Valuing Stock Options:

The Black-Scholes-Merton Model

The Black-Scholes-Merton Random Walk Assumption

• Consider a stock whose price is S

• In a short period of time of length the return on the stock is assumed to be normal with mean and standard deviation

• m is expected return and s is volatility

t

The Lognormal Distribution

The Volatility

• The volatility is the standard deviation of the continuously compounded rate of return in 1 year

• The standard deviation of the return in time Dt is

• If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?

t

Nature of Volatility

• Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed

• For this reason time is usually measured in “trading days” not calendar days when options are valued.

• Calendar days consists of total 365 days but from now on we are going to use trading days.

• There are 252 days in Trading Days

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

Assumptions of Black-Scholes-Merton Formula

1. Stock price behaviour corresponds to the lognormal model

2. There are no transactions costs or taxes. All securities are perfectly divisible.

3. There are no dividends on the stock during the life of the option. (We will change this assumption further in the chapter)

4. There are no riskless arbitrage opportunities.

5. Security trading is continuous.

6. Investors can borrow or lend at the same risk-free rate of interest.

7. The short-term risk-free rate of interest, r, is constant.

The Black-Scholes Formulas

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where

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

Properties of Black-Scholes Formula

• As S0 becomes very large c tends to S0 – Ke-rT and p tends to zero

• As S0 becomes very small c tends to zero and p tends to Ke-

rT – S0

• What happens as s becomes very large?

• What happens as T becomes very large?

Example

• 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, and the

• volatility is 20% per annum.

• What is the price of an European Call and European Put?

Implied Volatility

• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price

• The is a one-to-one correspondence between prices and implied volatilities

Dividends

• European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into the Black-Scholes-Merton formula

• Only dividends with ex-dividend dates during life of option should be included

• The “dividend” should be the expected reduction in the stock price on the ex-dividend date

Example with Dividends

Consider a European call option on a stock with ex-dividend dates in two months

and five months. The dividend on each ex-dividend date is expected to be $0.50.

The current share price is $40, the exercise price is $40, the stock price volatility is

30% per annum, the risk-free rate of interest is 9% per annum, and the time to

maturity is six months. What is the price of the European Call?