PRICING DERIVATIVES ON MULTISCALE DIFFUSIONS: AN EIGENFUNCTION EXPANSION APPROACH

Authors


  • The author of this paper would like to thank Stephan Sturm, Ronnie Sircar, and Jean-Pierre Fouque for helpful conversations. In addition, the author would like to thank two anonymous referees, whose comments vastly improved both the quality and readability of this manuscript. Work partially supported by NSF grant DMS-0739195.

Abstract

Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class of derivative-assets. The payoff of the derivative-assets may be path-dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative-assets: a vanilla option on a defaultable stock, a path-dependent option on a nondefaultable stock, and a bond in a short-rate model.

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