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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.