Numerical schemes for pricing Asian options under state-dependent regime-switching jump-diffusion models. (English) Zbl 1443.65199
Summary: We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 45K05 | Integro-partial differential equations |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91G60 | Numerical methods (including Monte Carlo methods) |
Keywords:
Asian options; regime-switching; jump-diffusion; system of partial integro-differential equations; parallel computingSoftware:
PDE2DReferences:
| [1] | Broadie, M.; Glasserman, P.; Kou, P., Connecting discrete and continuous path-dependent options, Finance Stoch., 3, 55-82 (1999) · Zbl 0924.90007 |
| [2] | Broadie, M.; Glasserman, P., Estimating security price derivatives using simulation, Manage. Sci., 42, 269-285 (1996) · Zbl 0881.90018 |
| [3] | Kemna, A. G.Z.; Vorsta, A. C.F., A pricing method for options based on average asset values, J. Bank. Finance, 113-129 (1990), March |
| [4] | Ingersoll, J. E., Theory of Financial Decision Making (1987), Rowman & Littlefield |
| [5] | Zvan, R.; Forsyth, P.; Vetzal, K., Robust numerical methods for PDE models of Asian options, J. Comput. Finance, 1, 39-78 (1998) |
| [6] | Vecer, J., A new PDE approach for pricing arithmetic average Asian options, J. Comput. Finance, 4, 105-113 (2001) |
| [7] | Rogers, L.; Shi, Z., The value of Asian options, J. Appl. Probab., 32, 1077-1088 (1995) · Zbl 0839.90013 |
| [8] | Zhang, J., A semi-analytical method for pricing and hedging continuously sample arithmetic average rate options, J. Comput. Finance, 5, 1-20 (2001) |
| [9] | Zhang, J., Pricing continuously sampled Asian option with perturbation method, J. Futures Mark., 23, 535-560 (2003) |
| [10] | German, H.; Yor, M., Bessel processes, Asian options, and perpetuities, Math. Finance, 3, 349-375 (1993) · Zbl 0884.90029 |
| [11] | Hamilton, J., Time Series Analysis (1994), Princeton University Press · Zbl 0831.62061 |
| [12] | Yao, D.; Zhang, Q.; Zhou, Z., A regime switching model for European options, (Yan, H.; Yin, G.; Zhang, Q., Stochastic processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems (2006), Springer) · Zbl 1136.91015 |
| [13] | Boyle, P.; Draviam, T., Pricing exotic options under regime switching, Insurance Math. Econom., 40, 267-282 (2007) · Zbl 1141.91420 |
| [14] | Yuen, F.; Yang, H., Pricing Asian option and equity-indexed annuities with regime switching by the trinomial tree method, North Amer. Actuary J., 14, 256-272 (2012) · Zbl 1219.91145 |
| [16] | Merton, R., Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3, 125-144 (1976) · Zbl 1131.91344 |
| [17] | d’Halluin, Y.; Forsyth, P. A.; Labahn, G., A semi-Lagrangian approach for American Asian options under jump diffusion, SIAM J. Sci. Comput., 27, 315-345 (2005) · Zbl 1149.65316 |
| [18] | Vecer, J.; Xu, M., Pricing Asian options in semi-martingale model, Quant. Finance, 4, 170-175 (2004) · Zbl 1405.91652 |
| [19] | Bayraktar, E.; Xing, H., Pricing Asian option for jump diffusion, Math. Finance, 21, 117-143 (2011) · Zbl 1229.91332 |
| [20] | Kou, S., A jump-distribution model for option pricing, Manage. Sci., 48, 1086-1101 (2002) · Zbl 1216.91039 |
| [21] | Cai, N.; Kou, S. G., Pricing Asian options under a hyper-exponential jump diffusion model, Oper. Res., 60, 64-77 (2012) · Zbl 1241.91111 |
| [22] | Weron, R.; Bierbrauer, M.; Trück, S., Modeling electricity prices: jump diffusion and regime switching, Physica A, 336, 39-48 (2004) |
| [23] | Siu, T., Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows, Appl. Math. Comput., 216, 3184-3190 (2010) · Zbl 1194.91088 |
| [24] | Zhang, X.; Elliott, R. J.; Siu, T. K., Stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance, SIAM J. Control Optim., 2012, 964-990 (2012) · Zbl 1244.93180 |
| [25] | Elliott, R.; Siu, T.; Chan, L.; Lau, J., Pricing options under a generalized Markov modulated jump-diffusion model, Stoch. Anal. Appl., 25, 821-843 (2007) · Zbl 1155.91380 |
| [26] | Costabile, M.; Leccadito, A.; Massab, I.; Russo, E., Option pricing under regime switching jump diffusion models, J. Comput. Appl. Math., 256, 152-167 (2014) · Zbl 1314.91206 |
| [27] | Florescu, I.; Liu, R.; Mariani, M.; Sewell, G., Numerical schemes for option pricing in regime-switching jump diffusion models, Int. J. Theor. Appl. Finance, 16, 1-25 (2013) · Zbl 1290.91180 |
| [28] | Yuen, F.; Yang, H., Option pricing in a jump-diffusion model with regime-switching, ASTIN Bull., 39, 515-539 (2009) · Zbl 1180.91298 |
| [29] | Siu, T. K.; Lau, J. W.; Yang, H., Pricing participating products under a generalized jump-diffusion model, J. Appl. Math. Stoch. Anal., 2008, 1-30 (2008) · Zbl 1141.91386 |
| [30] | Ramponi, A., Fourier transform methods for regime-switching jump-diffusions and the pricing of forward starting options, Int. J. Theor. Appl. Finance, 15, 1-26 (2012) · Zbl 1262.91071 |
| [31] | Boyarchenko, M.; Boyarchenko, S., Double barrier options in regime-switching hyper-exponential jump-diffusion models, Int. J. Theor. Appl. Finance, 14, 1005-1043 (2011) · Zbl 1233.91257 |
| [32] | Yin, G.; Zhu, C., A Hybrid Switching Diffusion: Properties and Applications (2010), Springer · Zbl 1279.60007 |
| [33] | Sewell, G., The Numerical Solution of Ordinary and Partial Differential Equations (2015), World Scientific · Zbl 1306.65002 |
| [34] | McKean, H., A free boundary problem for the heat equation arising from a problem in mathematical economics, Indust. Manag. Rev., 60, 33-36 (1965) |
| [35] | Merton, R. C., The theory of rational option pricing, Bell J. Econ. Manage. Sci., 4, 141-183 (1973) · Zbl 1257.91043 |
| [36] | Guo, X., Information and option pricing, Quant. Finance, 1, 38-44 (2000) · Zbl 1405.91619 |
| [37] | Elliott, R.; Chan, L.; Siu, T., Option pricing and Esscher transform under regime switching, Ann. Finance, 1, 423-432 (2005) · Zbl 1233.91270 |
| [38] | Runggaldier, W., A regime switching model for European options, (Rachev, S., Handbook of Heavy Tailed Distributions in Finance (2003), Elesevier/North-Holland) |
| [39] | Henderson, V.; Wojakowski, R., On the equivalence of floating and fixed-strike Asian options, J. Appl. Probab., 39, 391-394 (2002) · Zbl 1004.60042 |
| [40] | Hugger, J., A fixed strike Asian option and comments on its numerical solution, ANZIAM J., 45, 215-231 (2004) · Zbl 1097.91044 |
| [41] | Florescu, I.; Liu, R.; Mariani, M., Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump-diffusion model, Electron. J. Differential Equations, 2012, 1-12 (2012) · Zbl 1294.35171 |
| [42] | Francesco, M. D.; Pascucci, A., On a class of degenerate parabolic equations of Kolmogorov type, Appl. Math. Res. Express, 3, 77-116 (2005) · Zbl 1085.35086 |
| [43] | Barucci, E.; Polidoro, S.; Vespri, V., Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11, 475-497 (2001) · Zbl 1034.35166 |
| [44] | Krylov, N., Lectures on Elliptic and Parabolic Equations in Holder Spaces (1996), American Mathematical Society · Zbl 0865.35001 |
| [45] | Lieberman, G. M., Second Order Parabolic Differential Equations (1998), Springer |
| [46] | Cont, R.; Voltchkova, E., A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal., 43, 1596-1626 (2005) · Zbl 1101.47059 |
| [47] | Alziarya, B.; Decamps, J. P.; Koehlc, P. F., A PDE approach to Asian options: analytical and numerical evidence, J. Banking Finance, 21, 613-640 (1997) |
| [48] | Nguyen, D.; Yin, G.; Zhang, Q., A stochastic approximation approach for trend-following trading, Internat. Ser. Oper. Res. Management Sci., 209, 167-184 (2014) · Zbl 1418.91484 |
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