Asymptotic expansion for the transition densities of stochastic differential equations driven by the gamma processes. (English) Zbl 1498.60219
Summary: In this paper, enlightened by the asymptotic expansion methodology developed by C. Li [Ann. Stat. 41, No. 3, 1350–1380 (2013; Zbl 1273.62196)] and C. Li and D. Chen [J. Econom. 195, No. 1, 51–70 (2016; Zbl 1443.62361)], 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
References:
| [1] | Abate, J. & Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems10: 5-88. · Zbl 0749.60013 |
| [2] | Applebaum, D. (2009). Lévy processes and stochastic calculus. New York, NY: Cambridge University Press. · Zbl 1200.60001 |
| [3] | 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: 167-241. · Zbl 0983.60028 |
| [4] | Barndorff-Nielsen, O.E., Mikosch, T. & Resnick, S.I. (2001). Lévy processes: Theory and applications. Boston: Birkhäuser. · Zbl 0961.00012 |
| [5] | Cont, R. & Tankov, P. (2004). Financial modelling with jump processes. London: Chapman and Hall. · Zbl 1052.91043 |
| [6] | Cox, J.C., Ingersoll, J.E. Jr, & Ross, S.A. (1985). A theory of the term structure of interest rates. Econometrica53(2): 385-408. · Zbl 1274.91447 |
| [7] | Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica68(6): 1343-1376. · Zbl 1055.91524 |
| [8] | Eberlein, E., Madan, D., Pistorius, M., & Yor, M. (2013). A simple stochastic rate model for rate equity hybrid products. Applied Mathematical Finance20: 461-488. · Zbl 1396.91780 |
| [9] | Hayashi, M. (2008). Asymptotic expansions for functionals of a Poisson random measure. Journal of Mathematics of Kyoto University48(1): 91-132. · Zbl 1281.60051 |
| [10] | Hayashi, M. & Ishikawa, Y. (2012). Composition with distributions of Wiener-Poisson variables and its asymptotic expansion. Mathematische Nachrichten285: 619-658. · Zbl 1252.60051 |
| [11] | Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies6(2): 327-343. · Zbl 1384.35131 |
| [12] | Ishikawa, Y. (2013). Stochastic calculus of variations for jump processes. Berlin: Walter De Gruyter. · Zbl 1301.60003 |
| [13] | James, L.F., Kim, D., & Zhang, Z. (2013). Exact simulation pricing with gamma processes and their extensions. Journal of Computational Finance17: 3-39. |
| [14] | James, L.F., Müller, G., & Zhang, Z. (2017). Stochastic volatility models based on OU-gamma time change: Theory and estimation. Journal of Business & Economic Statistics36: 1-13. |
| [15] | Jeanblanc, M., Yor, M., & Chesney, M. (2009). Mathematical methods for financial markets. London, UK: Springer-Verlag. · Zbl 1205.91003 |
| [16] | Kanwal, R.P. (2004). Generalized functions: Theory and applications, 3rd ed. Boston: Birkhäuser. · Zbl 1069.46001 |
| [17] | Kawai, R. & Takeuchi, A. (2011). Greeks formulas for an asset price model with gamma processes. Mathematical Finance21(4): 723-742. · Zbl 1244.91092 |
| [18] | Kawai, R. & Takeuchi, A. (2010). Sensitivity analysis for averaged asset price dynamics with gamma processes. Statistics & Probability Letters80(1): 42-49. · Zbl 1186.60044 |
| [19] | Kloeden, P. & Platen, E. (1992). Numerical solutions of stochastic differential equations. Berlin: Springer-Verlag. · Zbl 0752.60043 |
| [20] | Kohatsu-Higa, A. & Takeuchi, A. (2019). Jump SDEs and the study of their densities. Singapore: Springer. · Zbl 1447.60001 |
| [21] | Kunita, H. (2019). Stochastic flows and jump-diffusions. Singapore: Springer. · Zbl 1447.60002 |
| [22] | Li, C. & Chen, D. (2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics195(1): 51-70. · Zbl 1443.62361 |
| [23] | Li, C. (2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics41: 1350-1380. · Zbl 1273.62196 |
| [24] | Madan, D.B. & Seneta, E. (1990). The variance gamma (VG) model for share market returns. Journal of Business63(4): 511-524. |
| [25] | Madan, D.B., Carr, P.P., & Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review2(1): 79-105. · Zbl 0937.91052 |
| [26] | Platen, E. & Bruti-Liberati, N. (2010). Numerical solution of stochastic differential equations with jumps in finance. Berlin: Springer-Verlag. · Zbl 1225.60004 |
| [27] | Ribeiro, C. & Webber, N. (2004). Valuing path-dependent options in the variance-gamma model by monte carlo with a gamma bridge. Journal of Computational Finance7(2): 81-100. |
| [28] | Schoutens, W. (2003). Lévy processes in finance: Pricing financial derivatives. New York: Wiley. |
| [29] | Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics5: 177-188. · Zbl 1372.91113 |
| [30] | 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. · Zbl 1156.60030 |
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