EC3070 FINANCIAL DERIVATIVESBLACK–SCHOLES OPTION PRICINGThe Differential EquationThe Black-Scholes model of option pricing as-sumes that the priceStof the underlying asset has a geometric Brownianmotion, which is to say thatdS=μSdt+σSdw,(1)whereμandσare constant parameters.Letf=f(S, t) be the price of a derivative, which might be a call optioncontingent on the priceSof the underlying asset.Then, according to Ito’slemma, there isdf=ΩμS∂f∂S+∂f∂t+σ2S212∂2f∂S2ædt+σS∂f∂Sdw.(2)Because the forcing function in both equations (1) and (2) is the sameWiener process, it is possible to construct a portfolio that eliminates risk, whichis to say that the Wiener incrementdwcan be eliminated from the equationexpressing the value of the portfolio.The relevant portfolio comprises∂f/∂Sunits of the asset and 1 unit ofthe derivative or option as a liability. The value of the portfolio at time
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