1. S.Mann, 2014

2. 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

3. Simulated
lognormal returns
and
Lognormal prices

4. S.Mann, 2014