a customer is faced with the problem
of managing the risk. This can be done by forming an appropriate hedging portfolio
as we have seen above. In order to do this they need to use the partial derivatives
of the option price with respect to dierent variables. Such partial derivatives play
an important role and have been assigned specic names, according to Greek letters;
for this reason they are known as the Greek letters or simply the Greeks. They are
dened as follows:
Delta: = @f
@S
Theta: = @f
@t
Gamma: = @2f
@S2
Vega: V = @f
@
Rho: P = @f
@r
As we discussed earlier in the course, one of the main uses of derivatives is for
the transfer and management of risk. Here we take an initial look at the concepts of
hedging in derivatives, and explore the use of the Greeks that we have dened.
11.2 Delta Hedging
The of a derivative or a portfolio is the sensitivity of it value with respect to move-
ment in the underlying. As we have seen it is the rst derivative with respect to the
underlying, Delta: = @f
@S
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Stochastic Methods in Finance 1
We have also seen in previous lectures that is plays an important role in determining
the pde for the price of a derivative. The delta of the derivative was the amount of
the stock that we needed to hold to hedge the derivative, i.e. to create a replicating
portfolio.
Consider an agent who holds a portfolio consisting of an amount of an underlying
(say a stock) and a derivative on the stock. The agent wishes to make the value of
his portfolio, P (S; t) say, immunized to small changes in the value of the stock, S. In
other words he wants
@P
@S
= 0;
and if this is the case we say the portfolio is delta neutral. One way to do this is to
add an amount of a derivative to the portfolio to make the portfolio delta neutral. If
the derivative has pricing function F (S; t), and an amount x of the derivative is added,
then the new portfolio, V (S; t) say, is worth
V (S; t) = P (S; t) + xF (S; t):
To make this new portfolio delta neutral we need @V
@S
= 0 which implies that
@V
@S
=
@P
@S
+ x
@F
@S
= 0
which gives
x = @P
@S
=
@F
@S
:
This is the amount of the derivative that needs to be added to a portfolio to make it
delta neutral.
Consider the special case in which the original portfolio consists of just a short
position in one derivative and no stock, so that P (S; t) = f(S; t) say. In other words
we are an agent who has sold a derivative. Suppose we wish to hedge our position
with the underlying asset by making it delta neutral. The asset to be added to the
portfolio is the stock so that the price of the asset is F (S; t) = S, and hence @F
@S
= 1.
We then have that the amount of the stock that needs to be added to obtain a delta
neutral position is
x = @P
@S
=
@F
@S
= @f
@S
:
In other words, the number of units of a stock needed to hedge one unit of a
derivative is @f
@S
, which is the delta of the derivative dened earlier, and also the
amount used in the construction of the riskless portfolio during the Black-Scholes-
Merton p.d.e. derivation.
For an agent who has sold an option, the is the amount of stock that needs to
be held to hedge his position. If the agent wants a perfectly hedged position he will
89 J Herbert UCL 2011-12
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need to constantly adjust this amount by re-balancing his portfolio, as delta @f
@S
itself
is constantly changing as S changes.
It can be shown that the delta for a European call is
=
@f
@S
= N(d1)
using the notation of the Black-Scholes formula (see exercise sheet). Deltas for call
options are always positive, which means that a long (buy) call should be hedged with
a short (sell) position in the underlying, and vice versa.
The delta of a European put can be similarly calculated as N(d1) 1, or simply
derived from the delta of a call (again see exercise sheet). Deltas for put options are
always negative, which means that a long put should be hedged with a long position
in the underlying, and vice versa.
Delta is between 0 and +1 for calls and between 0 and -1 for puts. The delta for
the underlying is always 1. A put option with a delta of 0.5 will drop 0.5 in price for
each 1 rise in the underlying (i.e. increasingly out-of-the-money), a call option with
the same delta will rise 0.5 instead (i.e. increasingly in-the-money).
Example
If, for example, the share price is 10 and the call option price is 1 and the delta of the
call option is 0.5, an investor who has sold 12 call option contracts (options to buy
1,200 shares) can delta-hedge his/her position by buying 0.5 x 1,200 = 600 shares. A
rise in share price will produce a loss of 0.5 x 1,200 = 600 on the call options but a
gain of 600 on the shares.
The delta of the portfolio can be determined by adding up all his/her positions.The
delta of the short option position is -0.5 x 1,200 = -600 and delta of the long share
position is 0.5 x 1,200 = 600,thus his/her position has a delta of zero, this is referred
as being delta neutral.
Unfortunately, delta-hedging only works for a short period of time during when
delta of the option is xed. The hedge will have to be readjusted periodically to re ect
changes in delta, which could be aected by the share price, time to expiry, risk-free
rate of return and volatility of the underlying. See exercises for an exploration of how
how delta changes with the underlying share price and time to expiry.
11.3 Gamma and gamma neutral portfolios
In theory a derivative can be perfectly hedged by continuous re-balancing. This is the
basis behind the no-arbitrage derivation of the option prices we have been looking at.
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In a continuously re-balanced hedge the value of the stock and the money holdings in
the bank will replicate the value of the derivative. This is analogous to the replicating
portfolio approach to derivative pricing that we saw in the binomial model.
In practice of course, any hedging will need to be done at discrete time points. The
following steps show how this can be done:
Sell one unit of the derivative at time t = 0 at price f(0; S).
Find ; and buy shares. Use the income from the derivative sale, and borrow
from the bank at the risk free rate if necessary.
Wait one time period (day, week, minute...). The stock price has moved, and so
the old is no longer correct.
Calculate the new current and adjust stock holdings, borrowing or lending
with the bank if necessary.
Repeat until time T , the maturity date.
In practice the discrete hedging suers from a trade o between balancing more
often to obtain a better hedge, but suering higher transaction costs as a result.
Of course the reason why we have to re-balance the portfolio constantly is that
delta @f
@S
itself is constantly changing as S changes. For this reason we may wish to
know how quickly our portfolio will become unbalanced as S changes, in other words
how quickly changes as S changes. This measure of the sensitivity of a portfolio's
to S is given by = @
@S
= @
2P
@S2
.
If Gamma is high then delta is very sensitive to the underlying price, and the
portfolio has to be re-balanced frequently, whereas a low Gamma means delta only
changes slowly and so to keep the portfolio delta-neutral adjustments to can be made
less frequently, and so we can keep the hedge for a longer period.
Consider also the change in value of a portfolio of shares and options, P . Using a
Taylor expansion and ignoring higher order terms we can write an approximation of
{ = S +t+ 1
2
S2
so that for a delta neutral portfolio with = 0 we can see that for positive gamma,
any changes in the underlying share price value over a short time period will result in
an increase in the value of the portfolio. The reverse is also true.
It is therefore desirable to keep a portfolio Gamma neutral as well as Delta neutral,
in other words, keep @
2P
@S2
= 0. Delta neutrality will protect against small movements
in the underlying between rebalancing, whereas gamma neutrality will also protect
against large movements in the underlying price between delta-hedge rebalances.
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To make the portfolio gamma neutral as well as delta neutral, we need to add to
our portfolio an amount of two derivatives, so that it is now worth
V (S; t) = P (S; t) + xFF + xGG;
where F and G are the two derivatives to be added in amounts given by xF and xG.
Dierentiating once and then again, and setting both results to zero for a delta and
gamma neutral portfolio, gives
P + xF F +xGG = 0
P + xFF + xGG = 0
which can be solved if we know Delta () and Gamma () for F;G and the current
portfolio P .
Of course one of the derivatives may be the underling asset itself, which has a Delta
of 1 and a Gamma of 0 (as @S
@S
= 1 and @1
@S
= 0). Then the value of the new portfolio
V is
V (S; t) = P (S; t) + xFF + xSS
which is delta and gamma neutral if
P + xFF + xS = 0
P + xFF = 0:
This can be solved (check this) to give
xF = P
F
xS =
FP
F
P :
11.4 Further reading
For a discussion of hedging and the Greeks see for example chapter 10 in Wilmott,
chapter 8 in Bjork, or chapter 13 in Hull (4th edition).
92 J Herbert UCL 2011-12
Chapter 12
Volatility
We have seen that in the Black-Scholes model the option value is a function of the
stock price S, of time t and time to expiry T , of the strike price X, and of volatility
of the underlying asset , of the risk-free interest rate r, and of volatility of the un-
derlying asset . The main challenge in applying the standard Black-Scholes model is
to determine the most appropriate volatility to use. The volatility has a big impact
on the price of the option, though it is a forward looking parameter in the sense that
it is the volatility of the underlying over the future life of the derivative (until the
derivative maturity date) that is important. Therefore, despite its importance on the
derivative price, the volatility is in some sense the parameter that can not be directly
observed.
In practice at least two methods can be used to estimate or gain a view on the
volatility: we can either estimate from historical data for the movements of the price
of the underlying asset, or we can use the volatility implied by the prices of similar
derivatives in the market. The two methods are illustrated below.
12.1 Estimating volatility from historical data
Earlier we derived the distribution for the dierences in log-prices. We observe data
at xed intervals (e.g. every week or every day); denote by Si the stock price at the
end of the i-th interval, and let be the length of the interval, in years (e.g. for daily
observations = 1=252, where we assume that there are 252 trading days per year).
Then we know that
ui = (logSi logSi1) = log

Si
Si1
!
N
"

2
2
!
; 2
#
Therefore, if we have n observations, we can compute
s =
vuut 1
n 1
nX
i=1
(ui u)2
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Stochastic Methods in Finance 1
where u is the sample average of the ui's, and then obtain an estimate of the volatility
as
^ =
sp

Note that the volatility in the Black-Scholes formula corresponds to the future
volatility of the underlying asset, and not the historic volatility. Therefore the historical
volatility may not always be the best indicator of the volatility appropriate for the
option price.
12.2 Implied Volatility
Another way to estimate is to use the volatility that the market is currently using
or implicitly assuming. This involves calculating the volatility implied by an option
price observed in a market for tradable options. By looking at other tradable, similar
derivatives to the one we are pricing (based on the same underlying), we can substitute
their prices into the Black-Scholes pricing formulas, and solve for to get the implied
volatility. In other words, the implied volatility is the volatility of the underlying
which, when substituted into the Black-Scholes formula, gives the theoretical price
equal to the market price the option is currently trading at. 1
The Black-Scholes pricing formula is not directly solvable for but can be solved
by a numerical root nding procedure. Implied volatilities can be used to monitor the
market's opinion about the volatility of a particular stock. One use for them is to
calculate implied volatilities from actively traded options on a certain stock and use
them to calculate the price of a less actively traded option on the same stock.
Note that the prices of deep in-the-money and deep out-of-the-money2 options can
be relatively insensitive to volatility. Implied volatilities from these options therefore
tend to be unreliable if used for options on the same underlying that are at the money.
This is one example of a volatility smile.
12.3 Volatility smiles
In practice if you calculate the implied volatility using the Black-Scholes formula for
many dierent strike prices and expiry dates on the same underlying stock, then we
nd that the implied volatility is not constant across these strikes and expiry dates.
If you plot the graph of implied volatility against strike price the shape is of a smile
1When using this approach, care needs to be taken to consider the impact of possible volatility
smiles - see the following section.
2We say that an option is deep in-the-money if exercising the option today would provide a high
payo (i.e. St >> K for a call option), and similarly an option is deep out-of-the-money if the stock
is a long way from giving the option a payo positive, i.e. St << K
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or a smirk (roughly). Of course, if the assumptions and theory of Black-Scholes all
stood exactly then the graph should be a straight line representing constant volatility
for the stock, regardless of the strike price of any option used for the calculation.
In equity options the implied volatility often decreases as the strike price increases
(see Figure 1).
Figure 12.1: Volatility smile for equity options
In foreign currency options we often see that the volatility becomes higher as the
option moves either in the money or out of the money (see Figure 2).
Figure 12.2: Volatility smile for foreign exchange options
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There are many suggested reasons for volatility smiles. One of the most important
involves a closer look at the assumptions about the movement of the underlying in the
Black-Scholes model. As we have seen, the assumption that the price of the underlying
follows geometric Brownian motion implies
changes in asset price follow a lognormal distribution
the volatility of the asset is constant
The price of the asset changes smoothly with no jumps (which comes from the
continuous property of Brownian motion).
If the movement of the prices of the underlying exhibit behaviour dierent from
these assumptions, and in particular if extreme price movements occur more often than
the lognormal distribution predicts, then we may observe a volatility smile, or other
distortion from a straight line.
The smile eect clearly means that it can be dangerous to automatically use implied
volatilities for underlying assets in pricing models for options on these assets which
are not so readily traded, as we would ideally like to do. Estimating and incorporating
the potential impact of the volatility smiles is a key component of derivative pricing,
though exploration of the many approaches for doing this is beyond the scope of this
introductory course.
Volatility smiles and fat tailed distributions
Suppose that the movement of the underlying asset, or at least the market's view
of it, is in fact dierent from this model. In particular, suppose that extreme price
movements occur more often than the lognormal distribution predicts, so that the
implied distribution has \fatter tails" than the model log normal distribution. Let's
consider the eect this would have on the implied volatility graph.
Deep out-of-the-money (call) options will have a high strike prices relative to the
current asset price, and will only pay out if there are relatively large movements in the
underlying that take it above the strike price. If these large movements are more likely
in the implied distribution than the model distribution due to fatter right-hand tails,
then the call option price will be higher under the implied distribution. This is because
the option is more likely to yield a positive payo under the implied distribution. Thus,
when this higher implied (market) price is put into the Black-Scholes formula, it will
yield a higher implied volatility.
If the there are also fatter left-hand tails, then a similar argument concerning
deep out-of-the-money put options for low strike prices will also give higher implied
volatility (again, the larger probability of asset downward movements needed for a
positive payo under the implied distribution gives rise to higher put option prices
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Stochastic Methods in Finance 1
and hence higher implied volatility). Put-call parity can be used to show that the
same high volatility will be used to price a deep in-the-money call option with the
same strike as the out-of-the-money put option.
We can therefore see that fatter tails in both the right and the left side of the
distribution will lead to higher implied volatility for high and low strike prices when
compared to implied volatility for options at-the-money. This will produce an upwards
curve to the right and left of the implied volatility graph, hence the \smile" we see in
Figure 2.
Possible causes of fat tailed distributions
As we have seen, if the movement of the underlying prices exhibits behaviour dierent
from these assumptions, and in particular if extreme price movements occur more often
than the lognormal distribution predicts, then we may observe a volatility smile, or
other distortion from a straight line.
This means that the implied distribution of price movements is not in fact lognormal
as assumed, but may be a distribution, for example, with \fatter tails". Why might
these fatter tails occur?
Consider options for foreign exchange. Exchange rate prices may be subject not
only to non-constant volatility, but also jumps in their movements, both up and down
(maybe in response to central bank announcements). This is inconsistent with the
continuity assumption for the movement of the underlying in the Black-Scholes model.
The eect of both a nonconstant volatility and jumps is that extreme outcomes become
more likely, i.e. a fatter tailed distribution in both the right and left hand side.
The case of equity options is usually slightly dierent. A look at Figure 1 shows
that there is higher volatility at lower strike prices compared with at-the-money strike
prices, but lower volatility at higher strike prices. This implies that the tails of the
equity movement distribution are fatter on the left side, but not on the right side
(follow through the above arguments to check this).
A possible explanation sometimes oered for this is that the volatility of an equity
will increase as the equity value decreases because its nancial leaverage will increase,
making it a more risky investment3 (do not worry if you are not familiar with this
concept, this leaverage explanation will not be examined).
A further possible explanation that has been given for the volatility smile in equity
options is what is referred to as \crashophobia". The argument is that traders are
concerned about the possibility of further market crashes and price options accordingly.
3A company's leaverage refers to the relative amounts of debt and equity that are used to nance
the rm. All other things equal, a rm is considered more risky if it contains more debt i.e. if it
has a higher level of borrowing. If the equity value falls, then the amount of debt relative to equity
becomes greater, which makes the rm more risky
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12.4 Further reading
For a discussion of possible causes of the volatility smile and the impact of fat-tailed
distributions, see Hull chapter 17 (in 4th edition).
More advanced models in which volatility is a stochastic parameter are described
Derivatives in nancial markets with stochastic volatility by Fouque, Papanicolaou
and Sircar, published by Cambridge University Press.
See also Rebonato - Volatility and Correlation for discussion of techniques for mod-
elling of volatility smiles.
98 J Herbert UCL 2011-12
Chapter 13
Risk-neutral pricing in continuous
time
We saw in the discrete binomial model that we could use a convenient interpretation
of our pricing formula in terms of \risk-neutral" probabilities. We now look at the
analogous result in continuous time. See the Binomial Model lecture as background.
13.1 The risk-neutral process
The principle of risk-neutral pricing is that, for purposes of valuing a derivative, we
can use a technique in which we proceed as if the expected stock price were increasing
at the risk-free interest rate r (ignoring entirely its real-world rate of increase ). We
then use discounted expected derivative payos under this probability distribution. We
have already seen this approach to derivative pricing in the Binomial Model lectures
earlier in the course. As we saw under the Binomial model, the reason we can use this
approach is essentially because it is possible to create a replicating portfolio for the
derivative.
Suppose that a stock price St follows a geometric Brownian motion, with drift rate
and variance rate 2 per unit time:
dS = S dt+ S dz: (13.1)
In particular, this implies E(St) = S0 e
t.
In the articial risk-neutral world, the stock price can be assumed to follow the
process
dS = r S dt+ S dz: (13.2)
Then we have E^(St) = S0 e
rt, where E^() denotes expectation under the process (13.2),
so that this does indeed imply that the expected stock price grows at the risk-free rate,
as required. Note that we keep the volatility parameter unchanged when moving
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Stochastic Methods in Finance 1
from the real world (13.1) to the risk-neutral world (13.2): this is analogous to keeping
the up and down steps the same in the binomial model.
Note that this is the same argument we used in section 5.6, when we determined
what the risk-neutral probabilities of a stock up movement must be in a risk-neutral
world under the binomial model, to give a expected rate of return equal to the risk-free
rate.
13.2 Risk-neutral pricing
Consider a derivative on the stock, and denote its value at time t, if the stock price is
s, by f(s; t). So its actual value at t will be ft = f(St; t), depending on the random
quantity St and thus itself random. Under risk-neutral pricing, its value at time t
should be the suitably discounted expectation, under the risk-neutral model (13.2), of
its value at some later time tm (in our applications tm will be a maturity time, but
that is not crucial). Thus, given that St = s, we should have
ft = f(s; t) = e
rT E^(ftm j St = s); (13.3)
where T = tm t. In cases where we know the function f(; tm) determining the value
ftm = f(Stm ; tm) of the derivative at tm as a function of the stock price at that time,
we can use (13.3) directly to calculate f(s; t) for all t tm. In section (??) below we
give a proof that this formula does satisfy the Black-Scholes dierential equation.
Moreover, it is clearly correct when t = tm i.e. at maturity. It follows that it must
indeed be the correct derivative pricing formula. In particular, the initial value of the
derivative is given by
f0 = e
rT E^(fT j S0); (13.4)
for any future time T .
13.2.1 Example: European call option
Consider a European call option with strike price X and exercise date T . At time T ,
this will have known value
fT = maxfST X; 0g: (13.5)
We can thus apply (13.4) to calculate its initial value.
To evaluate E^(fT j S0) = E^(maxfST X; 0g j S0), we rst recall that the distri-
bution of the future stock value ST is log-normal. In fact, starting from current stock
value S0, and using the growth-rate r appropriate to the risk-neutral world, we have
logST N

logS0 + (r 122)T; 2T

: (13.6)
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We can express this in terms of a standard normal variable U :
logST = logS0 + (r 122)T +
p
T U: (13.7)
In terms of U ,
fT = S0 e
(r 1
2
2)T e
p
T U X
if logST logX, viz if
U log

X
S0

(r 1
2
2)T

p
T
= d2;
otherwise fT = 0.
We thus obtain
E^(fT ) =
Z 1
d2
n
S0 e
(r 1
2
2)T e
p
T u X
o
1p
2
e
1
2
u2 du
= S0 e
rT
Z 1
d2
1p
2
e
1
2
(upT )2 duX N(d2) (13.8)
where N denotes the standard normal distribution function:
N(x) :=
Z x
1
1p
2
e
1
2
u2 du:
To evaluate the remaining integral in (13.8), substitute v = u+
p
T . It becomesR d1
1
1p
2
e
1
2
v2 dv; i.e. N(d1), where
d1 = d2 +
p
T
=
log

S0
X

+ (r + 1
2
2)T

p
T
:
Thus E^(fT ) = S0 e
rTN(d1)XN(d2), and hence, on applying (13.4),
f0 = S0N(d1)XerTN(d2): (13.9)
Equation (13.9) is the formula for pricing a European call option. A similar analysis
yields the following pricing formula for a European put option, having fT = maxfX
ST ; 0g:
f0 = Xe
rTN(d2) S0N(d1):
See also workshop 2 for a similar look at this problem.
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13.3 A useful log-normal distribution result
The following result is useful for pricing derivatives when the underlying asset follows a
log normal distribution. We have eectively already used it to derive the Black-Scholes
formula in the workshop 2 and example above.
If V follows a log normal distribution where the standard deviation of ln(V ) is = s,
then we can write
E[max(V X; 0)] = E[V ]N(d1)XN(d2): (13.10)
where
d1 =
ln (E[V ]=X) + s2=2
s
and
d2 =
ln (E[V ]=X) s2=2
s
Exercise: prove this result - this requires the same approach as the one used in
Workshop 2 to determine the expectation integral in the Black-Scholes derivation, but
with a very slight modication to the notation.
13.4 The risk-neutral Monte Carlo approach to deriva-
tive pricing
As we have discussed in previous lectures, the solution to the Black-Scholes-Merton
partial dierential equation with appropriate boundary conditions often has to be
calculated using numerical methods such as nite dierence techniques. It is, however,
sometimes easier to price exotic options through Monte Carlo simulation in a risk-
neutral framework.
This uses the risk-neutral valuation approach in continuous time that we have cov-
ered above. However, with some exotic options it may not be possible to calculate
algebraic expressions for the expected payos under the risk-neutral probability dis-
tribution. We need to use computer based-simulation to calculate the risk-neutral
expectations, and a technique called Monte Carlo simulation.
This involves using a computer to generate (pseudo) random numbers, and mod-
elling realisations of the risk-neutral process from these. A summary of the steps for
Monte Carlo based risk-neutral valuation is:
1) Simulate the risk-neutral random walk of the required time horizon up until the
maturity date, starting at today's value of the asset.
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2) For this realisation, calculate the derivative payo.
3) Repeat many times.
4) Calculate the average payo from all realisations.
5) Take the present value of the average for the option value (i.e. discount at the
risk-free rate).
Note that for European style path independent options step 1 is straightforward
as the jump to time T can be simulated in one go. For example under a Geometric
Brownian motion model for the underlying asset price we know that the movement of
the asset price from today to the maturity time will follow a log-normal distribution,
which can be easily simulated.
For path dependent options the value of the asset price up to the maturity price
may need to be tracked at small time intervals to produce accurate results. American
options are very hard to use Monte Carlo simulation for, as at each time point there
needs to be an assessment of whether early exercise should be made
13.4.1 Simulating Geometric Brownian motions
We now look more closely at step (1) above. We know from work above that the
risk-neutral process for the stock price also follows a Geometric Brownian motion,
with drift r. We can now also use the result we obtained earlier about the log normal
distribution price movements under Geometric Brownian motion. Equation 13.7 above
gave us
logST = logS0 + (r 122)T +
p
T :
where is a standard normal variable. If we can then simulate a standard normal
variable then we are able to simulate the log of the stock price at time T , and hence
by taking exponentials, the stock price itself.
13.4.2 Generating random variables
Many methods for (pseudo) random number generation provide samples from a uniform
distribution on [0; 1] (which we denote by U[0,1]). For the purpose of generating Monte
Carlo simulations we then usually need to turn this random variable into a sample
from the distribution we are interested in. For example in the above section we were
interested in a sample from the normal distribution.
One useful result that can be used to generate samples from a range of probability
distributions is as follows. If u U[0,1] is a sample from a uniform distribution and
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FX(x) an invertible cumulative probability density function for the random variable of
interest X then the transformation
y = F1(u)
will be a random sample from a distribution with density Fx.
So in the case of generating a sample from a standard normal distribution we can
use N1(u); where N() is the cumulative distribution function of a standard normal.
13.5 Appendix - Risk neutral pricing and the Black-
Scholes equation
Here we demonstrate that, under mild conditions, the formula 13.3 satises the Black-
Scholes dierential equation
rf = rs
@f
@s
+
@f
@t
+ 1
2
2s2
@2f
@s2
;
or equivalently, in terms of T = tm t (keeping tm xed),
rf = rs
@f
@s
@f
@T
+ 1
2
2s2
@2f
@s2
: (13.11)
To simplify formulas, we introduce Y = logStm , and dene h(x) = f(e
x; tm), so
that f(s; tm) = h(log s). Then ftm = h(Y ). In the risk-neutral world the distribution
of Y (starting from an initial stock-price S0) is given by 13.6 or 13.7, with T = tm.
The conditional distribution of Y given St (t < tm) is, similarly, obtained by replacing
S0 by St, and interpreting T as tm t, in the right-hand sides of 13.6 and 13.7. Thus,
given St = s, we can represent
Y = log s+ (r 1
2
2)T pT U (13.12)
with T = tm t. Thus we can write 13.3 as
f(s; t) = erT Efh(Y )g; (13.13)
where Y is given by 13.12, and the expectation is under the standard normal distri-
bution for U . We now dierentiate 13.13, assuming that we can take this operation
through the expectation, and using, from 13.12,
dY
ds
=
1
s
dY
dT
= (r 1
2
2) 1
2
T
1
2 U:
104 J Herbert UCL 2011-12
Stochastic Methods in Finance 1
We obtain:
@f
@s
= erT E

1
s
h0(Y )

(13.14)
@2f
@s2
= erT E

1
s2
h0(Y ) +
1
s2
h00(Y )

(13.15)
@f
@T
= rf + erT E
hn
(r 1
2
2) 1
2
T
1
2 U
o
h0(Y )
i
: (13.16)
Substituting these into the right-hand side of 13.11 and simplifying then gives
rf + 1
2
T
1
2 erT E fpTh00(Y ) + U h0(Y )g : (13.17)
To complete the analysis we need the following result.
Lemma:
Let k be a piecewise dierentiable real function such that k(u)(u)! 0 as juj ! 1
(where (u) = (1=
p
2) exp( 1
2
u2) denotes the standard normal density function),
and let U have a standard normal distribution. Then
EfU k(U)g = Efk0(U)g: (13.18)
Proof:
We can write
EfU k(U)g =
Z 1
1
u k(u)(u) du
=
Z 1
1
k(u)0(u) du;
since 0(u) = u(u). The result follows on integrating by parts. The same result
will hold if there are isolated points at which k is undened or non-dierentiable, since
we can split the range of integration into intervals inside each of which everything is
well-behaved, and then combine. This extension is needed for dealing with functions
such as 13.5, which is not dierentiable at X.
Now take k(u) = h0(y). We have k0(u) = h00(y)(dy=du) = pT h00(y). Thus, from
13.18,
E fpTh00(Y ) + U h0(Y )g = 0; (13.19)
so that 13.17 becomes rf; verifying that the Black-Scholes dierential equation 13.11
is satised.
105 J Herbert UCL 2011-12