BLACKSHO - EC3070 FINANCIAL DERIVATIVES BLACK–SCHOLES...

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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES BLACK–SCHOLES OPTION PRICING The Differential Equation The Black-Scholes model of option pricing assumes that the price St of the underlying asset has a geometric Brownian motion, which is to say that dS = µS dt + σ S dw, (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 Ω æ 2 ∂f ∂f ∂f 2 21 ∂ f df = µS + +σ S dt + σ S dw. (2) 2 ∂S ∂t 2 ∂S ∂S 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 t is given by ∂f Vt = St − ft ; (3) ∂S and the change in its value is ∂f dVt = dSt − dft . (4) ∂S Substituting (1) and (2) into the latter and cancelling various terms gives Ω æ 2 ∂f 2 21 ∂ f dVt = − +σ S dt, (5) ∂t 2 ∂S2 which does not involve the stochastic increment dw. On the instant, this portfolio must earn the same as a sum Vt invested in a riskless asset at a rate of return of r, which is to say that dVt = rVt dt. By putting (5) into the LHS of this and (3) into the RHS, we get Ω æ Ω æ 2 ∂f ∂f 2 21 ∂ f +σ S dt = r ft − St dt ∂t 2 ∂S2 ∂S (6) (7) whence ∂f 1 ∂2f ∂f + σ2 S 2 +r dSt = rft . 2 ∂t 2 ∂S ∂S This it the Black-Scholes differential equation for option pricing. 1 (8) ...
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This note was uploaded on 03/02/2012 for the course EC 3070 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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