The Explicit Laplace Transform for the Wishart Process | Journal of Applied Probability | Cambridge Core (original) (raw)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral, which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation-of-constants method, the linearization of the matrix Riccati ordinary differential equation, and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.

References

Ahdida, A. and Alfonsi, A. (2013). Exact and high-order discretization schemes for Wishart processes and their affine extensions. Ann. Appl. Prob. 23, 1025–1073.CrossRefGoogle Scholar

Anderson, B. D. O. and Moore, J. B. (1971). Linear Optimal Control. Prentice-Hall, Englewood Cliffs, NJ.CrossRefGoogle Scholar

Barndorff-Nielsen, O. E. and Stelzer, R. (2007). Positive-definite matrix processes of finite variation. Prob. Math. Statist. 27, 3–43.Google Scholar

Bäuerle, N. and Li, Z. (2013). Optimal portfolios for financial markets with Wishart volatility. J. Appl. Prob. 50, 1025–1043.Google Scholar

Bru, M.-F. (1991). Wishart processes. J. Theoret. Prob. 4, 725–751.Google Scholar

Buraschi, A., Porchia, P. and Trojani, F. (2010). Correlation risk and optimal portfolio choice. J. Finance 65, 393–420.Google Scholar

Christoffersen, P., Heston, S. and Jacobs, K. (2009). The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag. Sci. 55, 1914–1914.Google Scholar

Cuchiero, C. (2011). Affine and polynomial processes. , ETH Zürich.Google Scholar

Cuchiero, C., Filipović, D., Mayerhofer, E. and Teichmann, J. (2011). Affine processes on positive semidefinite matrices. Ann. App. Prob. 21, 397–463.Google Scholar

Da Fonseca, J. and Grasselli, M. (2011). Riding on the smiles. Quant. Finance 11, 1609–1632.Google Scholar

Da Fonseca, J., Grasselli, M. and Ielpo, F. (2011). Hedging (co)variance risk with variance swaps. Internat. J. Theoret. Appl. Finance 14, 899–943.Google Scholar

Da Fonseca, J., Grasselli, M. and Ielpo, F. (2014). Estimating the Wishart affine stochastic correlation model using the empirical characteristic function. Stud. Nonlinear Dynam. Econometrics 18, 253–289.Google Scholar

Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2007). Option pricing when correlations are stochastic: an analytical framework. Rev. Derivatives Res. 10, 151–180.Google Scholar

Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2008). A multifactor volatility Heston model. Quant. Finance 8, 591–604.Google Scholar

Donati-Martin, C., Doumerc, Y., Matsumoto, H. and Yor, M. (2004). Some properties of the Wishart processes and a matrix extension of the Hartman–Watson laws. Publ. Res. Inst. Math. Sci. 40, 1385–1412.Google Scholar

Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.CrossRefGoogle Scholar

Gnoatto, A. (2012). The Wishart short rate model. Internat. J. Theoret. Appl. Finance 15, 1250056.Google Scholar

Gourieroux, C. (2006). Continuous time Wishart process for stochastic risk. Econometric Rev. 25, 177–217.Google Scholar

Gourieroux, C. and Sufana, R. (2010). Derivative pricing with Wishart multivariate stochastic volatility. J. Bus. Econom. Statist. 28, 438–451.Google Scholar

Gourieroux, C., Monfort, A. and Sufana, R. (2010). International money and stock market contingent claims. J. Internat. Money Finance 29, 1727–1751.Google Scholar

Grasselli, M. and Tebaldi, C. (2008). Solvable affine term structure models. Math. Finance 18, 135–153.Google Scholar

Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.Google Scholar

Kang, C. and Kang, W. (2013). Transform formulae for linear functionals of affine processes and their bridges on positive semidefinite matrices. Stoch. Process. Appl. 123, 2419–2445.Google Scholar

Kučera, V. (1973). A review of the matrix Riccati equation. Kybernetika 9, 42–61.Google Scholar

Levin, J. J. (1959). On the matrix Riccati equation. Proc. Amer. Math. Soc. 10, 519–524.Google Scholar

Mayerhofer, E. (2013). On the existence of non-central Wishart distributions. J. Multivariate Anal. 114, 448–456.Google Scholar

Mayerhofer, E., Pfaffel, O. and Stelzer, R. (2011). On strong solutions for positive definite Jump diffusions. Stoch. Process. Appl. 121, 2072–2086.Google Scholar

Muhle-Karbe, J., Pfaffel, O. and Stelzer, R. (2012). Option pricing in multivariate stochastic volatility models of OU type. SIAM J. Financial Math. 3, 66–94.Google Scholar

Pigorsch, C. and Stelzer, R. (2009). On the definition, stationary distribution and second order structure of positive semidefinite Ornstein–Uhlenbeck type processes. Bernoulli 15, 754–773.Google Scholar

Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.Google Scholar

Quarteroni, A., Sacco, R. and Saleri, F. (2000). Numerical Mathematics (Texts Appl. Math. 37). Springer, New York.Google Scholar

Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion (Fundamental Principles Math. Sci. 293), 2nd edn. Springer, Berlin.Google Scholar

Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York.Google Scholar