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### Enterprise Risk Management Formula Book 8. Financial derivatives

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 :

 Call option Put option

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: