The Black-Scholes-Merton Model Chapter 13 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Random Walk We begin be considering the discrete-time random walk description: W(t+1)=W(t) + e(t+1); W(0)=W0 e: i.i.d. N(0,1) The variable t represent time and is measured in discrete integer increments. W(t) is the level of the cumulant of e(t); it is called a random walk because it appears that W takes random steps up and down through time. We would like to describe a process that has the same characteristics as the random walk but observed more frequently: W(t+Δ)=W(t) + e(t+Δ); W(0)=W0 e: i.i.d. N(0,Δ) Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

W(t+dt)=W(t) + e(t+dt); Random Walk Now consider the behavior of the process as Δ → dt W(t+dt)=W(t) + e(t+dt); W(0)=W0 e: i.i.d. N(0,dt) Define dW(t) = W(t+dt)-W(t). We heuristically define dt as the smallest positive real number such that dta=0, whenever a>1. Either of these processes, dW(t) or e(t+dt), is referred to as a white noise. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Wiener process A variable W follows a Wiener process if it has the following three properties: The change dW during a small period of time dt is dW=e dt1/2 where e has a standardized normal distribution N(0,1) The values of dW for any two short intervals of time are independent. dW is normally distributed with mean 0 and variance dt. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Random Walk Recall that dW may be thought of as a normally distributed random variable with mean zero and variance dt. We note three properties that follow by construction: 1.      E[dW(t)]=0, (mean) 2.      E(dW(t)dt]=E[dW(t)]dt=0 3.      E[dW(t)2]=dt, (variance) These properties give rise to two multiplication rules: Rule 1: dW(t)2=dt Rule 2: dW(t)dt=0 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

An Ito Process for Stock Prices: Geometric Brownian Motion where m is the expected return s is the volatility. The discrete time equivalent is Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Properties of S The process of S has the following properties 1. If S starts with a positive value, it will remain positive. 2.  S has an absorbing barrier at 0: Thus, if S hits 0, then S will remain at zero. 3. The conditional distribution of Su given St is lognormal. The conditional mean of ln(Su) for u>t is ln(St)+ μ(u-t) – (1/2) σ2(u-t) and the conditional standard deviation of ln(Su) is σ(u-t)1/2. ln(Su) is normally distributed. The conditional expected value of Su is St exp[μ(u-t)]. 4.   The variance of a forecast Su tends to infinity as u does. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Lognormal Property (Equations 13.2 and 13.3, page 282) It follows from this assumption that Since the logarithm of ST is normal, ST is lognormally distributed Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Example Consider a stock with an initial price of 40$, an expected return of 16% per annum, and a volatility of 20% per annum. From the above equation, the probability distribution of the stock price, ST, in six months time is given by lnST ~ Φ[lnS0 + (μ-σ2/2)T, σ(T)1/2], lnST ~ Φ[ln40 + (0.16-0.22/2)0.5, 0.2(0.5)1/2] lnST ~ Φ[3.759,0.141] There is a 95% probability that a normally distributed variable has a value within 1.96 standard deviations of its mean. Hence, with 95% confidence interval, 3.759  - 1.96*0.141< lnST <3.759 + 1.96*0.141 This can be written e(3.759-1.96*0.141) < ST < e(3.759+1.96*0.141) , or 32.55 < ST < 56.56 Thus, there is a 95% probability that the stock price in six months will lie between 32.55 and 56.56. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Lognormal Distribution Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Monte Carlo Simulation We can sample random paths for the stock price by sampling values for e Suppose m= 0.14, s= 0.20, and Dt = 0.01, then Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Monte Carlo Simulation – One Path (See Table 12.1, page 272) Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Itô’s Lemma (See pages 273-274) If we know the stochastic process followed by S, Itô’s lemma tells us the stochastic process followed by some function f (S, t ) Since a derivative security is a function of the price of the underlying and time, Itô’s lemma plays an important part in the analysis of derivative securities Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Taylor Series Expansion A Taylor’s series expansion of f(S, t) gives Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Application of Ito’s Lemma to a Stock Price Process And Since dS = μSdt + σS dW, We have that Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Concepts Underlying Black-Scholes The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Derivation of the Black-Scholes Differential Equation It follows that by choosing a portfolio of the stock and the derivative, the Wiener process can be eliminated. The appropriate portfolio is -1 : derivative + : shares The holder of this portfolio is short one derivative and long an amount of shares. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Derivation of the Black-Scholes Differential Equation continued Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Derivation of the Black-Scholes Differential Equation continued Substituting the above equations for Δf and ΔS to the last one, we have Because the equation does not involve ΔW, the portfolio must be riskless during time Δt. Thus, the portfolio must instantaneously earn the same rate of return as other short-term risk-free security. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Derivation of the Black-Scholes Differential Equation continued Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Differential Equation Any security whose price is dependent on the stock price satisfies the differential equation The particular security being valued is determined by the boundary conditions of the differential equation In the case of call options the boundary condition is ƒ = max(S – K, 0) when t =T In the case of put options the boundary condition is ƒ = max(K - S, 0) when t =T Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The Black-Scholes Formulas (See pages 295-297) Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

The N(x) Function N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x See tables at the end of the book Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Properties of Black-Scholes Formula As S0 becomes very large c tends to S – Ke-rT and p tends to zero As S0 becomes very small c tends to zero and p tends to Ke-rT – S Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Risk-Neutral Valuation The variable m does not appear in the Black-Scholes equation The equation is independent of all variables affected by risk preference The solution to the differential equation is therefore the same in a risk-free world as it is in the real world This leads to the principle of risk-neutral valuation Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Example Consider the situation where the stock price six months from the expiration of an option is 42. The exercise price of the option is 40. The risk-free interest rate is 10% per annum. The volatility is 20% per annum. This means that S0=42, X=40, r=01, σ=0.2 and T=0.5. Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Example (Continued) Then =0.7693-0.2 0.51/2=0.6278 Xe-rt =40 Xe-0.05 =38.049 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Example (Continued) Hence, if the option is a European call, its value c is given by c = 42 N(0.7693) - 38.049 N(0.6278) And if the option is a European put option, its value, p, is given by p = 38.049 N(-0.6278) – 42 N(-0.7693) Using the polynomial approximation, N(0.7693) = 0.7791 N(-0.7693) = 0.2209 N(0.6278) = 0.7349 N(-0.6278) = 0.2651 So that, c=4.76 and p=0.81 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005