Lognormal Returns and drift Extensions to BlackScholesMerton option
Lognormal Returns and “drift” Extensions to Black-Scholes-Merton option pricing • • • Lognormal return simulation (Weiner process) Risk-neutral “drift” Dividends options on Forwards/Futures (Black model) currencies (Garman-Kohlhagen) S. Mann, 2016
Lognormal returns: Weiner Process Model returns as S(T) = S(0) exp [ m. T + s √T x Z*(0, 1)] Where Z*(0, 1) = a random draw from the N(0, 1) distribution The expected Value of S(T) will be: E[ S(T)] = S(0)exp[ m. T + s 2 T/2]) In order for the expected return to be equal to the riskless rate, We must adjust m. If we set m = r - s 2/2 where r is the riskless rate, then we achieve our goal: For lognormal returns: E[ S(T)] = S(0)exp[ m. T + s 2 T/2]) = S(0)exp[ (r - s 2/2)T + s 2 T/2]) = S(0)exp[ r. T ]
Generalized risk-neutral Drift Risk-neutral pricing: prices are a “martingale” (expected value next period is today’s + riskless interest) 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 - s 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 – d and m = r – d - s 2/2 E[S(T)] = S(0) exp[ m. T + s 2 T/2]) = S(0) exp[ (r – d - s 2/2)T + s 2 T/2]) = S(0) exp [ (r-d)T]
Generalized Black-Scholes-Merton model (European Call): where d 1 = C = exp(-r. T)[S exp(b. T) N(d 1) - K N(d 2)] ln(S/K) + (b + s 2/2)T s T and d 2 = d 1 - s T e. g. , for non-dividend paying asset, set b = r “Black-Scholes” C = S N(d 1) - exp (-r. T) K N(d 2)
Constant dividend yield stock option (Merton, 1973) Generalized Black-Scholes-Merton model (European Call): set b = r - d where d = dividend yield then C = exp(-r. T) [ S exp{(r- d)T}N(d 1) - K N(d 2)] = S exp(-r. T + r. T -d. T) N(d 1) - exp(-r. T) K N(d 2) = S exp(-d. T) N(d 1) where d 1 = - exp(-r. T) K N(d 2) ln(S/K) + (r - d + s 2/2)T s T and d 2 = d 1 - s T
Black (1976) model: options on futures/forwards Expected price appreciation rate is zero: set b = 0, replace S with F then C = exp(-r. T) [ F exp(0 T) N(d 1) - K N(d 2)] =exp(-r. T) [ FN(d 1) - K N(d 2)] where ln(F/K) + (s 2 T/2) d 1 = s T Note that F = S exp [(r – d )T] and d 2 = d 1 - s 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 ($/FX) then C = exp(-r. T) [ S exp[(r - rf)T] N(d 1) - K N(d 2)] = exp(-rf. T) S N(d 1) - exp(-r. T) K N(d 2) = Zf(0, T) S N(d 1) - Z$(0, T) K N(d 2) where ln(S/K) + (r -rf + s 2 T/2) d 1 = and d 2 = d 1 - s T
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