MSING0104: Introduction to Quantitative
Finance
Dr. Wei Cui
Department of Economics
Centre for Finance
UCL
The 2nd part: Lecture 1
Introduction to Financial Derivatives
Introduction 1-0
Financial Derivatives
A derivative is a financial instrument that is derived from other
financial instruments and whose value depends on the values of
other underlying variables.
Other terminology:
Derivative = Derivative security = contingent claim
Interesting case:
Weather derivatives
Introduction 1-1
Different Prices
Spot price
A spot price is the price at which a financial instrument is traded
against immediate payment and immediate delivery.
Forward contract
A forward contract is an agreement to buy or sell an asset at a
future time against a price agreed on today.
Introduction 1-2
Positions
A long position is an agreement to buy an asset,
a short position is an agreement to sell an asset.
You can think about a short position in an asset as taking a credit
in that asset (because the vendor actually does not own it); the
investor closes his position by re-purchasing the asset in the future.
In terms of his expectations, the investor is ‘bearish’, i.e. he is
speculating on tumbling markets.
Introduction 1-3
Example: Closing a position
An investor decides to short sell 1000 stocks.He borrows the stocks
from the owner and sells them for S0. At time t > 0 the price is St ,
and he decides to close the position by re-purchasing the 1000
stocks for St and to turn them back to the former owner.(If at time
t0; 0 < t0 < t a dividend D was paid the short seller has to pay the
former owner 1000 · D).
Obviously, short selling is only profitable if St < S0.
Introduction 1-4
Forward price
Value of forward contract at time t VK ,T (St , τ)
Delivery price (exercise price) K
Spot price (price at time t) St
Maturity (expiration date) T
Time to maturity τ def= T − t
The forward price Ft is the delivery price at which the forward
contract has value zero, i.e. VFt ,T (St , τ) = 0
payoff from a long position in a forward contract: ST − K
payoff from a short position in a forward contract: K − ST
There is no cash flow between t and T !!
Introduction 1-5
Example
An investor longs a forward contract on September 1 to buy 1
million EUR in 90 days at FX rate (foreign exchange rate) to the
USD 1.20.
The investor is obliged to buy EUR 1 000 000 for USD 1 200 000.
If the FX rate rises to, say, 1.30 at the end of the 90 days the
investor will gain since the EUR 1 000 000 may be sold for
USD 1 300 000. Let the current spot rate be USD 1.10.
K = USD 1 200 000 T = November 30
St =USD 1 100 000 t = September 1
τ = 90 days
Introduction 1-6
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6
ST
Pay-off
K
short position
long position













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Figure 1: Value of the forward contract on the delivery day
Introduction 1-7
Future contract
Unlike forward contracts, future contracts are standardized products
that are traded on an organized exchange market on a variety of
products, e.g. pork bellies, live cattle, sugar, wool, lumber, ...
Futures are settled in cash by daily margining, i.e. gains and
losses are credited and deducted every day.
Forward contract
Transaction is at the delivery date, underlying asset changes
owners, usually non-standardized OTC products.
Introduction 1-8
All Kinds of Options
Options
The long position has the right to decide whether the transaction
takes place.
Call option
A call option is a contract between two parties that gives the buyer
the right to buy an asset at a specified time at price K .
Put option
A put is a contract between two parties that gives the buyer the
right to sell an asset at a specified time at price K .
Introduction 1-9
European style option
A European option can only be exercised at a certain time T .
American style option
An American option can be exercised at any time t ≤ T .
Introduction 1-10
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6
ST
Pay-off
min{K − ST , 0}
K
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Figure 2: Value of a short position in a call option on the delivery day
If ST < K , the buyer of the option will not exercise: payoff = 0
If ST > K , the buyer will exercise, loss to option writer = K − ST
Introduction 1-11
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6
ST
Yield
K
C0 @
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Figure 3: Yield of a short position in a call option on the delivery day
C0 is the price paid to buy a call option, the seller of an option
receives C0
Introduction 1-12
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6
ST
Yield
K
Straddle
long Call
long Put
−(C0 + P0)
p p p p p p p p p p p p p p p p p p p p p p p
p p p p p
p p p p p
p p p p p
p p
p p p p p p p p p p p p p p p p p pppppp
ppppp
ppppp
p
−C0
−P0
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Figure 4: Yield of a long position in a straddle on the delivery day
P0 is the price paid to buy a put option.
Straddle: portfolio of a call and a put with the same delivery
price K .
Introduction 1-13
Plain Vanilla options
European/American options
Exotic options
Tailored financial instruments (e.g. options on options, non
standard payoff functions, non standard exercise conditions ... )
Asian options
Strike price K depends on an average of St for 0 ≤ t ≤ T
Cliquet options
A cliquet option settles periodically and resets the strike at the level
of the underlying during the time of settlement.
Introduction 1-14
Knock-out options
A call option with a knock-out barrier becomes worthless when the
price of underlying asset is above the barrier
Consider a stock that is trading at $100. A trader buys a knock-out
call option with a strike price of $105 and a knock-out barrier of
$110, expiring in three months, for a premium payment of $2.
Assume that the price of a three-month plain-vanilla call option
with a strike price of $105 is $3.
Why the knock-out call, rather than a plain-vanilla call? While the
trader is obviously bullish on the stock, he/she is quite confident
that it has limited upside beyond $105. She/he is therefore willing
to sacrifice some upside in the stock in return for slashing the cost
of the option by 33% (i.e. $2 rather than $3).
Introduction 1-15
Bonds
Zero bonds
The owner of a zero bond buys the bond at t0 paying B0 and will
receive an amount of BT at a future time T . BT is called the
nominal value of the bond. BT is not random.
Coupon bonds
The owner of a coupon bond receives an amount of BT at a future
time T and additional payments at discrete times t0, . . . , tn before
T . The additional payments are called coupons.
Introduction 1-16
Interest rates
Discrete compounding
One payment per year: B(1)n = B0(1+ r)n, n = # of years
k payments per year: B(k)n = B0(1+ rk )nk ,
r
k = interest rate
per 1k years
Continuous compounding
k →∞, i.e. infinitely many payments per year
Bn = B0 · enr
The larger k, the smaller |Bn − B(k)n |.
In the remainder we will work with continuous compounding of
interest rates.
Introduction 1-17
Cost of carry
Cost of carry measures the storage cost plus the interest that is
paid to finance the asset less the income earned on the asset.
For example:
Non-dividend-paying stock: cost of carry equals interest rate
since there are no storage costs and no income is earned.
Coupon bond: interest less coupon payments
Introduction 1-18
A Quick Look at Put-Call-Parity
Price of a European call and put : CK ,T (St , τ), PK ,T (St , τ)
Proposition (Put-Call-Parity)
1. If Dt is the value of all earnings and costs related to the
underlying during the time period τ = T − t calculated at
time t, we have
CK ,T (St , τ) = PK ,T (St , τ) + St − Ke−rτ − Dt (1.1)
2. If we have continuous costs of carry, b = r − d , the value of
the Call option at time t is
CK ,T (St , τ) = PK ,T (St , τ) + Ste(b−r)τ − Ke−rτ (1.2)
Introduction 1-19
Remark
We show a graphic approach to understand the parity; Next week,
we will have a formal proof.
The proof of the proposition does not hold for American options.
The reason is the opportunity to exercise the option before T .
Arbitrage Theory (Optional) 2-0
Arbitrage and State Prices (Optional)
Arbitrage
Arbitrage exists, if there is a trading strategy which produces a
riskless profit with non zero probability.
Formally
Sates 1, 2, ...,S
N securities with payoff matrix D (dimension N by S)
The security prices are given by some q ∈ RN
A portfolio θ ∈ RN has market value q · θ and payoff D′θ ∈ RS .
An arbitrage is a portfolio θ with q · θ ≤ 0 and D′θ > 0 or
q · θ < 0 and D′θ ≥ 0
Arbitrage Theory (Optional) 2-1
State Prices
A state-price vector is a vector ψ ∈ Rs+ such that q = Dψ
Theorem: There is no arbitrage if and only if there is a state-price
vector.
Arbitrage Theory (Optional) 2-2
Risk-neutral Pricing Physical probability: p1 + p2 + ...+ pS = 1.
Let a state price be ψ1, ψ2, ..., ψS and ψ0 = ψ1 + ...+ ψS
Risk-neutral probability ψˆj = ψj/ψ0. Therefore, for an arbitrary
security i ,
qi
ψ0
= Eˆ [Di ] =
S∑
j=1
ψˆjDij
ψ0 is the discount (or inverse of the price) of risk-less bonds.
qi = ψ0Eˆ [Di ] follows the risk-neutral pricing
Arbitrage Theory (Optional) 2-3
Finding a state-price vector
An agent is defined by a strictly increasing utility function and
endowment
The constraint:
X (q, e) = {e + D′θ ∈ RS+ : θ ∈ RN , q · θ ≤ 0}
The problem:
max
c∈X(q,e)
U(c)
Theorem: Suppose that c∗ is a strictly positive solution and that
∂U(c∗) is a strictly positive vector. Then, there is some scalar
λ > 0 such that λ∂U(c∗) is a state-price vector.
Arbitrage Theory (Optional) 2-4
Equilibrium, Pareto Optimality, and
Complete Markets
m agents. Endowment e1, ..., em. Utility functions U1, ...,Um
An equilibrium is a collection (θ1, ..., θm, q) such that given the
price vector q, for each agent i , θi solves maxθ Ui(ei +D′θ) subject
to q · θ ≤ 0, and such that ∑ θi = 0.
Complete markets: span(D) = RS
Arbitrage Theory (Optional) 2-5
Equilibrium and A Representative Agent
Proposition: If markets are complete, the associated equilibrium
allocation is Pareto optimal
Under some conditions, any Pareto optimal allocation can be
implemented in equilibrium with complete markets
Implication: one can construct a representative-agent and work
with his/her asset pricing formula
Arbitrage Theory (Optional) 2-6
State-Price Beta Models
Proposition: A dividend-price pair (D, q) admits no arbitrage if
and only if there is some pi ≥ 0 ∈ RS such that q = E (Dpi)
pi is called a state-price deflator , density, and kernel
Using the property cov(x , y) = E [xy ]− E [x ]E [y ], we have
E[Rθ]− R0 = −cov(R
θ, pi)
E [pi]
Arbitrage Theory (Optional) 2-7
Suppose the return of market portfolio is denoted as R∗, then
E [Rθ]− R0 = βθ{E [R∗]− R0}
where βθ = cov(R∗,Rθ)/var(R∗)
β pricing model: quantity of risks and price of risks
Arbitrage Theory (Optional) 2-8
Basic assumption for the remainder:
No Arbitrage
⇓
prices of derivatives must not permit arbitrage
Arbitrage Theory (Optional) 2-9
Perfect Financial Market
debit interest rate = credit interest rate
no transaction costs
no taxes
short-selling is allowed
stocks are unlimitedly divisible
no arbitrage












































































































































































































































































































































































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