Hostname: page-component-54dcc4c588-sdd8f Total loading time: 0 Render date: 2025-10-01T10:51:53.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit pathwise expansion for multivariate diffusions with applications

Part of: Stochastic analysis Mathematical finance Markov processes

Published online by Cambridge University Press:  29 September 2025

Nan Chen ,
Xiangwei Wan  and
Nian Yang Open the ORCID record for Nian Yang [Opens in a new window]
Nan Chen*
Affiliation:
The Chinese University of Hong Kong
Xiangwei Wan*
Affiliation:
Shanghai Jiao Tong University
Nian Yang*
Affiliation:
Nanjing University
*
*Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Email: nchen@se.cuhk.edu.hk
**Postal address: Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China. Email: xwwan@sjtu.edu.cn
***Postal address: Department of Finance and Insurance, School of Business, Nanjing University, Nanjing, Jiangsu, China. Email: yangnian@nju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we introduce a unified framework based on the pathwise expansion method to derive explicit recursive formulas for cumulative distribution functions, option prices, and transition densities in multivariate diffusion models. A key innovation of our approach is the introduction of the quasi-Lamperti transform, which normalizes the diffusion matrix at the initial time. This transformation facilitates expansions using uncorrelated Brownian motions, effectively reducing multivariate problems to one-dimensional computations. Consequently, both the analysis and the computation are significantly simplified. We also present two novel applications of the pathwise expansion method. Specifically, we employ the proposed framework to compute the value-at-risk for stock portfolios and to evaluate complex derivatives, such as forward-starting options. Our method has the flexibility to accommodate models with diverse features, including stochastic risk premiums, stochastic volatility, and nonaffine structures. Numerical experiments demonstrate the accuracy and computational efficiency of our approach. In addition, as a theoretical contribution, we establish an equivalence between the pathwise expansion method and the Hermite polynomial-based expansion method in the literature.

Keywords

Pathwise expansionquasi-Lamperti transformHermite expansionportfolio value-at-riskforward-starting option

MSC classification

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H35: Computational methods for stochastic equations
Secondary: 60J60: Diffusion processes 91G20: Derivative securities 91G70: Statistical methods, econometrics

Information

Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 34
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70, 223–262.10.1111/1468-0262.00274CrossRefGoogle Scholar
Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Stat. 36, 906–937.10.1214/009053607000000622CrossRefGoogle Scholar
Aït-Sahalia, Y. (2009). Estimating and testing continuous-time models in finance: the role of transition densities. Ann. Rev. Financ. Econ. 1, 341–359.10.1146/annurev.financial.050808.114424CrossRefGoogle Scholar
Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financ. Econ. 83, 413–452.10.1016/j.jfineco.2005.10.006CrossRefGoogle Scholar
Aït-Sahalia, Y. and Kimmel, R. (2010). Estimating affine multifactor term structure models using closed-form likelihood expansions. J. Financ. Econ. 98, 113–144.10.1016/j.jfineco.2010.05.004CrossRefGoogle Scholar
Aït-Sahalia, Y., Li, C. and Li, C. X. (2021a). Closed-form implied volatility surfaces for stochastic volatility models with jumps. J. Econom. 222(1), 364–392.10.1016/j.jeconom.2020.07.006CrossRefGoogle Scholar
Aït-Sahalia, Y., Li, C. and Li, C. X. (2021b). Implied stochastic volatility models. Rev. Financ. Stud. 34(1), 394–450.10.1093/rfs/hhaa041CrossRefGoogle Scholar
Aït-Sahalia, Y. and Yu, J. (2006). Saddlepoint approximations for continuous-time Markov processes. J. Econom. 134, 507–551.10.1016/j.jeconom.2005.07.004CrossRefGoogle Scholar
Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15, 2422–2444.10.1214/105051605000000485CrossRefGoogle Scholar
Bompis, R. and Gobet, E. (2013). Asymptotic and non asymptotic approximations for option valuation. In T. Gerstner and P. Kloeden (eds.), Recent Developments in Computational Finance: Foundations, Algorithms and Applications. World Scientific, Singapore, pp. 159–241.10.1142/9789814436434_0004CrossRefGoogle Scholar
Chen, D. and Li, C. (2023). Closed form expansion for option price under stochastic volatility model with concurrent jumps. IISE Trans. 55(8), 781–793.10.1080/24725854.2022.2135797CrossRefGoogle Scholar
Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Oper. Res. 38(3), 591–616.10.1287/moor.2013.0585CrossRefGoogle Scholar
Choi, S. (2013). Closed-form likelihood expansions for multivariate time-inhomogeneous diffusions. J. Econom. 174, 45–65.10.1016/j.jeconom.2011.12.007CrossRefGoogle Scholar
Choi, S. (2015). Explicit form of approximate transition probability density functions of diffusion processes. J. Econom. 187, 57–73.10.1016/j.jeconom.2015.02.003CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.10.2307/1911242CrossRefGoogle Scholar
Duan, J. C. (1995). The GARCH option pricing model. Math. Finance 5, 13–32.10.1111/j.1467-9965.1995.tb00099.xCrossRefGoogle Scholar
Filipović, D., Mayerhofer, E. and Schneider, P. (2013). Density approximations for multivariate affine jump-diffusion processes. J. Econom. 176, 93–111.10.1016/j.jeconom.2012.12.003CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343.10.1093/rfs/6.2.327CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland.Google Scholar
Jones, C. S. (2003). The dynamics of stochastic volatility: evidence from underlying and options markets. J. Econom. 116(1), 181–224.10.1016/S0304-4076(03)00107-6CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.Google Scholar
Kruse, S. and Nögel, U. (2005). On the pricing of forward starting options in Heston’s model on stochastic volatility. Finance Stochastics 9, 233–255.10.1007/s00780-004-0146-3CrossRefGoogle Scholar
Kunitomo, N. and Takahashi, A. (2001). The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Finance 11, 117–151.10.1111/1467-9965.00110CrossRefGoogle Scholar
Kunitomo, N. and Takahashi, A. (2003). On the validity of the asymptotic expansion approach in contingent claim analysis. Ann. Appl. Probab. 13(3), 914–952.10.1214/aoap/1060202831CrossRefGoogle Scholar
Li, C. (2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Ann. Stat. 41, 1350–1380.10.1214/13-AOS1118CrossRefGoogle Scholar
Li, C. (2014). Closed-form expansion, conditional expectation, and option valuation. Math. Oper. Res. 39, 487–516.10.1287/moor.2013.0613CrossRefGoogle Scholar
Li, C., An, Y., Chen, D., Li, Q. and Si, N. (2016). Efficient computation of the likelihood expansions for diffusion models. IIE Trans. 48, 1156–1171.10.1080/0740817X.2016.1200201CrossRefGoogle Scholar
Li, C. and Chen, D. (2016). Estimating jump-diffusions using closed-form likelihood expansions. J. Econom. 195(1), 51–70.10.1016/j.jeconom.2016.07.001CrossRefGoogle Scholar
Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econom. 45, 7–38.10.1016/0304-4076(90)90092-8CrossRefGoogle Scholar
Qiao, C., Wan, X. and Yang, N. (2025). An option pricing model with double-exponential jumps in returns and GARCH diffusion in volatilities. Oper. Res. Lett. 25, 107253.10.1016/j.orl.2025.107253CrossRefGoogle Scholar
Takahashi, A. (1999). An asymptotic expansion approach to pricing contingent claims. Asia-Pac. Financ. Mark. 6, 115–151.10.1023/A:1010080610650CrossRefGoogle Scholar
Takahashi, A. (2015). Asymptotic expansion approach in finance. In P. Friz et al. (eds.), Large Deviations and Asymptotic Methods in Finance. Springer, Switzerland, pp. 345–411.10.1007/978-3-319-11605-1_13CrossRefGoogle Scholar
Wan, X. and Yang, N. (2021). Hermite expansion of transition densities and European option prices for multivariate diffusions with jumps. J. Econ. Dyn. Control 125, 104083.10.1016/j.jedc.2021.104083CrossRefGoogle Scholar
Watanabe, S. (1987). Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15, 1–39.10.1214/aop/1176992255CrossRefGoogle Scholar
Yang, N., Chen, N. and Wan, X. (2019). A new delta expansion for multivariate diffusions via the Itô-Taylor expansion. J. Econom. 209(2), 256–288.10.1016/j.jeconom.2019.01.003CrossRefGoogle Scholar
Yoshida, N. (1992a). Asymptotic expansion for statistics related to small diffusions. J. Jpn. Stat. Soc. 22(2), 139–159.Google Scholar
Yoshida, N. (1992b). Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin–Watanabe. Probab. Theory Relat. Fields 92, 275–311.10.1007/BF01300558CrossRefGoogle Scholar
Yu, J. (2007). Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese yuan. J. Econom. 141, 1245–1280.10.1016/j.jeconom.2007.02.003CrossRefGoogle Scholar