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ASYMPTOTIC EXPANSION FOR THE TRANSITION DENSITIES OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE GAMMA PROCESSES

Published online by Cambridge University Press:  17 September 2020

Fan Jiang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn
Xin Zang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn
Jingping Yang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China E-mails: jiangfmath@pku.edu.cn; xzang@pku.edu.cn; yangjp@math.pku.edu.cn

Abstract

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

Abate, J. & Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10: 588.CrossRefGoogle Scholar
Applebaum, D. (2009). Lévy processes and stochastic calculus. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society, B (Statistical Methodology) 63: 167241.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Mikosch, T. & Resnick, S.I. (2001). Lévy processes: Theory and applications. Boston: Birkhäuser.CrossRefGoogle Scholar
Cont, R. & Tankov, P. (2004). Financial modelling with jump processes. London: Chapman and Hall.Google Scholar
Cox, J.C., Ingersoll, J.E. Jr, & Ross, S.A. (1985). A theory of the term structure of interest rates. Econometrica 53(2): 385408.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): 13431376.CrossRefGoogle Scholar
Eberlein, E., Madan, D., Pistorius, M., & Yor, M. (2013). A simple stochastic rate model for rate equity hybrid products. Applied Mathematical Finance 20: 461488.CrossRefGoogle Scholar
Hayashi, M. (2008). Asymptotic expansions for functionals of a Poisson random measure. Journal of Mathematics of Kyoto University 48(1): 91132.Google Scholar
Hayashi, M. & Ishikawa, Y. (2012). Composition with distributions of Wiener-Poisson variables and its asymptotic expansion. Mathematische Nachrichten 285: 619658.CrossRefGoogle Scholar
Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2): 327343.CrossRefGoogle Scholar
Ishikawa, Y. (2013). Stochastic calculus of variations for jump processes. Berlin: Walter De Gruyter.CrossRefGoogle Scholar
James, L.F., Kim, D., & Zhang, Z. (2013). Exact simulation pricing with gamma processes and their extensions. Journal of Computational Finance 17: 339.CrossRefGoogle Scholar
James, L.F., Müller, G., & Zhang, Z. (2017). Stochastic volatility models based on OU-gamma time change: Theory and estimation. Journal of Business & Economic Statistics 36: 113.Google Scholar
Jeanblanc, M., Yor, M., & Chesney, M. (2009). Mathematical methods for financial markets. London, UK: Springer-Verlag.CrossRefGoogle Scholar
Kanwal, R.P. (2004). Generalized functions: Theory and applications, 3rd ed. Boston: Birkhäuser.CrossRefGoogle Scholar
Kawai, R. & Takeuchi, A. (2011). Greeks formulas for an asset price model with gamma processes. Mathematical Finance 21(4): 723742.Google Scholar
Kawai, R. & Takeuchi, A. (2010). Sensitivity analysis for averaged asset price dynamics with gamma processes. Statistics & Probability Letters 80(1): 4249.CrossRefGoogle Scholar
Kloeden, P. & Platen, E. (1992). Numerical solutions of stochastic differential equations. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Kohatsu-Higa, A. & Takeuchi, A. (2019). Jump SDEs and the study of their densities. Singapore: Springer.CrossRefGoogle Scholar
Kunita, H. (2019). Stochastic flows and jump-diffusions. Singapore: Springer.CrossRefGoogle Scholar
Li, C. & Chen, D. (2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 5170.CrossRefGoogle Scholar
Li, C. (2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 13501380.CrossRefGoogle Scholar
Madan, D.B. & Seneta, E. (1990). The variance gamma (VG) model for share market returns. Journal of Business 63(4): 511524.CrossRefGoogle Scholar
Madan, D.B., Carr, P.P., & Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review 2(1): 79105.CrossRefGoogle Scholar
Platen, E. & Bruti-Liberati, N. (2010). Numerical solution of stochastic differential equations with jumps in finance. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Ribeiro, C. & Webber, N. (2004). Valuing path-dependent options in the variance-gamma model by monte carlo with a gamma bridge. Journal of Computational Finance 7(2): 81100.CrossRefGoogle Scholar
Schoutens, W. (2003). Lévy processes in finance: Pricing financial derivatives. New York: Wiley.CrossRefGoogle Scholar
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5: 177188.CrossRefGoogle Scholar
Yor, M. (2007). Some remarkable properties of gamma processes. In Fu, M.C., Jarrow, R.A., Yen, J.-Y.J. and Elliott, R.J. (eds.). Advances in mathematical finance. Boston, MA: Springer, pp. 37–47.CrossRefGoogle Scholar