1. 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

2. 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 ]

3. 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]

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, 2010

5. 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

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 - dy)T]
S. Mann, 2010

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 ($/FX)
then
C = exp(-rT) [ S exp[(r - rf)T] N(d1) - K N(d2)]
= exp(-rfT) S N(d1) - exp(-rT) K N(d2)
= Bf(0,T) S N(d1) - B$(0,T) K N(d2)
where
ln(S/K) + (r -rf + s2T/2)
d1 = and d2 = d1 - sT
sT
S. Mann, 2010