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section3 (2) - PDE for Finance Notes Spring 2011 Section 3...

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PDE for Finance Notes, Spring 2011 – Section 3. 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 and corrections made in 2011. More about linear PDE’s: the heat equation on an interval; uniqueness via the maximum principle; solution by finite differences. Section 2 gave the solution formula for the heat equation in a half-space – the essential tool for pricing barrier options. The natural next topic is to consider the heat equation in an interval – the essential tool for pricing double-barrier options. Then we turn to the question of uniqueness, something we’ve been assuming up to now. The proof uses the maximum principle – a basic and powerful tool for obtaining qualitative information about solutions of parabolic equations. Finally we discuss the most basic finite-difference scheme for solving the linear heat equation on an interval. This PDE material is quite standard. Our treatment of the heat equation in an interval is by separation of variables; this can be found in most basic PDE books. Uniqueness via the maximum principle is also quite standard; students of finance may find it convenient to read this in Steele’s book (the relevant section is in the middle of the book but is quite self-contained). Our centered finite-difference scheme for the heat equation is again in most PDE books; students of finance will recognize its resemblance to a trinomial tree for solving the Black-Scholes PDE. Our separation-of-variables approach to the heat equation on an interval can be a useful tool for pricing certain classes of options. See D. Davydov and V. Linetsky, Pricing options on scalar diffusions: an eigenfunction expansion approach , Operations Research 51, 2003, 185-209, available through JSTOR. ************ The heat equation in an interval. A double-barrier option has both an upper and lower barrier. Its value satisfies the Black-Scholes PDE with appropriate boundary data at the barriers. If the underlying is lognormal then this problem can be reduced, as shown in Section 2, to solving the linear heat equation on an interval with appropriate initial and boundary data. For simplicity we focus on the case when the interval is 0 < x < 1. Thus we wish to solve: u t = u xx for t > 0 and 0 < x < 1 with initial data u = g at t = 0, and boundary data u = φ 0 at x = 0, u = φ 1 at x = 1. Consider first the case when the boundary condition is zero, i.e. φ 0 = φ 1 = 0. We will use the following basic fact: any function f ( x ) defined for 0 < x < 1 and vanishing at the endpoints x = 0 , 1 can be expanded as a Fourier sine series: f ( x ) = X k =1 a n sin( nπx ); (1) 1
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moreover the coefficients a n are determined by f via a n = 2 Z 1 0 f ( x ) sin( nπx ) dx. (2) If you accept (1), the justification of (2) is easy. It follows from the fact that the functions { sin( nπx ) } n =1 are orthogonal in the inner product ( f, g ) = R 1 0 f ( x ) g ( x ) dx , and each has norm 1 / 2.
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