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Unformatted text preview: Continuous Time Finance Notes, Spring 2004 – Section 10, April 7, 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. Jump-diffusion models. Merton was the first to explore option pricing when the underly- ing follows a jump-diffusion model. His 1976 article “Option pricing when underlying stock returns are discontinuous” (reprinted as Chapter 9 of his book Continuous Time Finance ) is a pleasure to read. My notes follow it – but the article contains much more information than I’m presenting here. Much has happened since 1976 of course; a recent reference is A. Lipton, Assets with jumps , RISK , Sept. 2002, 149-153. I will begin with an introduction to jump-diffusions. Then I’ll discuss option pricing using such models. This cannot be done using absence of arbitrage alone: when the underlying can jump the market is not complete, since there are two sources of noise (the diffusion and the jumps) but just one tradeable (the underlying). How, then, can we price options? Merton’s proposal (still controversial) was to assume that the extra randomness due to jumps is uncorrelated with the market – i.e. its β is zero. This means it can be made negligibile by diversification, and (by the Capital Asset Pricing Model) only the average effect of the jumps is important for pricing. The analogue of the Black-Scholes PDE for a jump-diffusion model is an integrodifferential equation. You may wonder how one could ever hope to solve it. In the constant-coefficient setting the Fourier transform is a convenient tool. That’s beyond the scope of this course. I’ve nevertheless included a discussion of the Fourier transform and its use in this setting, as enrichment reading for those who have sufficient background. ***************** 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. A one-dimensional diffusion solves dy = μ dt + σ dw . (Here μ and σ can be functions of y and t .) A jump-diffusion solves the same stochastic differential equation most of the time, but the solution occasionally jumps. We need to specify the statistics of the jumps. We suppose the occurrence of a jump is a Poisson process with rate λ . This means the jumps are entirely independent of one another....
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- Fall '11
- Finance, Ito, Merton