Download Risk

Lecture 7: Stochastic Process
• The following topics are covered:
– Markov Property and Markov Stochastic Process
– Wiener Process
– Generalized Wiener Process
– Ito Process
– The process for stock price
– Ito Lemma and applications
– Black-Scholes-Merton Model
L7: Stochastic Process
1
Markov Property and Markov Stochastic Process
• A Markov process is a particular type of stochastic process
where only the present value of a variable is relevant for
predicting the future.
– It implies that the probability distribution of the price is not dependent
on the particular path followed by the price in the past.
– It is consistent with the weak form of market efficiency.
• Markov stochastic process
– φ(m,v) denotes the normal distribution with man m and variance v
– Markov property highlights that distributions of asset moves in
different time are independent
– The variance of the change in the value of variable during 1 year
equals the variance of the change during the first 6 month
– The change during any time period of length T is φ(0,T)
L7: Stochastic Process
2
Wiener Process
Property 1: The change Δz during a small period of time Δt is
z   t , where  has a standard normal distribution φ(0,1)
–
It follows that Δz has a normal distribution with mean =0; variance= Δt;
standard deviation=sqrt(t).
Property 2: The value of Δz for any two different short intervals are
independent.
–
–
–
Consider the change of variable z during a relative long period of time T.
The change can be denoted by z(T)-z(0). This can be regarded as the change
in z small time intervals of length Δt, where N=T/ Δt. Then,
N
z (T )  z (0)    i t .
i 1
When Δt-> 0, any T leads to an infinite N.
•
–
The expected length of the path followed by z in any time interval is infinite
Then z(T)-z(0) is normally distributed, with mean=0, variance=t; standard
deviation = sqrt(T).
L7: Stochastic Process
3
Wiener process, aka standard Brownian motion
L7: Stochastic Process
4
Generalized Wiener Process
• A generalized Wiener process for a variable x can be defined as:
dx=adt+bdz
–
–
–
–
a is the drift rate of the stochastic process
b2=the variance rate of the stochastic process;
a and b are constant;
dz is a basic Wiener process with a drift rate of zero and a variance rate of 1.
• Mean (per unit) = a
• Variance (per unit) = b2
• Standard deviation = b
L7: Stochastic Process
5
Ito’s Process
dx  a( x, t )dt  b( x, t )dz
• The process for a stock:
– ds/s=μdt+σdz
• Discrete time Model:
S  St  S t
L7: Stochastic Process
6
Ito’s Lemma
•
Assume G=f(x,t)
dx  a( x, t )dt  b( x, t )dz
•
Ito Lemma:
G
G 1  2G 2
G
dG  (
a

b
)
dt

bdz
x
t 2 x 2
x
•
•
For stocks:
ds  Sdt  Sdz
Stochastic differential equation (SDE)
We have:
G
G 1  2G 2 2
G
dG  (
S 


S
)
dt

Sdz
x
t 2 x 2
x
L7: Stochastic Process
7
Interesting Property of SDE
L7: Stochastic Process
8
Deriving Ito Lemma
Using Taylor Expansion, we have:
G
1  2G 2 G
dG 
dx 
dx 
dt
2
x
2 x
t
dx  adt  bdz
dx 2  b 2 dz 2  b 2 dt
Insert dx and dx2 in dG we have Ito Lemma.
L7: Stochastic Process
9
The Lognormal Property
G=lnS
• We have:
2
– dG  (  
)dt  dz (12.17)
2
–
– S ~  ( t ,  2 t ) (13.1)
S
ln S T ~ [ln S 0 (  
2
2
)T ,  2T ]
L7: Stochastic Process
10
Black-Scholes-Merton Differential Equation (1)
• Stock price is assume to follow the following process:
dS  Sdt  Sdz
• Suppose that f is the price of a call option or other derivative contingent
on S.
df  (
f
f 1 f 2 2
f
S  
 S )dt  Sdz
S
t 2 S
S
• We construct a portfolio of the stock and the derivative (page 287):
f
f 
S
S
L7: Stochastic Process
11
Black-Scholes-Merton Differential Equation (2)
• We then have:
f 1  2 f 2 2
  ( 
 S )t
2
t 2 S
• The portfolio is risk free. Thus,
  rt
• Putting all together, we have:
f
f 1 2 2  2 f
 rS
  S
 rf
t
S 2
S 2
• Applying the following boundary conditions:
– f=max(S-K,0) when t=T for a call option
– f=max(K-S,0) when t=T for a put option
L7: Stochastic Process
12
Intuition for Riskless Portfolio
• Long stocks and short the call option
• The percentage between the numbers of stocks and
call options depends on the sensitivity of call price to
stock price
• Requires instantaneous rebalance
• Delta hedging
• Gamma
L7: Stochastic Process
13
Risk-Neutral Valuation
• Note that the variables that appear in the differential equation
are the current stock price, time, stock price volatility, and the
risk-free rate of return. All are independent of risk preferences.
• Why call option price does not reflect stock returns?
• In a world where investors are risk neutral, the expected return
on all investment assets is the risk-free rate of return.
• Pricing a forward contract.
– Value of a forward contract is ST-K at the maturity date
– Based on risk-neutral valuation, we have the value of a forward
contract: f=S0-Ke-rT
L7: Stochastic Process
14
Risk Neutral Valuation and
Black-Scholes Pricing Formulas
• Risk-neutral valuation:
– The expected value of the option at maturity in a risk-neutral world is:
^
E[max( ST  K ,0)]
– Call option price c is:
ce
 rT
^
E[max( ST  K ,0)]
– Assuming the underlying asset follows the lognormal distribution, we
have the Black-Scholes-Merton formula. See Appendix on page 307.
L7: Stochastic Process
15
L7: Stochastic Process
16
One-Step Binomial Model
Stock price is currently $20 and will move either up to $22 or down to $18 at
the end of 3 months.
Consider a portfolio consisting of a long position in ∆ shares of the stock and
a short position in one call option.
22∆-1=18 ∆ 
∆=0.25
L7: Stochastic Process
17
What is the option price today?
• Once the riskless portfolio is constructed, we
can evaluate the value of the call option.
• The value of the portfolio is _____
• The value of the portfolio today is _____
• The value of the call option is _____
L7: Stochastic Process
18
Generalization
fu and fd are option value at upper or lower tree
p is the risk-neutral probability
If we know the risk-neutral probability, we can
easily obtain option price.
L7: Stochastic Process
19
Risk Neutral Valuation Revisited
-- solve the problem without u and d
• Risk neutral probability can be applied to the
stock, thus
In our example:
Option price:
L7: Stochastic Process
20
Two-Step Binomial Trees
• See page 244.
L7: Stochastic Process
21
Solution
-- value of the option
L7: Stochastic Process
22
More on Ito Processes – Product Rule
dX t  a( X t , t )dt  b( X t , t )dz
dYt  a(Yt , t )dt  b(Yt , t )dz
L7: Stochastic Process
23
Martingality
L7: Stochastic Process
24
Ito Isometry
L7: Stochastic Process
25
Continuity
L7: Stochastic Process
26
Linearity
L7: Stochastic Process
27