Variations on a theme:extensions to Black-Scholes-Mertonoption pricing

• Dividends• options on Futures (Black model)• currencies (Garman-Kohlhagen)

Finance 70520, Spring 2001Risk Management & Financial EngineeringThe Neeley School S. Mann

Martingale pricing : risk-neutral Drift

Risk-neutral pricing: we model prices asMartingales with respect to the riskless return.

For lognormal evolution on asset with no dividends, this requires drift to be:

= r - 2/2

where r is riskless return

Since E[ S(T)] = S(0)exp[ T + 2T/2]) = S(0)exp[ (r - 2/2)T + 2T/2]) = S(0)exp[ rT ]

Generalized risk-neutral Drift

Risk-neutral pricing:Implication: all assets have same expected rate of return.Not implied: all assets have same rate of price appreciation.

(some pay income)Generalized drift:

= b - 2/2

where b is asset’s expected rate of price appreciation.

E.g. If asset’s income is continuous constant proportion y,then b = r - y

so that E[ S(T) ] = S(0) exp [ (r-y)T]

Generalized Black-Scholes-Merton

Generalized Black-Scholes-Merton model (European Call):

C = exp(-rT)[S exp(bT) N(d1) - K N(d2)]where

ln(S/K) + (b + 2/2)Td1 = and d2 = d1 - T

T

e.g., for non-dividend paying asset, set b = r

“Black-Scholes” C = S N(d1) - exp (-rT) K N(d2)

Constant dividend yield stock option (Merton, 1973)

Generalized Black-Scholes-Merton model (European Call):

set b = r - dy where dy = continuous dividend yieldthen

C = exp(-rT) [ S exp{(r- dy)T}N(d1) - K N(d2)] = S exp(-rT + rT -dyT) N(d1) - exp(-rT) K N(d2)

= S exp(-dyT) N(d1) - exp(-rT) K N(d2)

where ln(S/K) + (r - dy + 2/2)T

d1 = and d2 = d1 - T T

Black (1976) model: options on futures

Expected price appreciation rate is zero:

set b = 0, replace S with Fthen

C = exp(-rT) [ F exp(0T) N(d1) - K N(d2)] =exp(-rT) [ FN(d1) - K N(d2)]where

ln(F/K) + (2T/2)d1 = and d2 = d1 - T

T

Note that F = S exp [(r - dy)T]

Options on foreign currency (FX): Garman-Kohlhagen (1983)

Expected price appreciation rate is domestic interest rate, r , less foreign interest rate, rf.

set b = r - rf, Let S = Spot exchange rate ($/FX)then

C = exp(-rT) [ S exp[(r - rf)T] N(d1) - K N(d2)] = exp(-rfT) S N(d1) - exp(-rT) K N(d2) = Bf(0,T) S N(d1) - B$(0,T) K N(d2) where

ln(S/K) + (r -rf + 2T/2)d1 = and d2 = d1 - T

T