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EC3070 FINANCIAL DERIVATIVES BLACK–SCHOLES OPTION PRICING The Di ff erential Equation The Black-Scholes model of option pricing as- sumes that the price S t of the underlying asset has a geometric Brownian motion, which is to say that dS = μSdt + σ Sdw, (1) where μ and σ are constant parameters. Let f = f ( S, t ) be the price of a derivative, which might be a call option contingent on the price S of the underlying asset. Then, according to Ito’s lemma, there is df = Ω μS f S + f t + σ 2 S 2 1 2 2 f S 2 æ dt + σ S f S dw. (2) Because the forcing function in both equations (1) and (2) is the same Wiener process, it is possible to construct a portfolio that eliminates risk, which is to say that the Wiener increment dw can be eliminated from the equation expressing the value of the portfolio. The relevant portfolio comprises f/ S units of the asset and 1 unit of the derivative or option as a liability. The value of the portfolio at time
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