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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: 