Financial Mathematics – MATH 262
2023-2024
CHAPTER 1
Economical and financial background
1. Time value of money
The time value of money (TVM) is an economic principle that suggests present
day money is worth less than money in the future because of its earning power over
time. Put simply a pound today is worth more than a pound next year because
money can be invested today and earn interest. The time value of money relates
to three basic parameters: inflation, opportunity cost and risk.
Definition 1.1: Inflation, opportunity cost and risk
(1) Inflation is reducing what is known as the purchasing power of money
because it increases the prices of goods and services. Therefore, over time
the same amount of money can purchase fewer goods and services.
(2) Opportunity cost refers to the potential gain or loss on an investment
that someone gives up by taking alternative action.
(3) Risk relates to the investment risk that investors undertake when
putting their money into investment assets. (we will study investment risk
and returns in Chapter 2).
1.1. Future Value.
Money value fluctuates over time hence has different values in the future. This
is because one can invest today in an interest-bearing bank account or any other
investment and that money will grow/shrink due to the rate of return.
To evaluate the real worthiness of an amount of money today after a given
period of time, economic agents compound the amount of money at a given interest
rate. Compounding at the risk-free interest rate would correspond to the minimum
guaranteed future cash flow. If one wants to compare their change in purchasing
power, then they should use the real interest rate which incorporates inflation.
Definition 1.2: Future value
Future value (FV) is the amount to which a current investment will grow
over time when placed in an account that pays compound interest. The
process of going from today’s value, known as the present values (PV) to
the future value (FV) is called compounding.
Example 1.1
Suppose you deposit 100 GBP in a bank that pays 10% interest each year.
What is the future value of this deposit at the end of years 1,2 and 3?
3
4 1. ECONOMICAL AND FINANCIAL BACKGROUND
Solution
At the end of years 1,2 and 3 the investment will grow to 110 GBP, 121
GBP and 132.10 respectively as you can see in the following time-line.
Mathematical formulas for calculating future values.
In general, to find how much the investor earns at the end of year n we use the
compounding formula
FVn = PV (1 + r)n. (1)
where
• FVn = the future value at the end of year n,
• PV = the present value or initial value of the account (or initial invest-
ment),
• r = interest rate (per annum) paid by the bank in the account of the
investor, and
• n = the number of years.
Example 1.2
A company invests 2M GBP to clear a tract of land and plants some palm
trees. The trees will mature in 5 years, at which time the farm will have a
market value of 5M GBP. What is the expected annual rate of return for
the company’s investment?
Solution
We are given n = 5, FVn = 5M and PV = 2M . We aim at finding r, we
know from Equation (1) that
FVn = PV (1 + r)n
Hence FVnPV = (1 + r)n i.e., r =
n
√
FVn
PV − 1. So r = 5
√
5
2 − 1 = 0.2011
Care is needed with the units. The rate of interest should be represented as a
decimal in the compounding formula. The investment offers an opportunity
to gain 20.11% per annum.
Exercise 1.1
How long will it take to double a capital investment of C GBP attracting
interest at 6% compounded yearly?
1.2. Present Value.
1. TIME VALUE OF MONEY 5
Definition 1.3: Present Value
Present value, often called the discounted value, is a financial formula that
calculates how much a given amount of money received on a future date
is worth in today’s pounds. In other words, it computes the amount of
money that must be invested today to equal the payment or amount of cash
received on a future date.
The process of finding present values is called discounting and the interest
rate used to calculate present values is called the discount rate.
Example 1.3
Consider a riskless investment opportunity that will pay 133.10 GBP at the
end of 3 years. Suppose your local bank is currently offering 10 percent
interest on 3-year Certificates of Deposit (CD), and you regard the security
(i.e. the riskless investment) as being exactly as safe as a CD. How much
should you be willing to pay for investment in the security?
Solution
Representing cash flows. Let’s set up a time-line for the cash flows.
From the future value formula in (1), an initial amount of PV GBP invested
at 10% per year in the local bank would be worth 133.10 GBP at the end
of 3 years. So, we get
PV = 133.10(1.10)3 = 100.
Hence, 100 GBP is defined as the present value of the 133.10 GBP due in 3
years when the interest rate is 10%.
Using this defined value, we can make several conclusions:
1) If the cost of the investment in the security was less than 100 GBP, you
should buy it, because its price would then be less than the 100 GBP you
would have to spend on a similar-risk alternative to end up with 133.10 GBP
after 3 years.
2) If the investment costs more than 100 GBP, you should not buy it, be-
cause you would have to invest only 100 GBP in a similar-risk alternative
to end up with 133.10 GBP after 3 years.
3) If the cost was exactly 100 GBP, then you should be indifferent. There-
fore, 100 GBP is defined as the fair value of the investment.
Mathematical formulas for calculating present values.
In general, the present value of a future cash flow given the discount rate and
number of years in the future that the cash flow occurs can be computed using the
following equation.
PV = CFn(1 + r)n , (2)
6 1. ECONOMICAL AND FINANCIAL BACKGROUND
where
• CFn = the future cash flow occurring at the end of year n,
• PV = the present value,
• r = the interest or discount rate (per annum), and
• n = the number of years.
Example 1.4
Theresa will retire in 15 years. This year she wants to fund an amount of
18,000 GBP to become available in 15 years. How much does she have to
deposit into a pension plan earning 8% annually?
Solution
We are given n = 15, CFn = 18000 GBP and r = 8%. We aim at finding
PV , we know from Equation (2) that
PV = CFn(1 + r)n =
18000
(1 + 0.08)15 = 5674.35
2. Interest rates
An interest rate is a promised rate of return denominated in some unit of
account (GBP, USD, EURO, etc) over some time period (a month, a trimester, a
semester, a year, 10 years, or longer). The time period and the unit of the account
are indicated.
Definition 2.1: Nominal rates
The nominal rate rnom per year is the rate that is quoted by banks, brokers,
and other financial institutions. However, to be meaningful, the quoted
nominal rate must also include the number of compounding periods per
year.
The nominal interest rate is stated on a loan or investment without any ad-
justments for inflation. Suppose you deposit 100 GBP in a bank that pays 10%
interest each year. In one year’s time, you are guaranteed to collect 110 GBP in
cash, but at this time costs will be higher than one year ago. The concept of real
interest rate is useful to account for the impact of inflation.
2.1. Real rates. The real return on your investment, studied by Fisher, will
depend on what your money can buy in one year relative to what it could buy today
(i.e. it is an interest rate that has been adjusted to remove the effects of inflation).
The concept of real interest rate is useful to account for the impact of inflation. In
the case of a loan, it is this real interest that the lender effectively receives. For
example, if the lender is receiving 8 percent from a loan and the inflation rate is
also 8 percent, then the real rate of interest is zero.
Using the following notation
• i the inflation rate in a year
• r the interest rate in a year
• R the real interest rate in a year
then the relationship between R and r is described by compounding income using
the rate r (numerator) and then discounting outgoings using the rate of inflation
2. INTEREST RATES 7
(denominator):
1 +R = 1 + r1 + i ⇔ R =
1 + r
1 + i − 1 =
r − i
1 + i . (3)
Definition 2.2: Real rate
The real rate of return in a year is the rate of interest received above inflation
discounted to remove the effects of inflation over that year. It is given as
follows
R = r − i1 + i .
Remark 2.1
When the inflation rate i is low, the real interest rate is approximately given
by the nominal interest rate minus the inflation rate, i.e., R ≈ r − i.
Example 2.1
Let i be 4% in a year. With 10% nominal interest rate paid yearly, after
netting out the 4% depreciation in purchasing power of money, you are
left with a net growth in purchasing power of approximately 6% in the
future. Here 6%/(1 + 0.04) = 5.77% is called the real rate of interest for an
investment made today.
2.2. Periodic rate.
Definition 2.3: Periodic rate
The periodic rate is the rate charged by a lender or paid by a borrower each
period. It can be a rate per day, per week, per three-month, per six-month
period, per year, or per any other time interval. The periodic rate is given
as follows
rPER =
rnom
m
, (4)
where m denote the number of compounding periods a year, and rnom is
the nominal rate per annum.
The future value of an initial investment at a given interest rate compounded m
times per year at any point in the future can be found by applying the following
equation.
FVn = PV (1 +
rnom
m
)nm, (5)
where n is the number of years.
Example 2.2
Suppose that you invest 100 GBP in an account that pays a nominal rate of
10% per annum, compounded monthly. How much would you be paid after
3 years?
8 1. ECONOMICAL AND FINANCIAL BACKGROUND
Solution
We are given n = 3, m = 12, PV = 100 and r = 10%. We know from (5)
that
PV (1 + rnom
m
)nm = 100(1 + 0.112 )
3×12 = 134.82 GBP.
Exercise 2.1
Calculate the point in time at which some initial capital c has doubled, if
interest is compounded (a) monthly or (b) weekly, using an interest rate of
r per annum (p.a.). In particular give a numerical answer to the above for
r = 0.05 (i.e. r = 5%).
Definition 2.4: Effective annual rate
The Effective Annual Rate rEAR is the annual rate that produces the same
result as if we had compounded at a given periodic rate m times per year.
To calculate the Effective Annual Rate we use
rEAR = (1 +
rnom
m
)m − 1. (6)
The present value of a future cash flow given the discount rate (compounded m
times per year) and the n number of years in the future that the cash flow occurs
can be computed as follows
PV = CFn(1 + rnomm )nm
, (7)
the number (1 + rnomm )−nm is called the discount factor.
Remark 2.2
For the sake of simplicity, the nominal rate per annum rnom will be denoted
r in the sequel.
2.3. Continuous compounding.
The formula of the future value FVn at time n of a present value PV attracting
interest at a rate r > 0 compounded m times a year is given by
FVn = PV
(
1 + r
m
)nm
.
Assume that the compounded period becomes shorter and shorter i.e, m grows
larger and larger. Then, in the limit as m goes to infinity, we obtain:
lim
m→∞PV
(
1 + r
m
)nm
= PV lim
m→∞
(
1 + r
m
)nm
= PV lim
m→∞
[(
1 + r
m
)m]n
= PV [er]n = PV enr,
where we have used the fact that as m → ∞, (1 + rm)m ↗ er. This is known as
the continuous compounding with corresponding growth factor enr.
2. INTEREST RATES 9
Definition 2.5: Future values with continuous rates of interest.
The future value M(t) at time t > 0 of an initial (or present value) M(0)
with a nominal rate r with continuous compounding is given by
M(t) =M(0)ert. (8)
r is also called the continuous interest rate; t is the overall length of time
the interest is applied, it is expressed using the same time units as r (i.e.
years).
Example 2.3
An investor receives 1100 GBP in one year in return for an investment of
1000 GBP now. Calculate the nominal rate r per annum with
(a) Annual compounding
(b) Semi-annual compounding
(c) Monthly compounding
(d) Continuous compounding.
Solution
(a): With the annual compounding, using Equation (1), we have that:
1000(1 + r) = 1100.
Then, the interest rate is
r = 11001000 − 1 = 0.1
or 10% per annum.
(b): With the Semi-annual compounding, using Equation (5) , we have that:
1000
(
1 + r2
)2
= 1100
Then, the interest rate is
r = 2
(√
1100
1000 − 1
)
= 0.0976
or 9.76% per annum.
(c): With monthly compounding, using Equation (5), we get
1000
(
1 + r12
)12
= 1100.
Hence, the interest rate is
r = 12
(
12
√
1100
1000 − 1
)
= 0.0957
or 9.57% per annum.
(d): With continuous compounding, using Equation (8), we get 1000er =
1100. Hence, the interest rate is
r = ln 11001000 = 0.0953
or 9.53% per annum.
10 1. ECONOMICAL AND FINANCIAL BACKGROUND
Discounted value. "A dollar today is worth more than a dollar tomorrow."
Let r be the continuous interest rate. If we compare prices across time of the same
good, then we have to keep in mind that money is increasing with interest rate over
time. Thus, with one pound today we are able to buy more in the future.
To illustrate this, let us say we like to have one item of a good. We can either
buy the good at time t or at a later time T > t while the continuous interest rate
is r. The price right now is St while the later price is ST . We could either buy the
good at time t for St or, we can store this amount of money at the bank account
until time T and buy the good then. If we store the money, then it will grow to
Ste
r(T−t) while the later price of the good is ST . In particular, if we only need
to have the good at the later time T then we should compare Ster(T−t) to ST or,
equivalently, compare Ste−rt to ST e−rT .
The value
S˜t := Ste−rt (9)
is know as the discounted value of St. The number e−rt is called the discount
factor.
2.4. Some aspects of interest rates.
Here are some aspects of interest rates which we have overlooked:
• Risky vs. Riskless: Sometimes a counterparty which we lend money be-
comes insolvent and unable to repay us.
• Fixed vs. variable: In reality, interest rate may change over time. One
might need to use a function (r(t))t≥0 instead of a fixed value r ∈ R.
• Stochastic vs. deterministic: Future interest rates may additionally to
being different than today also be unknown. This might be modelled
better by a stochastic interest rate rather than deterministic interest rate.
• Term structure of interest rate: The mathematics of this is out of the
scope of this course. Roughly spoken, it is about standardized contracts
of lending money in the future.
• Stock and money markets (bond markets): There are two principal kinds
of markets, stock and money markets. In money markets, money is being
borrowed over time, to be repaid later at specified rates of interest and/or
specified payment times. In this lecture we will focus on stock markets
where stocks and shares (part-ownership in a company) are traded (bought
and sold).
3. Economics and Finance
Economics is largely concerned with questions such as supply and demand (for
everything: commodities, manufactures, capital, labour, etc.). Much of economics
deals with how prices are determined.
Finance is concerned with the borrowing and lending of money needed to engage
in economic activity. We will be dealing with the mathematical side of this. In
mathematical finance, one takes prices for shares/goods largely as given and mainly
deals with the following questions:
(1) How do we price derivatives (options)?
(2) How do we asses risk of a given financial transaction?
(3) How can we reduce the financial risk which we are exposed to if we make
a financial transaction?
4. MARKETS AND OPTIONS 11
Finance is a (small) part of economics.
Small economic agents are price takers. They have no power to influence prices
with their trades. Price makers are large economic agents and their actions on the
market have visible effects on the prices.
3.1. Trading prices. The price of common everyday items is accurately known
at any given time. Anyone trying to sell at a higher price than the ’going rate’ would
tend to lose market share to cheaper competitors, and eventually have to reduce
their prices towards the going rate. In contrast, items never bought or sold do not
have a price at all – they are priceless (e.g. Buckingham palace).
In between the two extremes, prices are known but not accurately. Typically,
one has market makers who offer goods at higher prices (ask price) than the prices
at which they are willing to buy (bid price) and market participants who accept
the prices of the market makers. The difference in the offered buy and sell prices
is known as bid-ask spread.
If large trades are made, then prices move. This can be made by modelling
outstanding orders and how they are executed, the totality of outstanding orders
is called Limit order book.
Small trades, one could say have no effect. In reality they do have an affect
which is simply very little. If a lot of small trades occur, then the effect becomes
visible. One may argue that each single effect is invisible and therefore that the price
moves continuously. This is typically modelled by continuous stochastic processes
(prototype: Brownian motion). In reality, the prices will move by a lot of tiny
jumps which is referred as jitter.
4. Markets and Options
This course is about the mathematics needed to model financial markets. These
are of several types:
• Stock markets.
• Bond markets, dealing in government bonds.
• Currency and foreign exchange markets.
• Futures and option markets, dealing in financial instruments derived from
the above.
Derivatives are very important in the world of finance. Futures and options are
traded actively on many exchanges throughout the world. Forward contracts, and
many different types of options are regularly traded outside exchanges by financial
institutions, fund managers, and corporate treasurers in what is termed the over-
the-counter market. Derivatives are also sometimes added to a bond or stock issue.
4.1. Derivatives.
Definition 4.1: Derivative
A derivative can be defined as a financial instrument whose value depends on
(or derives from) the values of other, more basic underlying variables. Very
often the variables underlying derivatives are the prices of traded assets.
A stock option, for example, is a derivative whose value is dependent on
the price of a stock. However, derivatives can be dependent on almost any
variable, from the price of hogs to the amount of snow falling at a certain
ski resort.
There is active trading in credit derivatives, electricity derivatives, weather
derivatives, and insurance derivatives. Many new types of interest rate, foreign
12 1. ECONOMICAL AND FINANCIAL BACKGROUND
exchange, and equity derivative products have been created. There have been many
new ideas in risk management and risk measurement. Capital investment appraisal
now often involves the evaluation of what are known as real options (these are the
options acquired by a company when it invests in real assets such as real estate,
plant, and equipment.)
4.2. Exchange-traded markets. A derivatives exchange is a market where
individuals trade standardized contracts that have been defined by the exchange.
Derivatives exchanges have existed for a long time. The Chicago Board of Trade
(CBOT) was established in 1848 to bring farmers and merchants together. Initially
its main task was to standardize the quantities and qualities of the grains that were
traded. Within a few years the first futures-type contract was developed. It was
known as a to-arrive contract. Speculators soon became interested in the contract
and found trading the contract to be an attractive alternative to trading the grain
itself. A rival futures exchange, the Chicago Mercantile Exchange (CME), was
established in 1919. Now futures exchanges exist all over the world.
The Chicago Board Options Exchange (CBOE www.cboe.com) started trading
call option contracts on 16 stocks in 1973. Options had traded prior to 1973 but the
CBOE succeeded in creating an orderly market with well-defined contracts. Put
option contracts started trading on the exchange in 1977. Like futures, options have
proved to be very popular contracts. Many other exchanges throughout the world
now trade options. The underlying assets include foreign currencies and futures
contracts as well as stocks and stock indices.
Traditionally derivatives traders have met on the floor of an exchange and
used shouting and a complicated set of hand signals to indicate the trades they
would like to carry out. This is known as the open outcry system. Exchanges have
largely replaced the open outcry system by electronic trading. This involves traders
entering their desired trades at a keyboard and a computer being used to match
buyers and sellers. The open outcry system has its advocates, but, as time passes,
it is becoming less and less used.
Electronic trading has led to a growth in high-frequency and algorithmic trad-
ing. This involves the use of computer programs to initiate trades, often without
human intervention, and has become an important feature of derivatives markets.
4.3. Over-the-counter markets. Not all derivatives trading is on exchanges.
Many trades take place in the over-the-counter (OTC) market. Banks, other large
financial institutions, fund managers, and corporations are the main participants
in OTC derivatives markets. Once an OTC trade has been agreed, the two parties
can either present it to a central counterparty (CCP) or clear the trade bilaterally.
A CCP is like an exchange clearing house. It stands between the two parties to the
derivatives transaction so that one party does not have to bear the risk that the
other party will default. When trades are cleared bilaterally, the two parties have
usually signed an agreement covering all their transactions with each other. The
issues covered in the agreement include the circumstances under which outstanding
transactions can be terminated, how settlement amounts are calculated in the event
of a termination, and how the collateral (if any) that must be posted by each side
is calculated.
Traditionally, participants in the OTC derivatives markets have contacted each
other directly by phone and email, or have found counterparties for their trades us-
ing an interdealer broker. Banks often act as market makers for the more commonly
traded instruments. This means that they are always prepared to quote a bid price
(at which they are prepared to take one side of a derivatives transaction) and an
offer price (at which they are prepared to take the other side). Prior to the credit
4. MARKETS AND OPTIONS 13
crisis, which started in 2007, OTC derivatives markets were largely unregulated.
Following the credit crisis and the failure of Lehman Brothers, we have seen the
development of many new regulations affecting the operation of OTC markets. The
purpose of the regulations is to improve the transparency of OTC markets, improve
market efficiency, and reduce systemic risk. The over-the-counter market in some
respects is being forced to become more like the exchange-traded market.
4.4. Forward and future contracts.
4.4.1. Forward contracts.
Definition 4.2: Forward contracts
A forward contract is a relatively simple derivative. It is an agreement to
buy or sell an asset at a certain future time for a certain price. It can be
contrasted with a spot contract, which is an agreement to buy or sell
an asset almost immediately. A forward contract is traded in the over-
the-counter market usually between two financial institutions or between a
financial institution and one of its clients.
Definition 4.3: Long and short positions
The party who agrees to buy the asset is said to be taking a long position
in the forward contract.
The party who agrees to sell the asset is in a short position in the forward
contract.
Forward contracts on foreign exchange are very popular. Most large banks
employ both spot and forward foreign-exchange traders.
Example 4.1
Table 1.1 below provides the quotes on the exchange rate between the British
pound (GBP) and the U.S. dollar (USD) that might be made by a large
international bank on August 16, 2001. The quote is for the number of USD
per GBP. The first quote indicates that the bank is prepared to buy GBP
(i.e., sterling) in the spot market (i.e., for virtually immediate delivery)
at the rate of $1.4452 per GBP and sell sterling in the spot market at
$1.4456 per GBP. The second quote indicates that the bank is prepared
to buy sterling in one month at $1.4435 per GBP and sell sterling in one
month at $1.4440 per GBP; the third quote indicates that it is prepared to
buy sterling in three months at $1.4402 per GBP and sell sterling in three
months at $1.4407 per GBP; and so on. These quotes are for very large
transactions. (As anyone who has traveled abroad knows, retail customers
face much larger spreads between bid and offer quotes than those in given
Table 1.1.)
14 1. ECONOMICAL AND FINANCIAL BACKGROUND
Example 4.2
We consider again Table 1.1 above. Forward contracts can be used to hedge
foreign currency risk. Suppose that on August 16, 2001, the treasurer of
a U.S. corporation knows that the corporation will pay £1 million in six
months (on February 16, 2002) and wants to hedge against exchange rate
moves. Using the quotes in Table 1.1, the treasurer can agree to buy £1 mil-
lion in six months forward at an exchange rate of $1.4359. The corporation
then has a long forward contract on GBP. It has agreed that on February
16, 2002, it will buy £1 million from the bank for $1.4359 million. The bank
has a short forward contract on GBP. It has agreed that on February 16,
2002, it will sell £1 million for $1.4359 million.
Consider the position of the corporation in the trade we have just described.
What are the possible outcomes? The forward contract obligates the corporation to
buy £1 million for $1,435,900. If the spot exchange rate rose to, say, 1.5000, at the
end of the six months the forward contract would be worth $64,100 (= $1,500,000
- $1,435,900) to the corporation. It would enable £1 million to be purchased at
1.4359 rather than 1.5000. Similarly, if the spot exchange rate fell to 1.4000 at
the end of the six months, the forward contract would have a negative value to
the corporation of $35,900 because it would lead to the corporation paying $35,900
more than the market price for the sterling.
Definition 4.4: Payoffs from Forward contracts
In general, the payoff from a long position in a forward contract on one unit
of an asset is
ST −K,
whereK is the delivery price and ST is the spot price of the asset at maturity
T of the contract (this is because the holder of the contract is obligated to
buy an asset worth ST for K).
Similarly, the payoff from a short position in a forward contract on one unit
of an asset is
K − ST .
4. MARKETS AND OPTIONS 15
Figure 1.1. illustrates the payoffs from the previous definition. These payoffs can
be positive or negative.
Remark 4.1
Because it costs nothing to enter into a forward contract, the payoff from
the contract is also the trader’s total gain or loss from the contract.
4.4.2. Futures contracts.
Definition 4.5: Future contracts
Like a forward contract, a futures contract is an agreement between two
parties to buy or sell an asset at a certain time in the future for a certain
price. Unlike forward contracts, futures contracts are normally traded on
an exchange. To make trading possible, the exchange specifies certain stan-
dardized features of the contract. As the two parties to the contract do not
necessarily know each other, the exchange also provides a mechanism that
gives the two parties a guarantee that the contract will be honored.
The largest exchanges on which futures contracts are traded are the Chicago Board
of Trade (CBOT) and the Chicago Mercantile Exchange (CME), which have now
merged to form the CME Group. On these and other exchanges throughout the
world, a very wide range of commodities and financial assets form the underlying
assets in the various contracts. The commodities include pork bellies, live cattle,
sugar, wool, lumber, copper, aluminum, gold, and tin. The financial assets include
stock indices, currencies, and Treasury bonds. Futures prices are regularly reported
in the financial press. Suppose that, on September 1, the December futures price of
gold is quoted as 1,380 GBP. This is the price, exclusive of commissions, at which
traders can agree to buy or sell gold for December delivery. It is determined in the
same way as other prices (i.e., by the laws of supply and demand).
4.5. Short-selling.
16 1. ECONOMICAL AND FINANCIAL BACKGROUND
Definition 4.6: Short-selling
This trade, usually simply referred to as ”shorting”, involves selling an asset
that is not owned. It is something that is possible for some, but not all,
investment assets. It involves borrowing assets (often securities such as
shares or bonds), usually through a broker, and immediately selling them.
The investor (or short seller) will later purchase the same number of the
same type of securities in order to return them to the lender (i.e. the original
owner of the assets). If the price has fallen in the meantime, the investor
will have made a profit equal to the difference. Conversely, if the price has
risen then the investor will bear a loss. The short seller must usually pay a
fee (handling fee) to borrow the securities (charged at a particular rate over
time, similar to an interest payment), and reimburse the lender for any cash
returns such as dividends they were due during the period of lease.
Remark 4.2
In later chapters we will make several idealistic assumptions on market mod-
els, thus when there is a short-selling we suppose that there is no fee to
borrow the securities and no dividends to be paid to the lender.
You can see above a schematic representation of physical short-selling in two
steps (we suppose that there are no dividends). The short seller borrows shares
and immediately sells them. The short seller then expects the price to decrease,
after which the short seller can profit by purchasing the shares to return to the
lender.
We will illustrate how short-selling works by considering a short sale of shares
of a stock as you can see in the following example.
4. MARKETS AND OPTIONS 17
Example 4.3
Suppose an investor instructs a broker to short 500 IBM shares. The broker
will carry out the instructions by borrowing the shares from someone (client)
who owns them and selling them in the market in the usual way. Then at
some later stage the investor will close out the position by purchasing 500
IBM shares in the market. These shares are then replaced in the account of
the client from whom the shares were borrowed. The investor takes a profit
if the stock price has declined and a loss if it has risen.
An investor with a short position must pay to the broker any income, such
as dividends or interest, that would normally be received on the securities
that have been shorted. The broker will transfer this to the account of the
client from whom the securities have been borrowed. Consider the position
of an investor who shorts 500 IBM shares in April when the price per share
is £120 and closes out the position by buying them back in July when the
price per share is £100. Suppose that a dividend of £1 per share is paid in
May. The investor receives 500×£120 = £60, 000 in April when the short
position is initiated. The dividend leads to a payment by the investor of
500 × £1 = £500 in May. The investor also pays 500 × £100 = £50, 000
when the position is closed out in July. The net gain is therefore
£60, 000−£500−£50, 000 = £9, 500
assuming there is no fee for borrowing the shares.
Remark 4.3
Consider a riskless asset (bank account) with interest rate r > 0. If our
bank deposit is positive, then we lend money to the bank while negative
values (overdraft) mean that we borrow money from the bank.
In many markets, risky assets such as stocks may be treated in the same
way. That means a positive number of shares indicates that we have this
number of shares while a negative number of shares means that we did a
short-selling of these shares (when we do a short-selling of shares we borrow
them as we have seen previously).
Therefore the mathematics around short-selling (which we will allow in all
our models) is rather simple. We simply allow for negative numbers in the
amount of shares held.
Remark 4.4
For many investors interest rates for lending or borrowing are not the same
but in this course we will always model them as equal!
4.6. Options.
18 1. ECONOMICAL AND FINANCIAL BACKGROUND
Definition 4.7: Option
An option is a financial instrument giving the right but not the obligation
to make a specified transaction at a specified price at a specified date. Exer-
cising the option means to take the right. Whether or not the option will be
exercised depends on the uncertain future, namely whether using this right
proves to be advantageous.
Definition 4.8: Long and short positions
The holder (buyer) of an option is said to have a long position in the option,
while the writer (seller) of an option takes a short position.
Definition 4.9: Various standard types of options
1) Call options give on the right (but not the obligation) to buy a specified
amount of a financial good (the underlying) at a specified price (exercise
price or strike price or strike).
2) Put options give on the right (but not the obligation) to sell a specified
amount of a financial good (the underlying) at a specified price (strike).
3) European options give the right to the transaction on the specified date,
the expiry date (also called maturity date).
4) American options give the right to the transaction at any time prior or
at the expiry date.
Options are traded both on exchanges and in the over-the-counter market.
European options are generally easier to analyze than American options, and some
of the properties of an American option are frequently deduced from those of its
European counterpart.
It should be emphasized that an option gives the holder the right to do some-
thing. The holder does not have to exercise this right. This is what distinguishes
options from forwards and futures, where the holder is obligated to buy or sell the
underlying asset.
Remark 4.5
Note that whereas it costs nothing to enter into a forward or futures contract,
there is a cost to acquiring an option.
Historically, the names European/ American (Asian, Bermudian, Russian, ...)
referred also to the area where these options were mainly traded. Today, American
options are predominant throughout the globe.
4.6.1. Payoffs from European put and call options.
• The case of a call option:
A call option on a stock (with price St at time t) is a derivative taking the form
of a contract between two parties, the holder (buyer) and the writer (seller). This
is specified by a maturity time T and a strike K. The holder pays a fee (which is
the option premium also called the option price) to the writer at inception of the
contract and then has the right but not the obligation to purchase the stock at
time T at price K. The writer, having accepted the premium, has the obligation to
deliver one unit of the stock in exchange for a paymentK, if the holder wish to make
this exchange. The option will be exercised if ST > K, because then the holder is
4. MARKETS AND OPTIONS 19
buying the stock for less than its worth; in principle (s)he could immediately sell it
in the market, making a profit of ST −K. (Often, options are ’cash settled’:
on exercise the writer pays the holder ST −K in cash). On the other hand,
if ST < K, the holder will not exercise and the option expires worthless.
Definition 4.10: Payoff from the call option
The payoff from the call option (also called the value of the call on its
maturity date) is
C(T ) = max{ST −K, 0} = (ST −K)+,
where (X)+ denotes the greater of X and 0.
• The case of a put option:
A put option on a stock (with price St at time t) is a derivative taking the form
of a contract between two parties, the holder (buyer) and the writer (seller). This
is specified by a maturity time T and a strike K. The holder pays a fee (which is
the option premium also called the option price) to the writer at inception of the
contract and then has the right but not the obligation to sell the stock at time T
at price K. The writer, having accepted the premium, has the obligation to buy
one unit of the stock in exchange for a payment K, if the holder wish to make this
exchange. The option will be exercised if K > ST , because then the holder is selling
the stock for more than its worth; in principle (s)he could immediately buy it in
the market, making a profit of K − ST . (Often, options are ’cash settled’: on
exercise the writer pays the holder K − ST in cash). On the other hand, if
K < ST , the holder will not exercise and the option expires worthless.
Definition 4.11: Payoff from the put option
The payoff from the put option (also called the value of the put on its
maturity date) is
P (T ) = max{K − ST , 0} = (K − ST )+,
where (X)+ denotes the greater of X and 0.
Remark 4.6
Options are referred to as in the money, at the money, or out the money.
A call option is in the money when ST > K, at the money when ST = K,
and out of the money when ST < K. A put option is in the money when
ST < K, at the money when ST = K, and out of the money when ST > K.
20 1. ECONOMICAL AND FINANCIAL BACKGROUND
Example 4.4: Call Options.
Consider the situation of an investor who buys a European call option with a
strike price of $60 to purchase one Microsoft share. Suppose that the current
stock price is $58, the expiration date of the option is in four months, and
the price of an option to purchase one share is $5. The initial investment is
$5. Because the option is European, the investor can exercise only on the
expiration date. If the stock price on this date is less than $60, the investor
will clearly choose not to exercise (there is no point in buying, for $60, a
share that has a market value of less than $60.) In these circumstances, the
investor loses the whole of the initial investment of $5. If the stock price is
above $60 on the expiration date, the option will be exercised. Suppose, for
example, that the stock price is $75. By exercising the option, the investor is
able to buy one share for $60. If the share is sold immediately, the investor
makes a gain of $15 ignoring transactions costs. When the initial cost of the
option is taken into account, the profit to the investor is $10.
Remark 4.7
When charts showing the profit or loss from options trading are produced,
the usual practice is to ignore the time value of money, so that the profit is
the final payoff minus the initial cost.
Figure 1.2 shows how the investor’s profit (also called net profit) or loss on an
option to purchase one share varies with the final stock price in the example. It
is important to realize that an investor sometimes exercises an option and makes
a loss overall. Suppose that in the example Microsoft’s stock price is $62 at the
expiration of the option. The investor would exercise the option for a gain of ($62
- $60) = $2 and realize a loss overall of $3 when the initial cost of the option is
taken into account. It is tempting to argue that the investor should not exercise
the option in these circumstances. However, not exercising would lead to an overall
loss of $5, which is worse than the $3 loss when the investor exercises. In general,
call options should always be exercised at the expiration date if the stock price is
above the strike price.
4. MARKETS AND OPTIONS 21
Example 4.5: Put options.
Whereas the purchaser of a call option is hoping that the stock price will
increase, the purchaser of a put option is hoping that it will decrease. Con-
sider an investor who buys a European put option to sell one share in IBM
with a strike price of $90. Suppose that the current stock price is $85, the
expiration date of the option is in three months, and the price of an option
to sell one share is $7. The initial investment is $7. Because the option is
European, it will be exercised only if the stock price is below $90 at the
expiration date. Suppose that the stock price is $75 on this date. The in-
vestor can buy one share for $75 and, under the terms of the put option, sell
the same share for $90 to realize a gain of $15 (again transactions costs are
ignored). When the $7 initial cost of the option is taken into account, the
investor’s profit is $8. There is no guarantee that the investor will make a
gain. If the final stock price is above $90, the put option expires worthless,
and the investor loses $7.
Figure 1.3 shows the way in which the investor’s profit (also called net profit) or loss
on an option to sell one share varies with the terminal stock price in this example.
Remark 4.8
In chapter 4, we will consider the results appearing in the following sections
with more details, our purpose here is to give a short intuitive introduction
about them.
4.7. Hedging. We consider an investor with some capital to invest. The
simplest model is that in which (s)he has two choices: to invest in a risk-free asset
(that is a unit of money in a bank account) or in a risky asset.
(1) Bank account: A capital that is not invested in the risky asset is placed
in the bank account where we assume a constant deterministic continuous
interest rate r > 0. Let B denote the initial investment, then the future
value of this investment at a later time T > 0 is BerT .
22 1. ECONOMICAL AND FINANCIAL BACKGROUND
(2) Risky asset: The initial price of the risky asset (e.g. a share) S0 is known
at time 0 and the now unknown price at time T is ST (typically modelled
by a random variable).
Example 4.6
For any x, y ∈ R a trader may form a portfolio consisting of x risk-free assets
and y shares. Such a portfolio is denoted by the pair (x, y). Let us suppose
in our model that the trader forms a portfolio (B,X) of assets at time 0 and
hold this portfolio, unchanged, until time T . Then, B +XS0 is the initial
value of the portfolio and BerT +XST is the value of the portfolio at time
T .
Definition 4.12: Trading and hedging strategies
A trading strategy is a rule chosen by the investor to update his portfolio
over time as new prices arrive.
A hedging strategy is a trading strategy with the aim of removing risk.
Forwards and options provide a natural hedging tool.
Example 4.7: Hedging using options
Imagine you have 10, 000 shares of a stock and the current price of the share
is 100 GBP. In order to ensure that you won’t loose substantial amount of
money in case the price of the stock drops at the end of the year, you buy
European put options (which mature at the end of the year) on these 10, 000
shares with a strike of 80 GBP per share (i.e. each European put option on
a share has a strike K = 80 GBP). The price of each European put option is
0.50 GBP (so this costs 5, 000 GBP). If the price of the share stays above 80
GBP, then you simply keep the shares (or sell them directly on the market).
If the price does fall below 80 GBP, then you use the option right to sell all
of them for 800, 000 GBP.
4.8. Arbitrage. Economic agents enter market for various reasons. The two
main reasons are to ensure oneself against risk or to speculate.
Everyone would, of course, like to have larger net profit for no risk. If that
would be possible, then people would exploit these opportunities and take money
from the market. This would continue until no more money is left in the market.
Definition 4.13: Arbitrage
Arbitage is the availability of riskless profit, i.e. it is an opportunity to make
money out of nothing. A trading strategy realizing more profit for no extra
risk is referred as arbitrage.
We will go back to this notion in Chapter 4 with a mathematical definition,
our purpose here is just to give an introduction to it.
Real markets do allow for arbitrage from time to time but as very quick traders
start to exploit them prices are adjusted and the arbitrage disappears. Slower
traders never have the chance to exploit market arbitrages. So, it is reasonable to
suppose that there is no arbitrage in the market.
4. MARKETS AND OPTIONS 23
Amarket with arbitrage opportunities is a disordered market; modelling it leads
to disordered models. Modelling a market without arbitrage opportunities leads to a
deep mathematical theory which enhanced the understanding of markets in various
ways. In this module (in Chapter 4), we will investigate the mathematical theory
of no-arbitrage (NA) markets. Despite idealizing assumptions (e.g. frictionless, i.e.
no transaction costs) this leads to a sufficiently realistic tool to price derivatives,
analyze risky positions and determine methods to reduce risk.
Theorem 4.1: Put-Call parity
Consider a market where a risky asset is traded at price S0 now and price
ST at a later time T > 0. Consider a European Call option and a European
Put option (the underlying of these options is the risky asset) are traded
at prices C0, P0 respectively both having strike K > 0 and maturity T , let
r be the continuous interest rate. Under (NA) we must have the Put-Call
parity:
S0 + P0 − C0 = Ke−rT . (10)
Equation (10) shows that the price of a European call with a certain exercise
price and exercise date can be deduced from the price of a European put with the
same exercise price and exercise date, and vice versa.
In this introductory chapter you can admit this result. But let us give an idea
about the proof of (10).
Proof : Put-Call parity
We recall that the values at time T of the European call option and the
European put option are given by CT = (ST −K)+ and PT = (K − ST )+
respectively. Now, I consider the following trading strategy str1: at time 0, I
buy the risky asset, I buy one European put option and I sell one European
call option with the above strike and maturity. The value of this trading
strategie at time 0 is V0 = S0+P0−C0 (we assume that V0 > 0). The value
of this trading strategy at time T is equal to:
VT = ST + PT − CT =
{
ST + 0− (ST −K) = K if ST > K,
ST + (K − ST )− 0 = K if ST ≤ K.
(11)
Thus, VT = K GBP at time T in any case while the initial payment is
V0 = S0 + P0 − C0.
I Consider now another trading strategy str2 which is given as follows: at
time 0, I invest Ke−rT in the bank account. Then, at time T , the cash
invested has grown to K. So, the value of str2 at time T is K.
We have found two strategies with the same payout at time T , namely K. If
one strategy is more expensive than the other, then we can sell the expensive
strategy (i.e. take the opposite position instead) and buy the cheap strategy
(i.e. take this strategy) which leads to an arbitrage as it is proved below.
24 1. ECONOMICAL AND FINANCIAL BACKGROUND
Proof : Put-Call parity continued
(1) If (S0 + P0 − C0) < K e−rT : starting with a capital 0 at time 0, I
borrow K e−rT from the bank (i.e. I sell str2). On the other hand, I sell
a European call option so I get C0 and I have already K e−rT (observe
that by the hypothesis S0 + P0 < C0 +K e−rT ), then I buy the risky asset
and I buy a European put option for S0 + P0 (so, as a consequence I took
the trading strategy str1) and I still have the positive amount of money
C0 + K e−rT − (S0 + P0). At time T , I get K GBP from str1 as we have
seen in (11), that I have to return to the bank (in order to repay the loan),
but I still have C0 +K e−rT − (S0 + P0) GBP with me which is a positive
profit. It is an arbitrage because starting with a capital 0 at time 0, it
is sure to get a positive profit at time T . Thus, under the assumption
of (NA) we deduce that the inequality (S0+P0−C0) < K e−rT does not hold.
(2) If (S0 + P0 − C0) > K e−rT : starting with a capital 0 at time 0, I take
the opposite position in str1, i.e. I do a short-selling of the risky asset and
I sell a European put option (so, I get S0+P0 GBP) and I buy a European
call option, then I have I0 := S0 + P0 − C0 GBP at time 0, then I invest
this amount in the bank account. At time T , the cash invested in the bank
account has grown to I0 erT GBP, I have also one European call option (its
value is CT ), but I have the engagement to return the risky asset to the
lender (so, I have to buy the risky asset from the market, which costs ST )
and I have an engagement with the owner of the put option if (s)he decides
to exercise the option (which costs PT for me), the total cost is ST + PT .
Finally, at time T , I have
I0 e
rT + CT − (ST + PT ) > K + CT − (ST + PT ) = K − (ST + PT − CT )
= 0,
where we have the last equality thanks to (11). Observe that I have the
positive amount I0 erT +CT − (ST +PT ) with me which is a positive profit.
It is an arbitrage because starting with a capital 0 at time 0, it is sure to get
a positive profit at time T . Thus, under the assumption of (NA) we deduce
that the inequality (S0 + P0 − C0) > K e−rT does not hold.
Finally, we proved that under (NA) we must have:
S0 + P0 − C0 = Ke−rT .
4. MARKETS AND OPTIONS 25
Example 4.8: One step binomial model
Let us assume that there is a risky asset with price S0 = 100 (GBP) right
now and only two possible prices at a later time T > 0, say 1 year, given by
ST =
{
150 with probability p,
70 with probability 1− p.
We assume p = 1/2 and zero interest rate r = 0 for simplicity. Now suppose
we are looking for the price C0 (at time 0) of a European call option with
maturity T and strike K = 110. The value of this option at time T is:
CT = (ST − 110)+. Thus, this option is paying 40 if the price of the risky
asset goes up and 0 if it goes down. Its expected discounted value is given
by
E
[
CT
erT
]
= E[CT ] = p · 40 + (1− p) · 0 = 20.
Let us suppose that this European call option is traded at time 0 at price
C0 = 20. We consider the following trading strategy:
Starting with a capital equal to 0 at time 0, we sell two European call options
for 20 each, take a loan of 60 from the bank and buy one share for 100. At
time T , the value of this strategy is:
VT = −2CT − 60 + ST =
{
−2 · 40− 60 + 150 = 10 if ST = 150,
−2 · 0− 60 + 70 = 10 if ST = 70.
So, there is a positive profit equal to 10. We see that we make money no
matter the outcome while we don’t have to pay anything initially. Thus,
there is arbitrage which is not realistic.
Remark 4.9
We shall later see in Chapter 4 that the only price for the option which is
compatible with (NA) is C0 = 15.
4.9. Idealised market. A market is a price model for a number n of traded
assets for different points of time. These prices can be deterministic or stochastic.
In later chapters we will make several idealistic assumptions on market models:
(1) Perfect liquidity: This means that assets are available at the market to
be bought or sold at any quantity. This includes the selling or buying of
very high position sizes as well as fractional position sizes.
(2) Short-selling allowed at any quantity: This means that a trader may sell
an asset that (s)he doesn’t have at any quantity.
(3) Linearity of prices: If we buy several shares, then the total price equals
the sum of the prices for each individual item. This extends to fractional
and negative quantities in the sense that q ∈ R shares cost q-times as
much as 1 share.
(4) Frictionless market: Prices for buying and selling are usually not the same
and there might be transaction costs. The additional cost incurred are
referred as friction. A frictionless market is a market where buy and sell
prices (for the same good) coincide and no transaction costs exist.
4.10. Complements. There are various types of risk. These include:
(1) Market risk. This is the risk that one’s current market position goes down
in value.
26 1. ECONOMICAL AND FINANCIAL BACKGROUND
(2) Credit risk. This is the risk that counterparties default and fail to fulfil a
contract, e.g. they sold us a European call option but become unable to
provide us with a share of the option.
(3) Operational risk. This is the risk that the own companies procedure’s fail
to be executed as planned. E.g. a faulty algorithm for a computer at a
digital stock exchange or a trader taking on his own accord more risky
positions than he is allowed to.
(4) Liquidity risk. The risk that a traded good starts to become untrade-
able. For instance, there might be a sudden oil shortage and buying of
oil becomes impossible (or extremely expensive). Someone might buy up
all shares from the market. Or, major banks become suddenly unwilling
to allow loans to large companies or other banks resulting in a money
shortness (This is what happened during the credit crunch 2007/2008).
(5) Model risk. To handle any real-world problem with large complexity one
needs precise mathematical models. Model risk refers to the risk of choos-
ing a model inappropriate for the problem at hand.
(6) Model uncertainty. This is the risk that model parameters are chosen
wrongly. In fact this is a special type of model risk.