Abstract
Analytical tractability is a desirable property of stochastic stock price models. Informally speaking, a stochastic model is analytically tractable if various important characteristics of the model can be represented explicitly or asymptotically in terms of standard functions of mathematical analysis. Classical stochastic volatility models (Hull-White, Stein-Stein, Heston) are analytically tractable. In this chapter, explicit formulas are obtained for Laplace transforms of mixing densities and Mellin transforms of stock price densities in classical stochastic volatility models. For example, an alternative proof of an explicit formula for the Laplace transform of the distribution density of an integrated geometric Brownian motion due to L. Alili and J.C. Gruet is given. Chapter 4 also contains an explicit formula for the stock price density in the correlated Hull-White model with driftless volatility obtained by Y. Maghsoodi. In addition, Chap. 4 provides explicit formulas for the Mellin transform of the stock price density in the correlated Heston and Stein-Stein models.
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References
Alili, L., Gruet, J. C., An explanation of a generalized Bboujerol’s identity in terms of hyperbolic geometry, in: Yor, M. (Ed.), Exponential Functionals and Principal Values Related to Brownian Motion, pp. 15–33, Biblioteca de la Revista Matemàtica Ibero-Americana, Madrid, 2007.
Barrieu, P., Rouault, A., Yor, M., A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options, Journal of Applied Probability 41 (2004), pp. 1049–1058.
Bougerol, Ph., Exemples des théorèmes locaux sur les groupes résolubles, Annales de l’Institut Henry Poincaré 19 (1983), pp. 369–391.
del Baño Rollin, S., Ferreiro-Castilla, A., Utzet, F., On the density of log-spot in Heston volatility model, Stochastic Processes and Their Applications 120 (2010), pp. 2037–2062.
Flajolet, P., Sedgewick, R., Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.
Gerhold, S., The Hartman–Watson distribution revisited: asymptotics for pricing Asian options, Journal of Applied Probability 48 (2011), pp. 892–899.
Gulisashvili, A., Stein, E. M., Asymptotic behavior of distribution densities in models with stochastic volatility, I, Mathematical Finance 20 (2010), pp. 447–477.
Gulisashvili, A., van Casteren, J. A., Non-Autonomous Kato Classes and Feynman–Kac Propagators, World Scientific, Singapore, 2006.
Hartman, P., Watson, G. S., “Normal” distribution functions on spheres and the modified Bessel functions, Annals of Probability 2 (1974), pp. 593–607.
Heston, S. L., A closed-form solution for options with stochastic volatility, with applications to bond and currency options, Review of Financial Studies 6 (1993), pp. 327–343.
Jefferies, B., Evolution Processes and the Feynman–Kac Formula, Kluwer Academic, Dordrecht, 1996.
Keller-Ressel, M., Moment explosions and long-term behavior of affine stochastic volatility models, Mathematical Finance 21 (2011), pp. 73–98.
Maghsoodi, Y., Exact solution of a martingale stochastic volatility option problem and its empirical evaluation, Mathematical Finance 17 (2007), pp. 249–265.
Matsumoto, H., Yor, M., Exponential functionals of Brownian motion, I: probability laws at fixed time, Probability Surveys 2 (2005), pp. 312–347.
Matsumoto, H., Yor, M., Exponential functionals of Brownian motion, II: some related diffusion processes, Probability Surveys 2 (2005), pp. 348–384.
Øksendal, B., Stochastic Differential Equations. An Introduction with Applications, 6th ed., Springer, Berlin, 2003.
Paris, R. B., Kaminski, D., Asymptotics and Mellin–Barnes Integrals, Cambridge University Press, Cambridge, 2001.
Pitman, J., Yor, M., A decomposition of Bessel bridges, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 59 (1982), pp. 425–457.
Revuz, D., Yor, M., Continuous Martingales and Brownian Motion, Springer, Berlin, 2004.
Schöbel, R., Zhu, J., Stochastic volatility with an Ornstein–Uhlenbeck process: an extension, European Finance Review 3 (1999), pp. 23–46.
Stein, E. M., Stein, J., Stock price distributions with stochastic volatility: an analytic approach, Review of Financial Studies 4 (1991), pp. 727–752.
Wenocur, M. L., Ornstein–Uhlenbeck process with quadratic killing, Journal of Applied Probability 27 (1990), pp. 707–712.
Yor, M., Loi de l’indice du lacet Brownien, et distribution de Hartman–Watson, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 53 (1980), pp. 71–95.
Yor, M., On some exponential functionals of Brownian motion, Advances in Applied Probability 24 (1992), pp. 509–531.
Yor, M., Sur les lois des fonctionells exponentielles du mouvement brownien, considérées en certain instants aléatoires, Comptes Rendus de l’Académie des Sciences de Paris 314 (1992), pp. 951–956.
Yor, M., Exponential Functionals of Brownian Motion and Related Processes, Springer, Berlin, 2001.
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Gulisashvili, A. (2012). Integral Transforms of Distribution Densities. In: Analytically Tractable Stochastic Stock Price Models. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31214-4_4
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DOI: https://doi.org/10.1007/978-3-642-31214-4_4
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