Black Scholes Option Pricing Model Finance (Derivative Securities) 312 Tuesday, 10 October 2006 Readings: Chapter 12

Random Walk Assumption  Consider a stock worth S  In a short period of time,  t, change in stock price is assumed to be normal with mean  S  t and standard deviation where   is expected return,  is volatility

Lognormal Property  Implies that ln S T is normally distributed with mean: and standard deviation :  Since logarithm of S T is normal, S T is lognormally distributed

Lognormal Property where Φ[m,s] is a normal distribution with mean m and standard deviation s

Lognormal Distribution

Expected Return  Expected value of stock price is S 0 e  T  Expected return on stock with continuous compounding is   –  2 /2  Arithmetic mean of returns over short periods of length  t is   Geometric mean of returns is  –  2 /2

Volatility  Volatility is standard deviation of the continuously compounded rate of return in one year  Standard deviation of return in time  t :  If stock price is $50 and volatility is 30% per year what is the standard deviation of the price change in one week? 30 x √(1/52) = 4.16% => 50 x = $2.08

Estimating Volatility  Using historical data: Take observations S 0, S 1,..., S n at intervals of  years Define the continuously compounded return as: Calculate the standard deviation, s, of the u i values Historical volatility estimate is:

Concepts Underlying Black Scholes  Option price and stock price depend on the same underlying source of uncertainty  Can form a portfolio consisting of the stock and option which eliminates this source of uncertainty Portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

Black Scholes Formulae

N ( x ) Tables  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

Black Scholes Properties  As S 0 becomes very large c tends to S – Ke –rT and p tends to zero  As S 0 becomes very small c tends to zero and p tends to Ke –rT – S

Black Scholes Example  Suppose that: Stock price in six months from expiration of option is $42 Exercise price of option is $40 Risk-free rate is 10%, volatility is 20% p.a.  What is the price of the option?

Black Scholes Example  Using Black Scholes: d 1 = , d 2 = Ke –rT = 40 e –0.1(0.5) = If European call, c = 4.76 If European put, p = 0.81  Thus, stock price must: rise by $2.76 for call holder to breakeven fall by $2.81 for put holder to breakeven

Risk-Neutral Valuation   does not appear in the Black-Scholes equation equation is independent of all variables affected by risk preference, consistent with the risk-neutral valuation principle  Assume the expected return from an asset is the risk-free rate  Calculate expected payoff from the derivative  Discount at the risk-free rate

Application to Forwards  Payoff is S T – K  Expected payoff in a risk-neutral world is Se rT – K  Present value of expected payoff is e –rT (Se rT – K) = S – Ke –rT

Implied Volatility  Implied volatility of an option is the volatility for which the Black-Scholes price equals the market price  One-to-one correspondence between prices and implied volatilities  Traders and brokers often quote implied volatilities rather than dollar prices

Effect of Dividends  European options on dividend-paying stocks valued by substituting stock price less PV of dividends into Black-Scholes formula  Only dividends with ex-dividend dates during life of option should be included  “Dividend” should be expected reduction in the stock price expected

Effect of Dividends  Suppose that: Share price is $40, ex-div dates in 2 and 5 months, with dividend value of $0.50 Exercise price of a European call is $40, maturing in 6 months Risk-free rate is 9%, volatility is 30% p.a.  What is the price of the call?

Effect of Dividends  PV of dividends 0.5 e –0.09(2/12) e –0.09(5/12) = S 0 is  d 1 = , N(d 1 ) =  d 2 = –0.0104, N(d 2 ) =  c = x – 40 e –0.09(0.5) x = $3.67

American Calls  American call on non-dividend-paying stock should never be exercised early  American call on dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date  Set American price equal to maximum of two European prices: The 1st European price is for an option maturing at the same time as the American option The 2nd European price is for an option maturing just before the final ex-dividend date