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Unformatted text preview: PDE for Finance Notes, Spring 2011 – Section 8 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use only in connection with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates made in 2011. Underlyings with jumps . We began the semester studying diffusions and the associated linear PDE’s, namely the backward and forward Kolmogorov equations. This section asks: what becomes of that story when the underlying process has jumps? The answer, briefly, is that the backward and forward Kolmogorov equations become integrodifferential equations. In general they must be solved numerically, but in the constant-coefficient setting they can be solved using the Fourier transform. One application (though perhaps not the most important one) is to option pricing. But be careful: when the underlying can jump the market is not complete; so arbitrage alone does not determine the value of options. Why, then, should an option have a well-defined price? The answer suggested by Merton: if the extra randomness due to jumps is uncorrelated with the market (i.e. if its β is zero) then its stochasticity can be made negligible by diversification, so that (by the Capital Asset Pricing Model) only the average effect of the jumps is important for pricing. For basic issues involving processes with jumps and option pricing, I strongly recommend Merton’s 1976 article “Option pricing when underlying stock returns are discontinuous,” J. Financial Economics 3, 1976, 125-144 (reprinted as Chapter 9 of the book Continuous Time Finance [a collection of Merton’s papers] and also available from Merton’s website http://www.people.hbs.edu/rmerton/optionpricingwhenunderlingstock.pdf). The Fourier Transform is discussed in many books; a fairly basic treatment can be found in H.F. Wein- berger, A First Course in Partial Differential Equations with Complex Variables and Trans- form Methods (an inexpensive Dover paperback). The jump-diffusions we’ll discuss here are the most basic of a huge family of processes with jumps that are sometimes used for option pricing. For other processes and a modern summary of the theory, see Rama Cont and Peter Tankov, Financial Modeling with Jump Processes, Chapman and Hall 2003 (a 2nd edition will be coming out soon). ******************** Jump-diffusion processes . The standard (constant-volatility) Black-Scholes model as- sumes that the logarithm of an asset price is normally distributed. In practice however the observed distributions are not normal – they have “fat tails,” i.e. the probability of a very large positive or negative change is (though small) much larger than permitted by a Gaussian. The jump-diffusion model provides a plausible mechanism for explaining the fat tails and their consequences....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
- Fall '11