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section2 (2)

# section2 (2) - PDE for Finance Notes Spring 2011 Section 2...

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PDE for Finance Notes, Spring 2011 – Section 2. Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates made in 2011. Solution formulas for the linear heat equation. Applications to barrier options. Section 1 was relatively abstract – we listed many PDE’s but solved just a few of them. This section has the opposite character: we discuss explicit solution formulas for the linear heat equation – both the initial value problem in all space and the initial-boundary-value problem in a halfspace. This is no mere academic exercise, because the (constant-volatility, constant-interest-rate) Black-Scholes PDE can be reduced to the linear heat equation. As a result, our analysis provides all the essential ingredients for valuing barrier options. The PDE material here is fairly standard – most of it can be found in John or Evans or Strauss (among other places). Our discussion of the half-space problem with initial condition 0 follows the treatment by John. For the financial topics (reduction of Black-Scholes to the linear heat equations; valuation of barrier options) see e.g. the “student guide” by Wilmott, Howison, and Dewynne. **************************** The heat equation and the Black-Scholes PDE. We’ve seen that linear parabolic equa- tions arise as backward Kolmogorov equations, determining the expected values of various payoffs. They also arise as forward Kolmogorov equations, determinining the probability distribution of the diffusing state. The simplest special cases are the backward and forward linear heat equations u t + 1 2 σ 2 Δ u = 0 and p s - 1 2 σ 2 Δ p = 0, which are the backward and forward Kolmogorov equations for dy = σdw , i.e. for Brownian motion scaled by a factor of σ . From a PDE viewpoint the two equations are equivalent, since v ( t, x ) = u ( T - t, x ) solves v t - 1 2 σ 2 Δ v = 0, and final-time data for u at t = T determines initial-time data for v at t = 0. This basic example has direct financial relevance, because the Black-Scholes PDE can be reduced to it by a simple change of variables. Indeed, the Black-Scholes PDE is V t + rsV s + 1 2 σ 2 s 2 V ss - rV = 0 . It is to be solved for t < T , with specified final-time data V ( s, T ) = Φ( s ). (Don’t be confused: in the last paragraph s was time, but here it is the “spatial variable” of the Black- Scholes PDE, i.e. the stock price.) We claim this is simply the standard heat equation u t = u xx written in special variables. To see this, consider the preliminary change of variables ( s, t ) ( x, τ ) defined by s = e x , τ = 1 2 σ 2 ( T - t ) , and let v ( x, τ ) = V ( s, t ). An elementary calculation shows that the Black-Scholes equation becomes v τ - v xx + (1 - k ) v x + kv = 0 1

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with k = r/ ( 1 2 σ 2 ). We’ve done the main part of the job: reduction to a constant-coefficient equation. For the rest, consider u ( x, t ) defined by v = e αx + βτ u ( x, τ ) where α and β are constants. The equation for v becomes an equation for u , namely ( βu + u τ ) - ( α 2 u + 2 αu x + u xx ) + (1 - k )( αu + u x ) + ku = 0 .
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section2 (2) - PDE for Finance Notes Spring 2011 Section 2...

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