Variations on a theme: extensions to Black-Scholes-Merton option pricing Dividends options on Forwards/Futures (Black model) currencies (Garman-Kohlhagen) S. Mann, 2010 S. Mann, 2010

Martingale pricing : risk-neutral Drift Risk-neutral pricing: we model prices as Martingales with respect to the riskless return. For lognormal evolution on asset with no dividends, this requires drift to be: m = r - s2/2 where r is riskless return Since E[ S(T)] = S(0)exp[ mT + s2T/2]) = S(0)exp[ (r - s2/2)T + s2T/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: m = b - s2/2 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 Black-Scholes-Merton model (European Call): C = exp(-rT)[S exp(bT) N(d1) - K N(d2)] where ln(S/K) + (b + s2/2)T d1 = and d2 = d1 - sT sT e.g., for non-dividend paying asset, set b = r “Black-Scholes” C = S N(d1) - exp (-rT) K N(d2) S. Mann, 2010

Constant dividend yield stock option (Merton, 1973) Generalized Black-Scholes-Merton model (European Call): set b = r - dy where dy = continuous dividend yield then 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 + s2/2)T d1 = and d2 = d1 - sT sT S. Mann, 2010

Black (1976) model: options on futures/forwards Expected price appreciation rate is zero: set b = 0, replace S with F then C = exp(-rT) [ F exp(0T) N(d1) - K N(d2)] =exp(-rT) [ FN(d1) - K N(d2)] where ln(F/K) + (s2T/2) d1 = and d2 = d1 - sT sT Note that F = S exp [(r - dy)T] S. Mann, 2010

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 + s2T/2) d1 = and d2 = d1 - sT sT S. Mann, 2010