L7: Stochastic Process 1 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 2 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 3 Wiener Process Property 1: The change Δz during a small period of time Δt is, 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, –. –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 4 Wiener process, aka standard Brownian motion

L7: Stochastic Process 5 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 –b 2 =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) = b 2 Standard deviation = b

L7: Stochastic Process 6 Ito’s Process The process for a stock: –ds/s=μdt+σdz Discrete time Model:

L7: Stochastic Process 7 Ito’s Lemma Assume G=f(x,t) Ito Lemma: For stocks: We have: Stochastic differential equation (SDE)

Interesting Property of SDE L7: Stochastic Process 8

9 Deriving Ito Lemma Using Taylor Expansion, we have: Insert dx and dx 2 in dG we have Ito Lemma.

L7: Stochastic Process 10 The Lognormal Property G=lnS We have: – (12.17) – – (13.1)

L7: Stochastic Process 11 Black-Scholes-Merton Differential Equation (1) Stock price is assume to follow the following process: Suppose that f is the price of a call option or other derivative contingent on S. We construct a portfolio of the stock and the derivative (page 287):

L7: Stochastic Process 12 Black-Scholes-Merton Differential Equation (2) We then have: The portfolio is risk free. Thus, Putting all together, we have: 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

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

L7: Stochastic Process 14 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 S T -K at the maturity date –Based on risk-neutral valuation, we have the value of a forward contract: f=S 0 -Ke -rT

L7: Stochastic Process 15 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: –Call option price c is: –Assuming the underlying asset follows the lognormal distribution, we have the Black-Scholes-Merton formula. See Appendix on page 307.

L7: Stochastic Process 16

One-Step Binomial Model L7: Stochastic Process 17 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

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 L7: Stochastic Process 19 f u and f d 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.

Risk Neutral Valuation Revisited -- solve the problem without u and d Risk neutral probability can be applied to the stock, thus L7: Stochastic Process 20 In our example: Option price:

Two-Step Binomial Trees See page 244. L7: Stochastic Process 21

Solution L7: Stochastic Process value of the option

More on Ito Processes – Product Rule 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