Lectures
Sergei Fedotov
20912 - Introduction to Financial Mathematics
No tutorials in the first week
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2010
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Lecture 1
1
Introduction
Elementary economics background
What is financial mathematics?
The role of SDE’s and PDE’s
2
Time Value of Money
3
Continuous Model for Stock Price
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General Information
Textbooks:
• J. Hull, Options, Futures and Other Derivatives, 7th Edition,
Prentice-Hall, 2008.
• P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial
Derivatives: A Student Introduction, Cambridge University Press, 1995
Assessment:
Test in a week 6: 20%
2 hours examination: 80%
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Elementary Economics Background
This course is concerned with mathematical models for financial markets:
• Stock Markets, such as NYSE(New York Stock Exchange), London
Stock Exchange, etc.
• Bond Markets, where participants buy and sell debt securities.
• Futures and Option Markets, where the derivative products are traded.
Example: European call option gives the holder the right (not obligation)
to buy underlying asset at a prescribed time T for a specified price E .
Option market is massive! More money is invested in options than in the
underlying securities. The main purpose of this course is to determine the
price of options.
Why stochastic differential equations (SDE’s) and partial differential
equations (PDE’s)?
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Time value of money
What is the future value V (t) at time t = T of an amount P invested or
borrowed today at t = 0?
• Simple interest rate:
V (T ) = (1 + rT )P
(1)
where r > 0 is the simple interest rate, T is expressed in years.
• Compound interest rate:
r mT
V (T ) = 1 +
P
m
where m is the number interest payments made per annum.
(2)
• Continuous compounding:
In the limit m → ∞, we obtain
V (T ) = e rT P
(3)
1 z
since e = limz→∞ 1 + z . Throughout this course the interest rate r
will be continuously compounded.
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Simple Model for Stock Price S(t)
Let S(t) represent the stock price at time t. How to write an equation for
this function?
• Return (relative measure of change):
where ∆S = S(t + δt) − S(t)
In the limit δt → 0 :
∆S
S
(4)
dS
S
(5)
• How to model the return?
Let us decompose the return into two parts: deterministic and stochastic
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Modelling of Return
Return:
dS
= µdt + σdW
(6)
S
where µ is a measure of the expected rate of growth of the stock price. In
general, µ = µ(S, t). In simpe models µ is taken to be constant
(µ = 0.1 ÷ 0.3).
• σdW describes the stochastic change in the stock price, where dW
stands for
∆W = W (t + ∆t) − W (t)
as ∆t → 0
• W (t) is a Wiener process
• σ is the volatility (σ = 0.2 ÷ 0.5)
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Stochastic differential equation for stock price
dS = µSdt + σSdW
(7)
• Simple case: volatility σ = 0
dS = µSdt
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(8)
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Wiener process
Definition. The standard Wiener process W (t) is a Gaussian process such
that
• W (t) has independent increments: if u ≤ v ≤ s ≤ t, then W (t) − W (s)
and W (v ) − W (u) are independent
• W (s + t) − W (s) is N(0, t) and W (0) = 0
Clearly
• EW (t) = 0 and EW 2 = t, where E is the expectation operator.
• The increment ∆W = W (t + ∆t) − W (t) can be written as
1
∆W = X (∆t) 2 , where X is a random variable with normal distribution
with zero mean and unit variance:
X ∼ N (0, 1)
• E∆W = 0 and E(∆W )2 = ∆t.
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Lecture 2
1
Properties of Wiener Process
2
Approximation for Stock Price Equation
3
Itô’s Lemma
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Wiener process
The probability density function for W (t) is
2
1
y
p(y , t) = √
exp −
2t
2πt
and P (a ≤ W (t) ≤ b) =
Rb
a
p(y , t)dy
• Simulations of a Wiener process:
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Approximation of SDE for small ∆t
• The increment ∆W = W (t + ∆t) − W (t) can be written as
1
∆W = X (∆t) 2 , where X is a random variable with normal distribution
with zero mean and unit variance: X ∼ N (0, 1)
• E∆W = 0 and E(∆W )2 = ∆t.
Recall: equation for the stock price is
dS = µSdt + σSdW ,
then
1
∆S ≈ µS∆t + σSX (∆t) 2
It means ∆S ∼ N µS∆t, σ 2 S 2 ∆t
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Examples
Example 1. Consider a stock that has volatility 30% and provides expected
return of 15% p.a. Find the increase in stock price for one week if the
initial stock price is 100.
Answer: ∆S = 0.288 + 4.155X
Note: 4.155 is the standard deviation
Example 2. Show that the return
µ∆t and variance σ 2 ∆t
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∆S
S
is normally distributed with mean
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Itô’s Lemma
We assume that f (S, t) is a smooth function of S and t.
Find df if dS = µSdt + σSdW
• Volatility σ = 0
df =
∂f
∂t dt
+
∂f
∂S dS
=
∂f
∂t
∂f
+ µS ∂S
dt
• Volatility σ 6= 0
Itô’s Lemma:
∂f
1 2 2 ∂2f
∂f
df = ∂f
+
µS
+
σ
S
dt + σS ∂S
dW
∂t
∂S
2
∂S 2
Example. Find the SDE satisfied by f = S 2 .
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Lecture 3
1
Distribution for ln S(t)
2
Solution to Stochastic Differential Equation for Stock Price
3
Examples
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Differential of ln S
Example 1. Find the stochastic differential equation (SDE) for
f = ln S
by using Itô’s Lemma:
∂f
1 2 2 ∂2f
∂f
df = ∂f
+
µS
+
σ
S
dW
dt + σS ∂S
2
∂t
∂S
2
∂S
We obtain
df =
σ2
µ−
2
dt + σdW
This is a constant coefficient SDE.
Integration from 0 to t gives
σ2
f − f0 = µ −
t + σW (t)
2
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since
W (0) = 0.
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Normal distribution for ln S(t)
We obtain for ln S(t)
ln S(t) − ln S0 =
σ2
µ−
2
t + σW (t)
where S0 = S(0) is the initial stock price.
• ln S(t) has a normal distribution with mean ln S0 + µ −
variance σ 2 t.
σ2
2
t and
Example 2. Consider a stock with an initial price of 40, an expected return
of 16% and a volatility of 20%.
Find the probability distribution of ln S in six months.
We have
σ2
2
ln S(T ) ∼ N ln S0 + µ −
T,σ T
2
Answer:
ln S(0.5) ∼ N (3.759, 0.020)
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Probability density function for ln S(t)
Recall that if the random variable X has a normal distribution with mean
µ and variance σ 2 , then the probability density function is
1
(x − µ)2
p(x) = √
exp −
2σ 2
2πσ 2
The probability density function of X = ln S(t) is
1
(x − ln S0 − (µ − σ 2 /2)t)2
√
exp −
2σ 2 t
2πσ 2 t
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Exact expression for stock price S(t)
Definition. The model of a stock dS = µSdt + σSdW is known as a
geometric Brownian motion.
The random function S(t) can be found from
σ2
ln(S(t)/S0 ) = µ −
t + σW (t)
2
Stock price at time t:
S(t) = S0 e
2
µ− σ2
t+σW (t)
Or
S(t) = S0 e
2
µ− σ2
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√
t+σ tX
20912
where
X ∼ N (0, 1)
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Lecture 4
1
Financial Derivatives
2
European Call and Put Options
3
Payoff Diagrams, Short Selling and Profit
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Derivatives
• Derivative is a financial instrument which value depends on the values of
other underlying variables. Other names are financial derivative, derivative
security, derivative product. A stock option, for example, is a derivative
whose value is dependent on the stock price. Examples: forward contracts,
futures, options, swaps, CDS, etc.
• Options are very attractive to investors, both for speculation and for
hedging
What is an Option?
Definition.
European call option gives the holder the right (not obligation) to buy
underlying asset at a prescribed time T for a specified (strike) price E .
European put option gives its holder the right (not obligation) to sell
underlying asset at a prescribed time T for a specified (strike) price E .
The question is ”what does this actually mean?”
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Example.
Consider a three-month European call option on a BP share with a strike
price E = 15 (T = 0.25). If you enter into this contract you have the right
but not the obligation to buy one share for E = 15 in a three months time.
Whether you exercise your right depends on the stock price in the market
at time T :
• If the stock price is above £15, say £25, you can buy the share for £15,
and sell it immediately for £25, making a profit of £10.
• If the stock price is below £15, there is no financial sense to buy it. The
option is worthless.
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Payoff Diagrams
We denote by C (S, t) the value of European call option and P(S, t) the
value of European put option
Definition. Payoff Diagram ia a graph of the value of the option position
at expiration t = T as a function of the underlying stock price S.
0, S ≤ E ,
Call price at t = T : C (S, T ) = max (S − E , 0) =
S − E, S > E,
Put price at t = T :
P(S, T ) = max (E − S, 0) =
E − S, S ≤ E ,
0, S > E ,
If a trader thinks that the stock price is on the rise, he can make money by
purchase a call option without buying the stock. If a trader believes the
stock price is on the decline, he can make money by buying put options.
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Profit of call option buyer
The profit (gain) of a call option holder (buyer) at time T is
max (S − E , 0) − C0 e rT , C0 − initial call price
Example:
Find the stock price on the exercise date in three months, for a European
call option with strike price £10 to give a gain (profit) of £14 if the option
is bought for £2.25, financed by a loan with continuously compounded
interest rate of 5%
Solution:
1
14 = S(T ) − 10 − 2.25 × e 0.05× 4 ,
S(T ) = 26.28
For the holder of European put option, the profit at time T is
.
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max (E − S, 0) − P0 e rT
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Portfolio and Short Selling
Definition. Short selling is the practice of selling assets that have been
borrowed from a broker with the intention of buying the same assets back
at a later date to return to the broker.
This technique is used by investors who try to profit from the falling price
of a stock.
Definition. Portfolio is the combination of assets, options and bonds.
We denote by Π the value of portfolio. Example: Π = 2S + 4C − 5P.
It means that portfolio consists of long position in two shares, long
position in four call options and short position in five put options.
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Option positions
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Straddle
Straddle is the purchase of a call and a put on the same underlying
security with the same maturity time T and strike price E .
The value of portfolio is Π = C + P
• Straddle is effective when an investor is confident that a stock price will
change dramatically, but is uncertain of the direction of price move.
Example of large profits: S0 = 40, E = 40, C0 = 2, P0 = 2
Let us find an expected return!!
Ans: 400
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Lecture 5
1
Trading Strategies: Straddle, Bull Spread, etc.
2
Bond and Risk-Free Interest Rate
3
No Arbitrage Principle
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Portfolio and Short Selling
Reminder from previous lecture 4.
Definition. Short selling is the practice of selling assets that have been
borrowed from a broker with the intention of buying the same assets back
at a later date to return to the broker. This technique is used by investors
who try to profit from the falling price of a stock.
Definition. Portfolio is the combination of assets, options and bonds. We
denote by Π the value of portfolio.
Examples.
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Option positions
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Trading Strategies Involving Options
Straddle is the purchase of a call and a put on the same underlying
security with the same maturity time T and strike price E .
The value of portfolio is Π = C + P
• Straddle is effective when an investor is confident that a stock price will
change dramatically, but is uncertain of the direction of price move.
• Short Straddle, Π = −C − P, profits when the underlying security
changes little in price before the expiration t = T .
Barings Bank was the oldest bank in London until its collapse in 1995. It
happened when the bank’s trader, Nick Leeson, took short straddle
positions and lost 1.3 billion dollars.
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Bull Spread
Bull spread is a strategy that is designed to profit from a moderate rise in
the price of the underlying security.
Let us set up a portfolio consisting of a long position in call with strike
price E1 and short position in call with E2 such that E1 < E2 .
The value of this portfolio is Πt = Ct (E1 ) − Ct (E2 ). At maturity t = T
ΠT =


0, S ≤ E1 ,
S − E1 , E1 ≤ S < E2 ,

E2 − E1 , S ≥ E2
• The holder of this portfolio benefits when the stock price will be above
E1 .
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Risk-Free Interest Rate
We assume the existence of risk-free investment. Examples: US
government bond, deposit in a sound bank. We denote by B(t) the value
of this investment.
Definition. Bond is a contact that yields a known amount F , called the
face value, on a known time T , called the maturity date. The authorized
issuer (for example, government) owes the holders a debt and is obliged to
pay interest (the coupon) and to repay the face value at maturity.
Zero-coupon bond involves only a single payment at T .
Return dB
B = rdt, where r is the risk-free interest rate.
If B(T ) = F , then B(t) = Fe −r (T −t) , where e −r (T −t) - discount factor
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No Arbitrage Principle
The key principle of financial mathematics is No Arbitrage Principle.
• There are never opportunities to make risk-free profit
• Arbitrage opportunity arises when a zero initial investment Π0 = 0 is
identified that guarantees non-negative payoff in the future such that
ΠT > 0 with non-zero probability.
Arbitrage opportunities may exist in a real market. But, they cannot last
for a long time.
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Lecture 6
1
No-Arbitrage Principle
2
Put-Call Parity
3
Upper and Lower Bounds on Call Options
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No Arbitrage Principle
The key principle of financial mathematics is No Arbitrage Principle.
• There are never opportunities to make risk-free profit.
• Arbitrage opportunity arises when a zero initial investment Π0 = 0 is
identified that guarantees non-negative payoff in the future such that
ΠT > 0 with non-zero probability.
Arbitrage opportunities may exist in a real market.
• All risk-free portfolios must have the same return: risk-free interest rate.
Let Π be the value of a risk-free portfolio, and dΠ is its increment during
a small period of time dt. Then
dΠ
= rdt,
Π
where r is the risk-less interest rate.
• Let Πt be the value of the portfolio at time t. If ΠT ≥ 0, then Πt ≥ 0
for t < T .
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Put-Call Parity
Let us set up portfolio consisting of long one stock, long one put and short
one call with the same T and E .
The value of this portfolio is Π = S + P − C .
The payoff for this portfolio is
ΠT = S + max (E − S, 0) − max (S − E , 0) = E
The payoff is always the same whatever the stock price is at t = T .
Using No Arbitrage Principle, we obtain
St + Pt − Ct = Ee −r (T −t) ,
where Ct = C (St , t) and Pt = P (St , t).
This relationship between St , Pt and Ct is called Put-Call Parity which
represents an example of complete risk elimination.
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Upper and Lower Bounds on Call Options
• Put-Call Parity (t = 0): S0 + P0 − C0 = Ee −rT .
It shows that the value of European call option can be found from the
value of European put option with the same strike price and maturity:
C0 = P0 + S0 − Ee −rT .
Therefore C0 ≥ S0 − Ee −rT since P0 ≥ 0 S0 − Ee −rT is the lower bound
for call option.
S0 − Ee −rT ≤ C0 ≤ S0
Let us illustrate these bounds geometrically.
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Examples
Example 1. Find a lower bound for six months European call option with
the strike price £35 when the initial stock price is £40 and the risk-free
interest rate is 5% p.a.
In this case S0 = 40, E = 35, T = 0.5, and r = 0.05.
The lower bound for the call option price is S0 − E exp (−rT ) , or
40 − 35 exp (−0.05 × 0.5) = 5.864
Example2. Consider the situation where the European call option is £4.
Show that there exists an arbitrage opportunity.
We establish a zero initial investment Π0 = 0 by purchasing one call for
£4 and the bond for £36 and selling one share for £40. The portfolio is
Π = C + B − S.
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Examples
At maturity t = T , the portfolio Π = C + B − S has the value:
36.911 − S, S ≤ 35
ΠT = max (S − E , 0) + 36 exp (0.05 × 0.5) − S =
1.911 S > 35
It is clear that ΠT > 0, therefore there exists an arbitrage opportunity.
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Lecture 7
1
Upper and Lower Bounds on Put Options
2
Proof of Put-Call Parity by No-Arbitrage Principle
3
Example on Arbitrage Opportunity
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Upper and Lower Bounds on Put Option
Reminder from lecture 6.
• Arbitrage opportunity arises when a zero initial investment Π0 = 0 is
identified that guarantees a non-negative payoff in the future such that
ΠT > 0 with non-zero probability.
• Put-Call Parity at time t = 0:
S0 + P0 − C0 = Ee −rT .
Upper and Lower Bounds on Put Option (exercise sheet 3):
Ee −rT − S0 ≤ P0 ≤ Ee −rT
Let us illustrate these bounds geometrically.
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Proof of Put-Call Parity
The value of European put option can be found as
P0 = C0 − S0 + Ee −rT .
Let us prove this relation by using No-Arbitrage Principle.
Assume that P0 > C0 − S0 + Ee −rT . Then one can make a riskless profit
(arbitrage opportunity).
We set up the portfolio Π = −P − S + C + B. At time t = 0 we
• sell one put option for P0 (write the put option)
• sell one share for S0 (short position)
• buy one call option for C0
• buy one bond for B0 = P0 + S0 − C0 > Ee −rT
The balance of all these transactions is zero, that is, Π0 = 0
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Proof of Put-Call Parity
At maturity t = T the portfolio Π = −P − S + C + B has the value
−(E − S) − S + B0 e rT , S ≤ E ,
ΠT =
= −E + B0 e rT
−S + (S − E ) + B0 e rT , S > E ,
Since B0 > Ee −rT , we conclude ΠT > 0. and Π0 = 0.
This is an arbitrage opportunity.
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Proof of Put-Call Parity
Now we assume that P0 < C0 − S0 + Ee −rT .
We set up the portfolio Π = P + S − C − B.
At time t = 0 we
• buy one put option for P0
• buy one share for S0 (long position)
• sell one call option for C0 (write the call option)
• borrow B0 = P0 + S0 − C0 < Ee −rT
The balance of all these transactions is zero, that is, Π0 = 0
At maturity t = T we have ΠT = E − B0 e rT . Since B0 < Ee −rT , we
conclude ΠT > 0.
This is an arbitrage opportunity!!!
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Example on Arbitrage Opportunity
Three months European call and put options with the exercise price £12
are trading at £3 and £6 respectively.
The stock price is £8 and interest rate is 5%. Show that there exists
arbitrage opportunity.
Solution:
The Put-Call Parity P0 = C0 − S0 + Ee −rT is violated, because
1
6 < 3 − 8 + 12e −0.05× 4 = 6.851
To get arbitrage profit we
• buy a put option for £6
• sell a call option for £3
• buy a share for £8
• borrow £11 at the interest rate 5%.
The balance is zero!!
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Example: Arbitrage Opportunity
The value of the portfolio Π = P + S − C − B at maturity T =
1
ΠT = E − B0 e rT = 12 − 11e 0.05× 4 ≈ 0.862.
1
4
is
1
Combination P + S − C gives us £12. We repay the loan £11e 0.05× 4 .
1
The balance 12 − 11e 0.05× 4 is an arbitrage profit £0.862.
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Lecture 8
1
One-Step Binomial Model for Option Price
2
Risk-Neutral Valuation
3
Examples
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One-Step Binomial Model
Initial stock price is S0 . The stock price can either move up from S0 to
S0 u or down from S0 to S0 d ( u > 1; d < 1).
At time T , let the option price be Cu if the stock price moves up, and Cd
if the stock price moves down.
The purpose is to find the current price C0 of a European call option.
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Riskless Portfolio
Now, we set up a portfolio consisting of a long position in ∆ shares and
short position in one call
Π = ∆S − C
• Let us find the number of shares ∆ that makes the portfolio Π riskless.
The value of portfolio when stock moves up is
∆S0 u − Cu
The value of portfolio when stock moves down is
∆S0 d − Cd
If portfolio Π = ∆S − C is risk-free, then ∆S0 u − Cu = ∆S0 d − Cd
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No-Arbitrage Argument
The number of shares is ∆ =
Cu −Cd
S0 (u−d) .
Because portfolio is riskless for this ∆, the current value Π0 can be found
by discounting: Π0 = (∆S0 u − Cu ) e −rT , where r is the interest rate.
On the other hand, the cost of setting up the portfolio is Π0 = ∆S0 − C0 .
Therefore ∆S0 − C0 = (∆S0 u − Cu ) e −rT .
Finally, the current call option price is
C0 = ∆S0 − (∆S0 u − Cu ) e −rT ,
where ∆ =
Cu −Cd
S0 (u−d)
(No-Arbitrage Argument).
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Risk-Neutral Valuation
Alternatively
C0 = e −rT (pCu + (1 − p)Cd ) ,
where
p=
(Risk-Neutral Valuation)
e rT − d
.
u−d
It is natural to interpret the variable 0 ≤ p ≤ 1 as the probability of an up
movement in the stock price, and the variable 1 − p as the probability of a
down movement.
Fair price of a call option C0 is equal to the expected value of its future
payoff discounted at the risk-free interest rate. For a put option P0 we
have the same result
P0 = e −rT (pPu + (1 − p)Pd ) .
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Example
A stock price is currently $40. At the end of three months it will be either
$44 or $36. The risk-free interest rate is 12%.
What is the value of three-month European call option with a strike price
of $42? Use no-arbitrage arguments and risk-neutral valuation.
In this case S0 = 40, u = 1.1, d = 0.9, r = 0.12, T = 0.25, Cu = 2,
Cd = 0.
No-arbitrage arguments: the number of shares
∆=
Cu − Cd
2−0
=
= 0.25
S0 u − S0 d
40 × (1.1 − 0.9)
and the value of call option
C0 = S0 ∆ − (S0 u∆ − Cu ) e −rT =
40 × 0.25 − (40 × 1.1 × 0.25 − 2) × e −0.12×0.25 = 1.266
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Example
Risk-neutral valuation: one can find the probability p
p=
e rT − d
e 0.12×0.25 − 0.9
=
= 0.6523
u−d
1.1 − 0.9
and the value of call option
C0 = e −rT [pCu + (1 − p)Cd ] = e −0.12×0.25 [0.6523 × 2 + 0] = 1.266
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Lecture 9
1
Risk-Neutral Valuation
2
Risk-Neutral World
3
Two-Steps Binomial Tree
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Risk-Neutral Valuation
Reminder from Lecture 8. Call option price:
C0 = e −rT (pCu + (1 − p)Cd ) ,
where p =
e rT −d
u−d .
No-Arbitrage Principle: d < e rT < u.
In particular, if d > e rT then there exists an arbitrage opportunity. We
could make money by taking out a bank loan B0 = S0 at time t = 0 and
buying the stock for S0 .
We interpret the variable 0 ≤ p ≤ 1 as the probability of an up movement
in the stock price.
This formula is known as a risk-neutral valuation.
The probability of up q or down movement 1 − q in the stock price plays
no role whatsoever! Why???
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Risk Neutral Valuation
Let us find the expected stock price at t = T :
Ep [ST ] = pS0 u + (1 − p)S0 d =
e rT −d
u−d S0 u
+ (1 −
e rT −d
u−d )S0 d
= S0 e rT .
This shows that stock price grows on average at the risk-free interest rate
r . Since the expected return is r , this is a risk-neutral world.
In Real World: E [ST ] = S0 e µT . In Risk-Neutral World: Ep [ST ] = S0 e rT
Risk-Neutral Valuation: C0 = e −rT Ep [CT ]
The option price is the expected payoff in a risk-neutral world, discounted
at risk-free rate r .
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Two-Step Binomial Tree
Now the stock price changes twice, each time by either a factor of u > 1
or d < 1. We assume that the length of the time step is ∆t such that
T = 2∆t. After two time steps the stock price will be S0 u 2 , S0 ud or S0 d 2 .
The call option expires after two time steps producing payoffs Cuu , Cud
and Cdd respectively.
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Option Price
The purpose is to calculate the option price C0 at the initial node of the
tree.
We apply the risk-neutral valuation backward in time:
Cu = e −r ∆t (pCuu + (1 − p)Cud ) , Cd = e −r ∆t (pCud + (1 − p)Cdd ) .
Current option price:
Sergei Fedotov (University of Manchester)
C0 = e −r ∆t (pCu + (1 − p)Cd ) .
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Option Price
Substitution gives
C0 = e −2r ∆t p 2 Cuu + 2p(1 − p)Cud + (1 − p)2 Cdd ,
where p 2 , 2p(1 − p) and (1 − p)2 are the probabilities in a risk-neutral
world that the upper, middle, and lower final nodes are reached.
Finally, the current call option price is
C0 = e −rT Ep [CT ] , T = 2∆t.
The current put option price can be found in the same way:
P0 = e −2r ∆t p 2 Puu + 2p(1 − p)Pud + (1 − p)2 Pdd
or
P0 = e −rT Ep [PT ] .
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Two-Step Binomial Tree Example
Consider six months European put with a strike price of £32 on a stock
with current price £40. There are two time steps and in each time step
the stock price either moves up by 20% or moves down by 20%. Risk-free
interest rate is 10%. Find the current option price.
We have u = 1.2, d = 0.8, ∆t = 0.25, and r = 0.1.
r∆t −d
0.1×0.25 −0.8
= e 1.2−0.8
= 0.5633.
Risk-neutral probability p = e u−d
The possible stock prices at final nodes are 40 × (1.2)2 = 57.6,
40 × 1.2 × 0.8 = 38.4, and 40 × (0.8)2 = 25.6.
We obtain Puu = 0, Pud = 0 and Pdd = 32 − 25.6 = 6.4.
Thus the value of put option is
P0 = e −2×0.1×0.25 × 0 + 0 + (1 − 0.5633)2 × 6.4 = 1.1610.
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Lecture 10
1
Binomial Model for Stock Price
2
Option Pricing on Binomial Tree
3
Matching Volatility σ with u and d
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Binomial model for the stock price
Continuous random model for the stock price: dS = µSdt + σSdW
The binomial model for the stock price is a discrete time model:
• The stock price S changes only at discrete times ∆t, 2∆t, 3∆t, ...
• The price either moves up S → Su or down S → Sd with d < e r ∆t < u.
• The probability of up movement is q.
Let us build up a tree of possible stock prices. The tree is called a binomial
tree, because the stock price will either move up or down at the end of
each time period. Each node represents a possible future stock price.
We divide the time to expiration T into several time steps of duration
∆t = T /N, where N is the number of time steps in the tree.
Example: Let us sketch the binomial tree for N = 4.
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Stock Price Movement in the Binomial Model
We introduce the following notations:
• Snm is the n-th possible value of stock price at time-step m∆t.
Then Snm = u n d m−n S00 , where n = 0, 1, 2, ..., m.
S00 is the stock price at the time t = 0. Note that u and d are the same at
every node in the tree.
For example, at the third time-step 3∆t, there are four possible stock
prices: S03 = d 3 S00 , S13 = ud 2 S00 , S23 = u 2 dS00 and S33 = u 3 S00 .
At the final time-step N∆t, there are N + 1 possible values of stock price.
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Call Option Pricing on Binomial Tree
We denote by Cnm the n-th possible value of call option at time-step m∆t.
• Risk Neutral Valuation (backward in time):
m+1
Cnm = e −r ∆t pCn+1
+ (1 − p)Cnm+1 .
Here 0 ≤ n ≤ m and p =
e r∆t −d
u−d .
• Final condition: CnN = max SnN − E , 0 , where n = 0, 1, 2, ..., N, E is the
strike price.
The current option price C00 is the expected payoff in a risk-neutral world,
discounted at risk-free rate r : C00 = e −rT Ep [CT ] .
Example: N = 4.
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Matching volatility σ with u and d
We assume that the stock price starts at the value S0 and the time step is
∆t. Let us findthe expected stock price, E [S] , and the variance of the
return, var ∆S
S , for continuous and discrete models.
• Expected stock price: Continuous model: E [S] = S0 e µ∆t .
On the binomial tree: E [S] = qS0 u + (1 − q)S0 d.
First equation: qu + (1 − q)d = e µ∆t .
• Variance of the return: Continuous model: var ∆S
= σ 2 ∆t (Lecture2)
S
∆S On the binomial tree: var S =
q(u − 1)2 + (1 − q)(d − 1)2 − [q(u − 1) + (1 − q)(d − 1)]2 =
qu 2 + (1 − q)d 2 − [qu + (1 − q)d]2 .
Recall: var [X ] = E X 2 − [E (X )]2 .
Second equation: qu 2 + (1 − q)d 2 − [qu + (1 − q)d]2 = σ 2 ∆t.
Third equation: u = d −1 .
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Matching volatility σ with u and d
From the first equation we find q =
e µ∆t −d
u−d .
This is the probability of an up movement in the real world.
Substituting this probability into the second equation, we obtain
e µ∆t (u + d) − ud − e 2µ∆t = σ 2 ∆t.
Using u = d −1 , we get
1
e µ∆t (u + ) − 1 − e 2µ∆t = σ 2 ∆t.
u
This equation can be reduced to the quadratic equation. (Exercise sheet 4,
part 5).
√
√
√
One can obtain u ≈ e σ ∆t ≈ 1 + σ ∆t and d ≈ e −σ ∆t .
These are the values of u and d obtained by Cox, Ross, and Rubinstein in
1979.
Recall: e x ≈ 1 + x for small x.
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Lecture 11
1
American Put Option Pricing on Binomial Tree
2
Replicating Portfolio
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American Option
• An American Option is one that may be exercised at any time prior to
expire (t = T ).
• We should determine when it is best to exercise the option.
• It is not subjective! It can be determined in a systematic way!
• The American put option value must be greater than or equal to the
payoff function.
If P < max(E − S, 0), then there is obvious arbitrage opportunity.
We can buy stock for S and option for P and immediately exercise the
option by selling stock for E .
E − (P + S) > 0
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American Put Option Pricing on Binomial Tree
We denote by Pnm the n-th possible value of put option at time-step m∆t.
• European Put Option:
m+1
Pnm = e −r ∆t pPn+1
+ (1 − p)Pnm+1 .
Here 0 ≤ n ≤ m and the risk-neutral probability p =
e r∆t −d
u−d .
• American Put Option:
n
o
m+1
Pnm = max max(E − Snm , 0), e −r ∆t pPn+1
+ (1 − p)Pnm+1 ,
where Snm is the n-th possible value of stock price at time-step m∆t.
• Final condition: PnN = max E − SnN , 0 , where n = 0, 1, 2, ..., N, E is the
strike price.
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Example: Evaluation of American Put Option on Two-Step
Tree
We assume that over each of the next two years the stock price either
moves up by 20% or moves down by 20%. The risk-free interest rate is 5%.
Find the value of a 2-year American put with a strike price of $52 on a
stock whose current price is $50.
In this case u = 1.2, d = 0.8, r = 0.05, E = 52.
Risk-neutral probability:
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p=
e 0.05 −0.8
1.2−0.8
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= 0.6282
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Replicating Portfolio
The aim is to calculate the value of call option C0 .
Let us establish a portfolio of stocks and bonds in such a way that the
payoff of a call option is completely replicated.
Final value: ΠT = CT = max (S − E , 0)
To prevent risk-free arbitrage opportunity, the current values should be
identical. We say that the portfolio replicates the option.
The Law of One Price: Πt = Ct .
Consider replicating portfolio of ∆ shares held long and N bonds held
short.
The value of portfolio: Π = ∆S − NB. A pair (∆, N) is called a trading
strategy.
How to find (∆, N) such that ΠT = CT and Π0 = C0 ?
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Example: One-Step Binomial Model.
Initial stock price is S0 . The stock price can either move up from S0 to
S0 u or down from S0 to S0 d. At time T , let the option price be Cu if the
stock price moves up, and Cd if the stock price moves down.
The value of portfolio: Π = ∆S − NB.
When stock moves up:
∆S0 u − NB0 e rT = Cu .
When stock moves down: ∆S0 d − NB0 e rT = Cd .
We have two equations for two unknown variables ∆ and N.
Current value: C0 = ∆S0 − NB0 .
Prove: C0 = e −rT (pCu + (1 − p)Cd ) , where p =
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e rT −d
u−d .
(Exercise sheet 5)
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Lecture 12
1
Black-Scholes Model
2
Black-Scholes Equation
The Black-Scholes model for option pricing was developed by Fischer
Black, Myron Scholes in the early 1970s. This model is the most
important result in financial mathematics.
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Black - Scholes model
The Black-Scholes model is used to calculate an option price using: stock
price S, strike price E , volatility σ, time to expiration T , and risk-free
interest rate r .
This model involves the following explicit assumptions:
• The stock price follows a Geometric Brownian motion with constant
expected return and volatility: dS = µSdt + σSdW . .
• No transaction costs.
• The stock does not pay dividends.
• One can borrow and lend cash at a constant risk-free interest rate.
• Securities are perfectly divisible (i.e. one can buy any fraction of a share
of stock).
• No restrictions on short selling.
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Basic Notation
We denote by V (S, t) the value of an option. We use the notations
C (S, t) and P(S, t) for call and put when the distinction is important.
• The aim is to derive the famous Black-Scholes Equation:
∂V
∂2V
1
∂V
+ σ 2 S 2 2 + rS
− rV = 0.
∂t
2
∂S
∂S
Now, we set up a portfolio consisting of a long position in one option and
a short position in ∆ shares.
The value is Π = V − ∆S.
• Let us find the number of shares ∆ that makes this portfolio riskless.
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Itô’s Lemma and Elimination of Risk
The change in the value of this portfolio in the time interval dt:
dΠ = dV − ∆dS, where dS = µSdt + σSdW .
Using Itô’s Lemma:
∂V
1 2 2 ∂2V
∂V
∂V
dV =
+ σ S
dt + σS
dW ,
+ µS
2
∂t
2
∂S
∂S
∂S
we find
∂V
1 2 2 ∂2V
∂V
∂V
dΠ =
+ σ S
+ µS
− µS∆ dt + σS
− ∆σS dW .
∂t
2
∂S 2
∂S
∂S
The main question is how to eliminate the risk!!!
We can eliminate the random component in dΠ by choosing ∆ =
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∂V
∂S
2010
.
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Black-Scholes Equation
This choice
portfolio Π = V − S ∂V
∂S whose increment
results in a risk-free
2
∂V
1 2 2∂ V
is dΠ = ∂t + 2 σ S ∂S 2 dt.
• No-Arbitrage Principle: the return from this portfolio must be rdt.
dΠ
Π
= rdt or
∂V
∂t
2V
∂S 2
+ 12 σ2 S 2 ∂
V −S ∂V
∂S
= r or
∂V
∂t
2
+ 12 σ 2 S 2 ∂∂SV2 = r V − S ∂V
∂S .
Thus, we obtain the Black-Scholes PDE:
∂V
∂2V
1
∂V
+ σ 2 S 2 2 + rS
− rV = 0.
∂t
2
∂S
∂S
Scholes received the 1997 Nobel Prize in Economics. It was not awarded
to Black in 1997, because he died in 1995.
Black received a Ph.D. in applied mathematics from Harvard University.
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The European call value C (S, t)
If PDE is of backward type, we must impose a final condition at t = T .
For a call option, we have C (S, T ) = max(S − E , 0).
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Lecture 13
1
Boundary Conditions for Call and Put Options
2
Exact Solution to Black-Scholes Equation
∂C
1
∂2C
∂C
+ σ 2 S 2 2 + rS
− rC = 0.
∂t
2
∂S
∂S
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Boundary Conditions
We use C (S, t) and P(S, t) for call and put option. Boundary conditions
are applied for zero stock price S = 0 and S → ∞.
• Boundary conditions for a call option:
C (0, t) = 0 and C (S, t) → S as S → ∞.
The call option is likely to be exercised as S → ∞
• Boundary conditions for a put option:
P(0, t) = Ee −r (T −t) .
We evaluate the present value of E .
P(S, t) → 0 as S → ∞.
As stock price S → ∞, then put option is unlikely to be exercised.
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Exact solution
The Black-Scholes equation
∂C
1
∂2C
∂C
+ σ 2 S 2 2 + rS
− rC = 0
∂t
2
∂S
∂S
with appropriate final and boundary conditions has the explicit solution:
C (S, t) = SN (d1 ) − Ee −r (T −t) N (d2 ) ,
where
1
N (x) = √
2π
Z
x
y2
e − 2 dy
(cumulative normal distribution)
−∞
and
√
ln (S/E ) + r + σ 2 /2 (T − t)
√
d1 =
, d2 = d1 − σ T − t.
σ T −t
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The European call value C (S, t) by K. Rubash
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Example
Calculate the price of a three-month European call option on a stock with
a strike price of $25 when the current stock price is $21.6 The volatility is
35% and risk-free interest rate is 1% p.a.
In this case S0 = 21.6, E = 25, T = 0.25, σ = 0.35 and r = 0.01.
The value of call option is C0 = S0 N (d1 ) − Ee −rT N (d2 ) .
First, we compute the values of d1 and d2 :
d1 =
ln(S0 /E )+(r +σ2 /2)T
√
σ T
=
ln( 21.6
+(0.01+(0.35)2 ×0.5))×0.25
25 )
√
0.35× 0.25
√
√
d2 = d1 − σ T = −0.7335 − 0.35 × 0.25 ≈ −0.9085
≈ −0.7335
Since
we obtain
N (−0.7335) ≈ 0.2316,
N (−0.9085) ≈ 0.1818,
C0 ≈ 21.6 × 0.2316 − 25 × e −0.01×0.25 × 0.1818 = 0.4689
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The Limit of Higher Volatility
Let us find the limit
lim C (S, t).
σ→∞
We know that
C (S, t) = SN (d1 ) − Ee −r (T −t) N (d2 ) ,
where d1 =
ln(S/E )+(r +σ2 /2)(T −t)
√
σ T −t
=
ln(S/E )
√
σ T −t
+
√
r T −t
σ
+
√
σ T −t
.
2
In the limit σ → ∞, d1 → ∞.
√
Since d2 = d1 − σ T − t, in the limit σ → ∞, d2 → −∞
Thus limσ→∞ N (d1 ) = 1 and limσ→∞ N(d2 ) = 0.
Therefore limσ→∞ C (S, t) = S.
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This is an upper bound for call option!!!
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Lecture 14
1
∆-Hedging
2
Greek Letters or Greeks
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Delta for European Call Option
Let us show that ∆ =
∂C
∂S
= N (d1 ) .
First, find the derivative ∆ = ∂C
∂S by using the explicit solution for the
European call C (S, t) = SN (d1 ) − Ee −r (T −t) N (d2 ).
∆=
∂C
∂d1
∂d2
= N (d1 ) + SN ′ (d1 )
− Ee −r (T −t) N ′ (d2 )
=
∂S
∂S
∂S
N (d1 ) + (SN ′ (d1 ) − Ee −r (T −t) N ′ (d2 ))
∂d1
.
∂S
We need to prove
(SN ′ (d1 ) − Ee −r (T −t) N ′ (d2 )) = 0.
See Problem Sheet 6.
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Delta-Hedging Example
Find the value of ∆ of a 6-month European call option on a stock with a
strike price equal to the current stock price ( t = 0). The interest rate is
6% p.a. The volatility σ = 0.16.
Solution: we have ∆ = N (d1 ) , where d1 =
ln(S0 /E )+(r +σ2 /2)T
1
σ(T ) 2
We find
d1 =
and
(0.06 + (0.16)2 × 0.5)) × 0.5
√
≈ 0.3217
0.16 × 0.5
∆ = N (0.3217) ≈ 0.6262
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Delta for European Put Option
Let us find Delta for European put option by using the put-call parity:
S + P − C = Ee −r (T −t) .
Let us differentiate it with respect to S
We find 1 +
∂P
∂S
−
∂C
∂S
∂P
∂S
=
∂C
∂S
− 1 = N (d1 ) − 1.
Therefore
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= 0.
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Greeks
The option value:
V = V (S, t | σ, r , T ).
Greeks represent the sensitivities of options to a change in underlying
parameters on which the value of an option is dependent.
• Delta:
∆ = ∂V
∂S measures the rate of change of option value with respect to
changes in the underlying stock price
• Gamma:
2
Γ = ∂∂SV2 = ∂∆
∂S measures the rate of change in ∆ with respect to changes
in the underlying stock price.
Problem Sheet 6:
Γ=
Sergei Fedotov (University of Manchester)
∂2C
∂S 2
=
′ (d )
N√
1
,
Sσ T −t
20912
where N ′ (d1 ) =
√1 e −
2π
d12
2
.
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Greeks
• Vega,
∂V
∂σ ,
measures sensitivity to volatility σ.
One can show that
∂C
∂σ
√
= S T − tN ′ (d1 ).
• Rho:
ρ=
∂V
∂r
measures sensitivity to the interest rate r .
One can show that ρ =
∂C
∂r
= E (T − t)e −r (T −t) N(d2 ).
The Greeks are important tools in financial risk management. Each Greek
measures the sensitivity of the value of derivatives or a portfolio to a small
change in a given underlying parameter.
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Lecture 15
1
Black-Scholes Equation and Replicating Portfolio
2
Static and Dynamic Risk-Free Portfolio
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Replicating Portfolio
The aim is to show that the option price V (S, t) satisfies the
Black-Scholes equation
∂V
1
∂2V
∂V
+ σ 2 S 2 2 + rS
− rV = 0.
∂t
2
∂S
∂S
Consider replicating portfolio of ∆ shares held long and N bonds held
short. The value of portfolio: Π = ∆S − NB. Recall that a pair (∆, N) is
called a trading strategy.
How to find (∆, N) such that Πt = Vt ?
• SDE for a stock price S(t) :
dS = µSdt + σSdW .
• Equation for a bond price B (t) : dB = rBdt.
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Derivation of the Black-Scholes Equation
By using the Ito’s lemma, we find the change in the option value
∂V
∂V
1 2 2 ∂2V
∂V
dV =
+ µS
+ σ S
dW .
dt + σS
2
∂t
∂S
2
∂S
∂S
By using self-financing requirement dΠ = ∆dS − NdB, we find the change
in portfolio value
dΠ = ∆dS−NdB = ∆(µSdt+σSdW )−rNBdt = (∆µS−rNB)dt+∆σSdW
Equating the last two equations dΠ = dV , we obtain
∆=
∂V
∂S
, −rNB =
∂V
∂t
Since NB = ∆S − Π =
equation
∂V
+
∂t
Sergei Fedotov (University of Manchester)
2
+ 12 σ 2 S 2 ∂∂SV2 .
S ∂V
∂S − V , we get the classical Black-Scholes
1 2 2 ∂2V
∂V
σ S
+ rS
− rV = 0.
2
2
∂S
∂S
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Static Risk-free Portfolio
Let me remind you a Put-Call Parity. We set up the portfolio consisting of
long position in one stock, long position in one put and short position in
one call with the same T and E .
The value of this portfolio is Π = S + P − C .
The payoff for this portfolio is
ΠT = S + max (E − S, 0) − max (S − E , 0) = E
The payoff is always the same whatever the stock price is at t = T .
Using No Arbitrage Principle, we obtain
St + Pt − Ct = Ee −r (T −t) ,
where Ct = C (St , t) and Pt = P (St , t).
This is an example of complete risk elimination.
Definition: The risk of a portfolio is the variance of the return.
Sergei Fedotov (University of Manchester)
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2010
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Dynamic Risk-Free Portfolio
Put-Call Parity is an example of complete risk elimination when we carry
out only one transaction in call/put options and underlying security.
Let us consider the dynamic risk elimination procedure.
We could set up a portfolio consisting of a long position in one call option
and a short position in ∆ shares.
The value is Π = C − ∆S.
We can eliminate the random component in Π by choosing
∆=
∂C
.
∂S
This is a ∆-hedging! It requires a continuous rebalancing of a number of
shares in the portfolio Π.
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Lecture 16
1
Option on Dividend-paying Stock
2
American Put Option
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Option on Dividend-paying Stock
We assume that in a time dt the underlying stock pays out a dividend
D0 Sdt,
where D0 is a constant dividend yield.
Now, we set up a portfolio consisting of a long position in one call option
and a short position in ∆ shares.
The value is Π = C − ∆S.
The change in the value of this portfolio in the time interval dt:
dΠ = dC − ∆dS − ∆D0 Sdt.
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Itô’s Lemma and Elimination of Risk
Using Itô’s Lemma:
dC =
∂C
∂2C
1
+ σ2 S 2 2
∂t
2
∂S
dt +
∂C
dS
∂S
we find
dΠ =
1
∂2C
∂C
∂C
+ σ 2 S 2 2 − D0 S
∂t
2
∂S
∂S
dt +
∂C
dS − ∆dS
∂S
We can eliminate the random component in dΠ by choosing ∆ =
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20912
∂C
∂S .
2010
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Modified Black-Scholes Equation
This choice
Π = C − S ∂C
∂S whose increment
results in a 2risk-free portfolio
∂C
1 2 2∂ C
∂C
is dΠ = ∂t + 2 σ S ∂S 2 − D0 S ∂S dt.
• No-Arbitrage Principle: the return from this portfolio must be rdt.
dΠ
Π
= rdt or
∂C
∂t
2
∂C
+ 12 σ 2 S 2 ∂∂SC2 − D0 S ∂C
∂S = r C − S ∂S .
Thus, we obtain the modified Black-Scholes PDE:
∂C
1
∂2C
∂C
+ σ 2 S 2 2 + (r − D0 )S
− rC = 0.
∂t
2
∂S
∂S
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Solution to Modified Black-Scholes Equation
Let us find the solution to modified Black-Scholes equation in the form
C (S, t) = e −D0 (T −t) C1 (S, t).
We prove that C1 (S, t) satisfies the Black-Scholes equation with r
replaced by r − D0 .
If we substitute C (S, t) = e −D0 (T −t) C1 (S, t) into the modified
Black-Scholes equation, we find the equation for C1 (S, t) in the form
∂C1 1 2 2 ∂ 2 C1
∂C1
+ (r − D0 )
+ σ S
− (r − D0 )C1 = 0
2
∂t
2
∂S
∂S
. The auxiliary function C1 (S, t) is the value of a European Call option
with the interest rate r − D0 .
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Solution to Modified Black-Scholes Equation
Problem sheet 7: show that the modified Black-Scholes equation has the
explicit solution for the European call
C (S, t) = Se −D0 (T −t) N (d10 ) − Ee −r (T −t) N (d20 ) ,
where
d10
√
ln (S/E ) + r − D0 + σ 2 /2 (T − t)
√
=
, d20 = d10 − σ T − t.
σ T −t
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American Put Option
Recall that An American Option is one that may be exercised at any time
prior to expire (t = T ).
• The American put option value must be greater than or equal to the
payoff function.
If P < max(E − S, 0), then there is obvious arbitrage opportunity.
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American Put Option
American put problem can be written as as a free boundary problem.
We divide the price axis S into two distinct regions:
0 ≤ S < Sf (t) and Sf (t) < S < ∞,
where Sf (t) is the exercise boundary. Note that we do not know a priori
the value of Sf (t).
When 0 ≤ S < Sf (t), the early exercise is optimal: put option value is
P(S, t) = E − S.
When S > Sf (t), the early exercise is not optimal, and P(S, t) obeys the
Black-Scholes equation.
The boundary conditions at S = Sf (t) are
P (Sf (t), t) = max (E − Sf (t), 0) ,
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20912
∂P
(Sf (t), t) = −1.
∂S
2010
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Lecture 17
1
Bond Pricing with Known Interest Rates and Dividend Payments
2
Zero-Coupon Bond Pricing
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Bond Pricing Equation
• Definition. A bond is a contract that yields a known amount (nominal,
principal or face value) on the maturity date, t = T . The bond pays the
dividends at fixed times.
If there is no dividend payment, the bond is known as a zero-coupon bond.
Let us introduce the following notation
V (t) is the value of the bond, r (t) is the interest rate, K (t) is the
coupon payment.
Equation for the bond price is
dV
= r (t)V − K (t).
dt
The final condition:
V (T ) = F .
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Zero-Coupon Bond Pricing
Let us consider the case when the dividend payment K (t) = 0.
The solution of the equation
as
dV
dt
= r (t)V with V (T ) = F can be written
Z
V (t) = F exp −
t
Let us show this by integration
ln V (T ) − ln V (t) =
RT
t
Sergei Fedotov (University of Manchester)
dV
V
r (s)ds or
T
r (s)ds .
= r (t)dt from t to T .
ln(V (t)/F ) = −
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RT
t
r (s)ds.
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Example
A zero coupon bond, V , issued at t = 0, is worth V (1) = 1 at t = 1. Find
the bond price V (t) at time t < 1 and V (0), when the continuous interest
rate is
r (t) = t 2 .
Solution: T = 1; one can find
Z 1
Z
r (s)ds =
t
1
s 2 ds =
t
1 1 3
− t
3 3
Therefore
Z 1
3
t
1
V (t) = exp −
r (s)ds = exp
−
,
3
3
t
1
V (0) = exp −
= 0.7165
3
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Bond Pricing for K (t) > 0
Let us consider the case when the dividend payment K (t) > 0. The
solution of the equation dV
dt = r (t)V − K (t) can be written as
Z T
V (t) = F exp −
r (s)ds + V1 (t) ,
t
R
T
where V1 (t) = C (t) exp − t r (s)ds .
To find C (t) we need to substitute V1 (t) into the equation
dV1
= r (t)V1 − K (t).
dt
(tutorial exercise 8)
The explicit solution is
Z T
Z
V (t) = exp −
r (s)ds
F+
t
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t
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T
K (y ) exp
Z
y
T
r (s)ds dy .
2010
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Lecture 18
1
Measure of Future Values of Interest Rate
2
Term Structure of Interest Rate (Yield Curve)
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Measure of Future Value of Interest Rate
Let us consider the case when the dividend payment K (t) = 0. Recall that
the solution of the equation dV
dt = r (t)V with V (T ) = F is
Z T
V (t) = F exp −
r (s)ds .
t
Let us introduce the notation V (t, T ) for the bond prices. These prices
are quoted at time t for different values of T .
Let us differentiate V (t, T ) with respect to T :
R
T
∂V
=
F
exp
−
r
(s)ds
(−r (T )) = −V (t, T )r (T ),
∂T
t
since
∂
∂T
RT
t
r (s)ds = r (T ). Therefore,
r (T ) = −
1
∂V
.
V (t, T ) ∂T
This is an interest rate at future dates (forward rate).
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Term Structure of Interest Rate (Yield Curve)
We define
ln(V (t, T )) − ln V (T , T )
T −t
as a measure of the future values of interest rate, where V (t, T ) is taken
from financial data.
Y (t, T ) = −
Y (t, T ) = −
R
ln(F exp − tT r (s)ds )−ln F
T −t
=
1
T −t
RT
t
r (s)ds
is the average value of the interest rate r (t) in the time interval [t, T ].
Bond price V (t, T ) can be written as V (t, T ) = Fe −Y (t,T )(T −t) .
Term structure of interest rate (yield curve):
ln(V (0, T )) − ln V (T , T )
1
=
Y (0, T ) = −
T
T
Z
T
r (s)ds
0
is the average value of interest rate in the future.
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Example
Assume that the instantaneous interest rate r (t) is
r (t) = r0 + at,
where r0 and a are constants.
Bond price:
V (t, T ) = Fe −
RT
t
r (s)ds
= Fe −
RT
t
(r0 +as)ds
a
= Fe −r0 (T −t)− 2 (T
2 −t 2
).
Term structure of interest rate:
1
Y (0, T ) =
T
Sergei Fedotov (University of Manchester)
Z
T
r (s)ds = r0 +
0
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aT
.
2
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Risk of Default
There exists a risk of default of bond V (t) when the principal are not paid
to the lender as promised by borrower.
How to take this into account?
Let us introduce the probability of default p during one year T = 1. Then
V (0) e r = V (0) (1 − p)e (r +s) + p × 0.
The investor has no preference between two investments.
The positive parameter s is called the spread w.r.t to interest rate r .
Let us find it.
s = − ln(1 − p) = p + o(p).
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Lecture 19
1
Asian Options
2
Derivation of PDE for Option Price
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Asian Options
• Definition. Asian option is a contract giving the holder the right to
buy/sell an underlying asset for its average price over some prescribed
period.
Asian call option final condition:
1
V (S, T ) = max S −
T
We introduce a new variable:
Z t
I (t) =
S(t)dt
or
0
Z
T
0
S(t)dt, 0 .
dI
= S(t).
dt
• Final condition can be rewritten as
I
V (S, I , T ) = max S − , 0 .
T
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Asian Options
Asian option price V (S, I , t) is the function of three variables.
Let us derive the equation for this price by using the standard portfolio
consisting of a long position in one call option and a short position in ∆
shares.
The value is Π = V − ∆S.
The change in the value of this portfolio in the time interval dt:
dΠ = dV − ∆dS.
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Itô’s Lemma and Elimination of Risk
Using Itô’s Lemma:
∂V
∂V
∂V
1 2 2 ∂2V
dV =
+ σ S
dt +
dS +
dI
2
∂t
2
∂S
∂S
∂I
we find
dΠ =
∂V
1
∂2V
+ σ2 S 2 2
∂t
2
∂S
dt +
∂V
∂V
dS − ∆dS +
dI .
∂S
∂I
We can eliminate the random component in dΠ by choosing ∆ =
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∂V
∂S
2010
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Modified Black-Scholes Equation
By using No-Arbitrage Principle, and the equation
dI = Sdt
we can obtain the modified Black-Scholes PDE for the Asian option price:
∂V
1
∂2V
∂V
∂V
+ σ 2 S 2 2 + rS
− rV + S
= 0.
∂t
2
∂S
∂S
∂I
The value of Asian option must be calculated numerically (no analytical
solution!)
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