Black and Scholes Formula

For European Options

Stock Price Dynamics

• Suppose that the price of the stock satisfies:

tttt dWSdtSdS is the expected return. is the volatility.– Both are constant.

• Value of S at moment T:

TWT

T eSS )2

1(

0

2

Lognormal Distribution

• Graphic representation:

0

Bond Price Dynamics

• There is a bond or checking account that satisfies:

rTT eBB 0

rdtBdB tt

– r is the continuously paid interest rate.– It is constant.

• The price of the bond at moment T is:

European Call Option Dynamics

• Consider an European call on S with strike price X and maturity at T.

• The price C will be a function of time (or time left to maturity) and S: C(S,t).

• By Ito’s Lemma:

dtSS

tSCdS

S

tSCdt

t

tSC

tSdC

222

2 ),(

2

1),(),(

),(

European Call Dynamics (cont.)

• The previous expression is equivalent to:

dWSdtSS

tSC

dtSS

tSCdt

t

tSCtSdC

),(

),(

2

1),(),( 22

2

2

• Suppose we form a portfolio with the option and the stock but without uncertainty term:– That portfolio would be riskless.– Its expected return should be the riskfree rate.

BS differential equation

• After constructing such portfolio we are left with:

rCSrS

CS

S

C

t

C

22

2

2

2

1

• Subject to the following condition at maturity:

)0,(),( XSMaxTSC TT

Black and Scholes formula

• Solution to the previous equation:

)()( tdNXedΝSC rt

• Where:

t

trXSd

)2

1()/log( 2

• r is the continuously compounded interest. is the volatility of the return.

Black and Scholes (cont.)

• N(d) is the cumulative normal distribution:

d

0

Black and Scholes (cont.)

• N(d) is the “delta” or number of shares (smaller than one) needed to replicate it.

• e-rtX is the present value of X.• Price of the European put: we can get it from the put-

call parity:

)()( dΝSdtNXeP rt

Risk-neutral valuation

• Suppose that the stock satisfies the following dynamics:

tttt dWSrdtSdS

• BS is the result of:

XSEeC TrT ,0max

• As in the binomial case.

• This will allow simple numerical methods.

Assumptions of BS

• Continuous and constant interest rate.

• Constant expected return : It does not appear in the BS formula.

• Constant standard deviation :Very restrictive.

• Frictionless markets.

• Unlimited borrowing/shortselling possibilities.

Graph: European, American call

Call option price

X Stock price

S-X

Graph: American put

Put option price

X Stock price

X-S

Early exercise

Computing volatility is the only parameter not directly

observable.

• Typically, estimated from past data.

• Volatility of the return, not of the price:

t

ttt

tt

t

t

tt

S

SSS

SdS

dS

S

SS

11

1

logloglog

)(log

Computing volatility (cont.)

• We compute the standard deviation of previous expression (say s).

• We then derive by adjusting the time period.

• For example, if we have considered daily returns:

sσ

s...sssσ

365

365 22222

Implied volatility

• Concept:– Consider all the observed values.– Including the price of the option.– It is the volatility for which the BS formula

would yield that price.

• In some markets, implied volatility quoted (instead of price of option).

• Provide information about the market:Different options on same stock can differ.

European options with dividends

• We assume the dividend and date of payment are known.

• Dividend is a “riskless component” of price of stock.

• We subtract the present value of the dividend and apply BS to the rest.

American options with dividends

• For put options, it could be optimal to exercise before maturity, with or without dividends:– With dividends, only after dividend is paid, if

around dividend date.

• For calls, only can be before dividend is paid, but, if dividend is too small, it is not optimal:– From put-call parity, if:

)(XPVXD It will not be optimal to exercise early.

Black’s approximation for calls

• We need:– Estimate of the dividend.– Date to be paid.

• Two different prices are computed:– Value if held until maturity.– Value if early exercise.

• We pick the maximum of them.

Black’s approximation (cont.)

A If held until maturity:1 Compute:

)(* DPVSS

2 Compute Black and Scholes with S* instead of S.

Black’s approximation (cont.)

B If early exercise:

1 Compute S* (as before).

2 Use the Black and Scholes formula but:– With S* instead of S.– With the time to dividend payment instead of

time to maturity.– With strike price X-D.