S.Mann, 2014

Black-Scholes-Merton Model Assumptions: Asset pays no dividends; European call No taxes or transaction costs Constant interest rate over option life Lognormal returns: ln(1+r ) ~ N (m , s) reflect limited liability -100% is lowest possible stable return variance over option life C = S N(d1 ) - K Z(0,t) N(d2 ) ln (S/K) + (r + s2/2 )t d1 = s t d2 = d1 - s t N( x) = Standard Normal [~N(0,1)] Cumulative density function: N(x) = area under curve left of x; e.g., N(0) = .5 coding: (excel) N(x) = NormSdist(x) N(d1 ) = Call Delta (D) = call hedge ratio = change in call value for small change in asset value = slope of call: first derivative of call with respect to asset price Z(0,T) = PV of $1.00 S.Mann, 2014

Simulated lognormal returns and Lognormal prices

S.Mann, 2014