Pseudo-diffusions and Quadratic term structure models
The non-gaussianity of processes observed in financial markets and relatively good performance of gaussian models can be reconciled by replacing the Brownian motion with Levy processes whose Levy densities decay as exp(-lambda|x|) or faster, where lambda>0 is large. This leads to asymptotic pricing models. The leading term, P0, is the price in the Gaussian model with the same instantaneous drift and variance. The first correction term depends on the instantaneous moments of order up to three, that is, the skewness is taken into account, the next term depends on moments of order four (kurtosis) as well, etc. In empirical studies, the asymptotic formula can be applied without explicit specification of the underlying process: it suffices to assume that the instantaneous moments of order greater than two are small w.r.t. moments of order one and two, and use empirical data on moments of order up to three or four. As an application, the bond pricing problem in the non-Gaussian quadratic term structure model is solved. For pricing of options near expiry, a different set of asymptotic formulas is developed; they require more detailed specification of the process, especially of its jump part. The leading terms of these formulas depends on the jump part of the process only, so that they can be used in empirical studies to identify the jump characteristics of the process.
References listed on IDEAS
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- Dong-Hyun Ahn & Robert F. Dittmar, 2002. "Quadratic Term Structure Models: Theory and Evidence," Review of Financial Studies, Society for Financial Studies, vol. 15(1), pages 243-288, March.
- Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53.
- Ahn, Dong-Hyun & Dittmar, Robert F. & Gallant, A. Ronald & Gao, Bin, 2003. "Purebred or hybrid?: Reproducing the volatility in term structure dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 147-180.
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- Li Chen & H. Vincent Poor, 2002. "A General Characterization of Quadratic Term Structure Models," Finance 0211008, EconWPA.
- George Chacko, 2002. "Pricing Interest Rate Derivatives: A General Approach," Review of Financial Studies, Society for Financial Studies, vol. 15(1), pages 195-241, March.
- Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
- Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
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