Enterprise Risk Management Formula Book
8. Financial derivatives
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8.1 Forward prices
The no arbitrage (fair) forward price which parties should
agree to exchange a security at time 
if it is
priced 
now and is
entitled to fixed income of present value 
in the
meantime is:
where 
is the
interest rate (continuously compounded).
If instead it pays dividends at a rate 
(continuously
compounded) then the forward prices is:
8.2 Black-Scholes formulae
Geometric Brownian motion for a security (stock) price 
involves
The partial differential equation satisfied by values of
payoffs involving such security prices is:
where is the
interest rate, is the
dividend yield (both continuously compounded) and 
is
the security price volatility.
Garman-Kohlhagen formulae for values at time 
of
European-style put and call options with strike price 
maturing
at time :
where
We then have 
, i.e. put-call
parity.
Technically the Black-Scholes formulae
are special cases of the Garman-Kohlhagen formulae for stocks that pay no
dividend, i.e. have 
, although
in practice the two names are normally treated as interchangeable.
The Black-Scholes option pricing formulae can also be
derived as the limit of binomial trees (lattices) with movements 
or 
with an
up-step probability 
and a
down-step probability 
where:
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