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 :


Call option


Put option





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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