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The Explicit Laplace Transform for the Wishart Process

Part of: Probabilistic methods, simulation and stochastic differential equations Mathematical economics Stochastic analysis

Published online by Cambridge University Press:  30 January 2018

Alessandro Gnoatto  and
Martino Grasselli
Alessandro Gnoatto*
Affiliation:
LMU München
Martino Grasselli*
Affiliation:
Università di Padova, Pole Universitaire Léonard de Vinci and Quanta Finanza S.R.L.
*
∗ Postal address: Mathematisches Institut, LMU München, Theresienstrasse 39, D-80333 München, Germany. Email address: gnoatto@mathematik.uni-muenchen.de
∗∗ Postal address: Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy. Email address: grassell@math.unipd.it
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Abstract

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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.

Keywords

Affine processWishart processordinary differential equationLaplace transform

MSC classification

Primary: 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 640 - 656
Copyright
© Applied Probability Trust 

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., Cieslak, A. and Trojani, F. (2008). Correlation risk and the term structure of interest rates. Working paper. Available at http://www.andreaburaschi.com/research.htm.Google Scholar
Buraschi, A., Porchia, P. and Trojani, F. (2010). Correlation risk and optimal portfolio choice. J. Finance 65, 393–420.Google Scholar
Chiarella, C., Hsiao, C.-Y. and To, T.-D. (2010). Risk premia and Wishart term structure models. Working paper. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1573184.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. (2003). Wishart quadratic term structure models. Working paper. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=757307.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
Leippold, M. and Trojani, F. (2010). Asset pricing with matrix Jump diffusions. Working paper. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1274482.Google Scholar
Levin, J. J. (1959). On the matrix Riccati equation. Proc. Amer. Math. Soc. 10, 519–524.Google Scholar
Mayerhofer, E. (2012). Wishart processes and Wishart distributions: an affine processes point of view. CIMPA lecture notes. Available at http://arxiv.org/abs/1201.6634.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
Spreij, P. and Veerman, E. (2010). The affine transform formula for affine Jump-diffusions with a general closed convex state space. Preprint. Available at http://arxiv.org/abs/1005.1099.Google Scholar
Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York.Google Scholar