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Unformatted text preview: Lecture 16 The BlackScholes PDE Consider a special case of Theorem 15.3 with a contingent claim Y = g ( S T ) — a European style derivative written on the stock price S T . Denote its time t value by P ( t,S t ), and the partial derivatives by ∂P ( t,S t ) ∂t = P t , ∂P ( t,S t ) ∂x = P x , ∂ 2 P ( t,S t ) ∂x 2 = P xx respectively. Itˆo’s formula yields e rt d [ B 1 t P ( t,S t )] = P t dt rP ( t,S t ) dt + P x dS t + 1 2 P xx ( dS t ) 2 = P x σS t dW t + • P t r P ( t,S t ) + P x rS t + 1 2 P xx σ 2 S 2 t ‚ dt, in which the risk neutral dynamics dS t = rS t dt + σS t dW t is adopted. Since B 1 t P ( t,S t ) is a Qmartingale, the coefficient of dt in the drift term has to be zero. Therefore, P ( t,S t ) must satisfy the BS PDE ∂P ( t,x ) ∂t + rx ∂P ( t,x ) ∂x + 1 2 σ 2 x 2 ∂ 2 P ( t,x ) ∂x 2 r P ( t,x ) = 0 (16.1) with the terminal condition P ( T,x ) = g ( x )....
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 Spring '08
 Staff
 Boundary value problem, Mathematical finance, riskneutral measure, BS PDE

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