Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) ...
View Full Document

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.

Ask a homework question - tutors are online