A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

Abstract

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.

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Notes

  1. 1.

    A higher number of regime-switching states implies more model parameters such that calibration algorithms might not converge to the correct solution when the parsimony of the model is destroyed.

References

  1. 1.

    Almendral, A., Oosterlee, C.W.: Numerical valuation of options with jumps in the underlying. Appl. Numer. Math. 53(1), 1–18 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bastani, A.F., Ahmadi, Z., Damircheli, D.: A radial basis collocation method for pricing American options under regime-switching jump-diffusion models. Appl. Numer. Math. 65, 79–90 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Biswas, A., Goswami, A., Overbeck, L.: Option pricing in a regime switching stochastic volatility model. Stat. Probab. Lett. 138, 116–126 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bo, L., Tang, D., Wang, Y.: Optimal investment of variance-swaps in jump-diffusion market with regime-switching. J. Econ. Dyn. Control 83, 175–197 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Boyle, P., Draviam, T.: Pricing exotic options under regime switching. Insur. Math. Econ. 40, 267–282 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods—Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)

    Google Scholar 

  8. 8.

    Chang, Y., Choi, Y., Park, J.Y.: A new approach to model regime switching. J. Econom. 196, 127–143 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Chen, F., Shen, J., Yu, H.: A new spectral element method for pricing European options under the Black–Scholes and Merton jump diffusion models. J. Sci. Comput. 52, 499–518 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Chen, S.N., Hsu, P.P.: Pricing and hedging barrier options under a Markov-modulated double exponential jump diffusion-CIR model. Int. Rev. Econ. Finance 56, 330–346 (2018)

  11. 11.

    Costabile, M., Leccadito, A., Massabó, I., Russo, E.: Option pricing under regime-switching jump-diffusion models. J. Comput. Appl. Math. 256, 152–167 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Dang, D.M., Nguyen, D., Sewell, G.: Numerical schemes for pricing Asian options under state-dependent regime-switching jump-diffusion models. Comput. Math. Appl. 71, 443–458 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Deelstra, G., Latouche, G., Simon, M.: On barrier option pricing by erlangization in a regime-switching model with jumps. J. Comput. Appl. Math. 371, 112606 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Dilloo, M.J., Tangman, D.Y.: A high-order finite difference method for option valuation. Comput. Math. Appl. 74, 652–670 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    El-Baghdady, G.I., El-Azab, M.S., El-Beshbeshy, W.S.: Legendre–Gauss–Lobatto pseudo-spectral method for one-dimensional advection–diffusion equation. Eur. J. Oper. Res. 2(1), 29–35 (2015)

    MATH  Google Scholar 

  16. 16.

    Elias, R.S., Wahab, M.I.M., Fang, L.: A comparison of regime-switching temperature modeling approaches for applications in weather derivatives. Eur. J. Oper. Res. 232(3), 549–560 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  18. 18.

    Fu, J., Wei, J., Yang, H.: Portfolio optimization in a regime-switching market with derivatives. Eur. J. Oper. Res. 233(1), 184–192 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Garreau, P., Kopriva, D.: A spectral element framework for option pricing under general exponential Lévy processes. J. Sci. Comput. 57, 390–413 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Godin, F., Lai, V.S., Trottier, D.A.: Option pricing under regime-switching models: novel approaches removing path-dependence. Insur. Math. Econ. 87, 130–142 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Hamilton, J.D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    He, X.J., Zhu, S.P.: How should a local regime-switching model be calibrated? J. Econ. Dyn. Control 78, 149–163 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Huang, J., Ju, L., Wua, B.: Fast compact exponential time differencing method for semilinear parabolic equations with Neumann boundary conditions. Appl. Math. Lett. 94, 257–265 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Huang, Y., Forsyth, P.A., Labahn, G.: Methods for pricing American options under regime switching. SIAM J. Sci. Comput. 33(5), 2144–2168 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Ikonen, S., Toivanen, J.: Operator splitting methods for American option pricing. Appl. Math. Lett. 17(7), 809–814 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Jacka, S.D., Ocejo, A.: On the regularity of American options with regime-switching uncertainty. Stoch. Process Appl. 128, 803–818 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Jackson, K.R., Jaimungal, S., Surkov, V.: Fourier space time-stepping for option pricing with Lévy models. J. Comput. Finance 12(2), 1–29 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Jang, B.G., Kim, K.T.: Optimal reinsurance and asset allocation under regime switching. J. Bank. Finance 56, 37–47 (2015)

    Article  Google Scholar 

  29. 29.

    Jang, B.G., Tae, H.W.: Option pricing under regime switching: integration over simplexes method. Finance Res. Lett. 24, 301–312 (2018)

    Article  Google Scholar 

  30. 30.

    Kassam, A., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26(4), 1214–1233 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Khaliq, A.Q.M., Kleefeld, B., Liu, R.H.: Solving complex PDE systems for pricing American options with regime-switching by efficient exponential time differencing schemes. Numer. Methods Partial Differ. Equ. 29(1), 320–336 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Ladyzenskaja, O., Solonnikov, V., Uraltseva, N.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

  33. 33.

    Leduc, G., Zeng, X.: Convergence rate of regime-switching trees. J. Comput. Appl. Math. 319, 56–76 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Lee, Y.: Financial options pricing with regime-switching jump-diffusions. Comput. Math. Appl. 68, 392–404 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Liu, R.H.: Regime-switching recombining tree for option pricing. Int. J. Theor. Appl. Finance 13(3), 479–499 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Liu, R.H., Zhao, J.L.: A lattice method for option pricing with two underlying assets in the regime-switching model. J. Comput. Appl. Math. 250, 96–106 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Luo, P., Yang, Z.: Real options and contingent convertibles with regime switching. J. Econ. Dyn. Control 75, 122–135 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Ma, J., Wang, H.: Convergence rates of moving mesh methods for moving boundary partial integro-differential equations from regime-switching jump–diffusion Asian option pricing. J. Comput. Appl. Math. 370, 112598 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Ma, J., Zhu, T.: Convergence rates of trinomial tree methods for option pricing under regime-switching models. Appl. Math. Lett. 39, 13–18 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Naik, V.: Option valuation and hedging strategies with jumps in the volatility of asset returns. J. Finance 48(5), 1969–1984 (1993)

    Article  Google Scholar 

  42. 42.

    Quarteroni, A.: Numerical Models for Differential Problems. MS&A. Springer, Milano (2014)

    Google Scholar 

  43. 43.

    Rambeerich, N., Pantelous, A.A.: A high order finite element scheme for pricing options under regime switching jump diffusion processes. J. Comput. Appl. Math. 300, 83–96 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Rambeerich, N., Tangman, D.Y., Lollchund, M.R., Bhuruth, M.: High-order computational methods for option valuation under multifactor models. Eur. J. Oper. Res. 224, 219–226 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Ramponi, A.: Fourier transform methods for regime-switching jump-diffusions and the pricing of forward starting options. Int. J. Theor. Appl. Finance 15(5), 1250037 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Reisinger, C., Witte, J.H.: On the use of policy iteration as an easy way of pricing American options. SIAM J. Financ. Math. 3(1), 459–478 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Schmelzer, T., Trefethen, L.N.: Evaluating matrix functions for exponential integrators via Carathéodory–Fejér approximation and contour integrals. Electron. Trans. Numer. Anal. 29, 1–18 (2007)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Schwendener, A.: The estimation of financial markets by means of a regime-switching model. PhD dissertation, University of St. Gallen, Zurich, Switzerland (2010)

  49. 49.

    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Google Scholar 

  50. 50.

    Shen, Y., Siu, T.K.: Asset allocation under stochastic interest rate with regime-switching. Econ. Model. 29(4), 1126–1136 (2012)

    Article  Google Scholar 

  51. 51.

    Tangman, D.Y., Gopaul, A., Bhuruth, M.: Exponential time integration and Chebychev discretisation schemes for fast pricing of options. Appl. Numer. Math. 58(9), 1309–1319 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Tour, G., Tangman, D.Y.: Option pricing under a Markov modulated model using a cubic B-spline collocation method. Int. J. Bus. Intell. Data Min. 9(4), 356–370 (2014)

    Article  Google Scholar 

  53. 53.

    Tour, G., Thakoor, N., Khaliq, A.Q.M., Tangman, D.Y.: COS method for option pricing under a regime-switching model with time-changed Lévy processes. Quant. Finance 18(4), 673–692 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Tour, G., Thakoor, N., Tangman, D.Y., Bhuruth, M.: A high-order RBF-FD method for option pricing under regime-switching stochastic volatility models with jumps. J. Comput. Science 35, 25–43 (2019)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Welfert, B.D.: Generation of pseudo-spectral differentiation. SIAM J. Numer. Anal. 34, 1640–1657 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Willyard, M.: Adaptive spectral element methods to price American options. PhD dissertation, The Florida State University, Thallahassee, USA (2011)

  57. 57.

    Wilmott, P., Howinson, S., Dewynne, J.: The Mathematics of Financial Derivatives. Cambridge University Press, New York (1995)

    Google Scholar 

  58. 58.

    Yousuf, M., Khaliq, A.Q.M., Alrabeei, S.: Solving complex PIDE systems for pricing American option under multi-state regime switching jump-diffusion model. Comput. Math. Appl. 75(8), 2989–3001 (2018)

  59. 59.

    Yue, T.: Spectral element method for pricing European options and their Greeks. PhD dissertation, Duke University, Durham, USA (2012)

  60. 60.

    Yuen, F.L., Yang, H.: Option pricing with regime switching by trinomial tree method. J. Comput. Appl. Math. 233, 1821–1833 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Yuen, F.L., Yang, H.: Pricing Asian options and equity-indexed annuities with regime switching by the trinomial tree method. North Am. Actur. J. 14(2), 256–277 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  62. 62.

    Zhu, W.: A spectral element method to price single and multi-asset European options. PhD dissertation, The Florida State University, Tallahassee, USA (2007)

  63. 63.

    Zhu, W., Kopriva, D.A.: A spectral element method to price European options. I. Single asset with and without jump diffusion. J. Sci. Comput. 39, 222–243 (2009)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Jingtang Ma.

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The research of Geraldine Tour was supported by a postgraduate research scholarship from the Tertiary Education Commission. The work of Jingtang Ma was supported by National Natural Science Foundation of China (Grant No. 11671323), Program for New Century Excellent Talents in University of China (Grant No. NCET-12-0922), and the Fundamental Research Funds for the Central Universities of China (JBK1805001). The work was done when J. Ma visited University of Mauritius in August 2019 and D. Y. Tangman visited SWUFE in December 2019.

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Tour, G., Thakoor, N., Ma, J. et al. A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps. J Sci Comput 83, 61 (2020). https://doi.org/10.1007/s10915-020-01252-7

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Keywords

  • Spectral element method
  • Option pricing
  • Regime-switching
  • Exponential time integration
  • Merton jump-diffusion model

Mathematics Subject Classification

  • 65R20
  • 91G20
  • 91G60
  • 91G80