Black Scholes Black Scholes Option Pricing Option Pricing

ModelModel Finance (Derivative Securities) 312

Tuesday, 10 October 2006

Readings: Chapter 12

Random Walk Random Walk AssumptionAssumption

Consider a stock worth SIn a short period of time, t, change in

stock price is assumed to be normal with mean St and standard deviation

where is expected return, is volatility

tS

Lognormal PropertyLognormal Property

Implies that ln ST is normally distributed with mean:

and standard deviation:

Since logarithm of ST is normal, ST is lognormally distributed

TS )2/(ln 20

T

Lognormal PropertyLognormal Property

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

TTS

S

TTSS

T

T

,)2(ln

or

,)2(lnln

2

0

20

Lognormal DistributionLognormal Distribution

E S S e

S S e e

TT

TT T

( )

( ) ( )

0

02 2 2

1

var

Expected ReturnExpected Return

Expected value of stock price is S0eT

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

VolatilityVolatility

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 0.0416 = $2.08

t

Estimating VolatilityEstimating Volatility

Using historical data:• Take observations S0, S1, . . . , Sn at intervals of years

• Define the continuously compounded return as:

• Calculate the standard deviation, s , of the ui values

• Historical volatility estimate is:

uS

Sii

i

ln1

sˆ

Concepts Underlying Black Concepts Underlying Black ScholesScholes

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 FormulaeBlack Scholes Formulae

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where

NN((xx) ) TablesTables

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 PropertiesBlack Scholes Properties

P

A

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

Black Scholes ExampleBlack 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 ExampleBlack Scholes Example

Using Black Scholes:• d1 = 0.7693, d2 = 0.6278

• Ke–rT = 40e–0.1(0.5) = 38.049 • 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 ValuationRisk-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 ForwardsApplication to Forwards

Payoff is ST – K

Expected payoff in a risk-neutral world is SerT – K

Present value of expected payoff is

e–rT(SerT – K) = S – Ke–rT

Implied VolatilityImplied 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 DividendsEffect 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 DividendsEffect 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 DividendsEffect of Dividends

PV of dividends• 0.5e–0.09(2/12) + 0.5e–0.09(5/12) = 0.9741

• S0 is 39.0259

d1 = 0.2017, N(d1) = 0.5800

d2 = –0.0104, N(d2) = 0.4959

c = 39.0259 x 0.5800 – 40e–0.09(0.5) x 0.4959

= $3.67

American CallsAmerican 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