Variations on a theme: extensions to Black-Scholes-Merton option pricing

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Variations on a themeextensions to
Black-Scholes-Mertonoption pricing

• Dividends
• options on Forwards/Futures (Black model)
• currencies (Garman-Kohlhagen)

S. Mann, 2010
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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
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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 assets
expected rate of price appreciation. E.g. If
assets income is continuous constant proportion
y, then b r - y so that E S(T) S(0)
exp (r-y)T
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Generalized Black-Scholes-Merton
Generalized 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)
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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)TN(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
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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
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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