1. 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, 2017
S. Mann, 2017

2. Lognormal returns: Weiner Process
Model returns as
S(T) = S(0) exp [ mT + 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[ mT + s2T/2])
In order for the expected return to be equal to the riskless rate,
We must adjust m. If we set
m = r - s2/2
where r is the riskless rate, then we achieve our goal:
For lognormal returns:
= S(0)exp[ (r - s2/2)T + s2T/2])
= S(0)exp[ rT ]
S. Mann, 2017

3. 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 - s2/2
where b is asset’s expected rate of price appreciation.
E.g. If asset’s income payout rate (dividend/yield) is d,
then b = r – d and m = r – d - s2/2
E[S(T)] = S(0) exp[ mT + s2T/2])
= S(0) exp[ (r – d - s2/2)T + s2T/2])
= S(0) exp [ (r-d)T]

4. 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, 2017

5. 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(-rT) [ S exp{(r- d)T}N(d1) - K N(d2)]
= S exp(-rT + rT -dT) N(d1) - exp(-rT) K N(d2)
= S exp(-dT) N(d1) - exp(-rT) K N(d2)
where
ln(S/K) + (r - d + s2/2)T
d1 = and d2 = d1 - sT
sT
S. Mann, 2017

6. 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 – d )T]
S. Mann, 2017

7. 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 ($/FX)
then
C = exp(-rT) [ S exp[(r - rf)T] N(d1) - K N(d2)]
= exp(-rfT) S N(d1) - exp(-rT) K N(d2)
= Zf(0,T) S N(d1) - Z$(0,T) K N(d2)
where
ln(S/K) + (r -rf + s2T/2)
d1 = and d2 = d1 - sT
sT
S. Mann, 2017