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section1 (2) - PDE for Finance Notes Spring 2011 Section 1...

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PDE for Finance Notes, Spring 2011 – Section 1. 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. Links between stochastic differential equations and PDE. A stochastic differential equation, together with its initial condition, determines a diffusion process. We can use it to define a deterministic function of space and time in two fundamentally different ways: (a) by considering the expected value of some “payoff,” as a function of the initial position and time; or (b) by considering the probability of being in a certain state at a given time, given knowl- edge of the initial state and time. Students of finance will be familiar with the Black-Scholes PDE, which amounts to an example of (a). Thus in studying topic (a) we will be exploring among other things the origin of the Black-Scholes PDE. The basic mathematical ideas here are the backward Kolmogorov equation and the Feynman-Kac formula . Viewpoint (b) is different from (a), but not unrelated. It is in fact dual to viewpoint (a), in a sense that we will make precise. The evolving probability density solves a different PDE, the forward Kolmogorov equation – which is actually the adjoint of the backward Kolmogorov equation. It is of interest to consider how and when a diffusion process crosses a barrier. This arises in thinking subjectively about stock prices (e.g. what is the probability that IBM will reach 200 at least once in the coming year?). It is also crucial for pricing barrier options. Prob- abilistically, thinking about barriers means considering exit times . On the PDE side this will lead us to consider boundary value problems for the backward and forward Kolmogorov equations. For a fairly accessible treatment of much of this material see Gardiner (the chapter on the Fokker-Planck Equation). Parts of my notes draw from Oksendal, however the treatment there is much more general and sophisticated so not easy to read. Our main tool will be Ito’s formula, coupled with the fact that any Ito integral of the form R b a f dw has expected value zero. (Equivalently: m ( t ) = R t a f dw is a martingale.) Here w is Brownian motion and f is non-anticipating. The stochastic integral is defined as the limit of Ito sums i f ( t i )( w ( t i +1 - w ( t i )) as Δ t 0. The sum has expected value zero because each of its terms does: E [ f ( t i )( w ( t i +1 ) - w ( t i ))] = E [ f ( t i )] E [ w ( t i +1 ) - w ( t i )] = 0. ******************** Expected values and the backward Kolmogorov equation . Here’s the most basic version of the story. Suppose y ( t ) solves the scalar stochastic differential equation dy = f ( y, s ) ds + g ( y, s ) dw, 1
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and let u ( x, t ) = E y ( t )= x [Φ( y ( T ))] be the expected value of some payoff Φ at maturity time T > t , given that y ( t ) = x . Then u solves u t + f ( x, t ) u x + 1 2 g 2 ( x, t ) u xx = 0 for t < T , with u ( x, T ) = Φ( x ) . (1) The proof is easy: for any function φ ( y, t ), Ito’s lemma gives d ( φ ( y ( s ) , s )) = φ y dy + 1 2 φ yy dydy + φ s ds = ( φ s + y + 1 2 g 2 φ yy ) dt + y dw.
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