Journal of Scientific Computing

, Volume 75, Issue 2, pp 733–761 | Cite as

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

  • Duy-Minh Dang
  • Luis Ortiz-Gracia


We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.


Shannon wavelets Dimension reduction Jump diffusions 

Mathematics Subject Classification

62P05 60E10 65T60 


  1. 1.
    Ahlip, R., Rutkowski, M.: Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quant. Finance 13, 955–966 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alizadeh, S., Brandt, M., Diebold, F.: Range-based estimation of stochastic volatility models. J. Finance 57, 1047–1091 (2002)CrossRefGoogle Scholar
  3. 3.
    Andersen, L., Piterbarg, V.: Moment explosions in stochastic volatility models. Finance Stoch 11, 29–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andersen, T.G., Benzoni, L., Lund, J.: An empirical investigation of continuous-time equity return models. J. Finance 57, 1239–1284 (2002)CrossRefGoogle Scholar
  5. 5.
    Bakshi, G., Cao, C., Zhiwu, C.: Empirical performance of alternativee option pricing models. J. Finance 52, 2003–2049 (1997)CrossRefGoogle Scholar
  6. 6.
    Bates, D.: Jumps and stochastic volatility: exchange rate process implicit in Deutsche Mark options. Rev. Financ. Stud. 9(1), 69–107 (1996)CrossRefGoogle Scholar
  7. 7.
    Berthe, E., Dang, D.M., Ortiz-Gracia, L.: A Shannon wavelet method for pricing foreign exchange options under the Heston multi-factor CIR model. Working paper, School of Mathematics and Physics, University of Queensland (2017)Google Scholar
  8. 8.
    Brigo, D., Mercurio, F.: Interest Rate Models—Theory and Practice, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  9. 9.
    Broadie, M., Kaya, O.: Exact simulation of stochastic volatility and other affine jump difusion processes. Oper. Res. 54, 217–231 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cattani, C.: Shannon wavelets theory. Mathe. Probl. Eng. Article ID 164808 (2008)Google Scholar
  11. 11.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall, London (2004)zbMATHGoogle Scholar
  12. 12.
    Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985a)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985b)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cozma, A., Reisinger, C.: A mixed Monte Carlo and PDE variance reduction method for foreign exchange options under the Heston-CIR model. J. Comput. Finance 20, 109–149 (2017)Google Scholar
  15. 15.
    Dang, D.M.: A multi-level dimension reduction Monte-Carlo method for jump-diffusion models. J. Comput. Appl. Math. 324, 49–71 (2017)Google Scholar
  16. 16.
    Dang, D.M., Christara, C., Jackson, K.: GPU pricing of exotic cross-currency interest rate derivatives with a foreign exchange volatility skew model. J. Concurr. Comput. Pract. Exp. 26, 1609–1625 (2014)CrossRefGoogle Scholar
  17. 17.
    Dang, D.M., Christara, C., Jackson, K., Lakhany, A.: An efficient numerical PDE approach for pricing foreign exchange interest rate hybrid derivatives. J. Comput. Finance 18(4), 1–55 (2015a)CrossRefGoogle Scholar
  18. 18.
    Dang, D.M., Jackson, K.R., Mohammadi, M.: Dimension and variance reduction for Monte-Carlo methods for high-dimensional models in finance. Appl. Math. Finance 22(6), 522–552 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dang, D.M., Jackson, K.R., Sues, S.: A dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models. Appl. Math. Finance 24(3), 175–215 (2017). doi: 10.1080/1350486X.2017.1358646
  20. 20.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fang, F., Oosterlee, C.W.: A novel pricing method for European options based on Fourier-Cosine series expansions. SIAM J. Sci. Comput. 31, 826–848 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fang, F., Oosterlee, C.W.: A Fourier-based valuation method for Bermudan and barrier options under Heston’s model. SIAM J. Financ. Math. 2, 439–463 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gatheral, J.: The Volatility Surface: A Practitioner\(^{\prime }\)s Guide. Wiley Finance, New York (2006)Google Scholar
  25. 25.
    Gearhart, W.B., Shultz, H.S.: The function \(\sin (x)/x\). Coll. Math. J. 21, 90–99 (1990)Google Scholar
  26. 26.
    Giles, M.B.: Multi-level Monte Carlo path simulation. Oper. Res. 56, 607–617 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Grzelak, L.A., Oosterlee, C.W.: The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives. Quant. Finance 11, 1647–1663 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Grzelak, L.A., Oosterlee, C.W.: On cross-currency models with stochastic volatility and correlated interest rates. Appl. Math. Finance 19, 1–35 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Fianan. Math. 2, 255–286 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Haastrecht, A.V., Lord, R., Pelsseri, A., Schrager, D.: Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic. Insur. Math. Econ. 45, 436–448 (2009)CrossRefzbMATHGoogle Scholar
  31. 31.
    Haastrecht, A.V., Pelsser, A.: Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quant. Finance 11, 665–691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Haentjens, T., In’t Hout, K.J.: Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation. J. Comput. Finance 16(1), 83–110 (2012)CrossRefGoogle Scholar
  33. 33.
    Heston, S.: A closed form solution for options with stochastic volatility with applications to bond and currency options. Rev. Finan. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  34. 34.
    Hull, J., White, A.: One factor interest rate models and the valuation of interest rate derivative securities. J. Financ. Quant. Anal. 28(2), 235–254 (1993)CrossRefGoogle Scholar
  35. 35.
    Jamshidian, F., Zhu, Y.: Scenario simulation: theory and methodology. Finance Stoch. 13, 4367 (1997)zbMATHGoogle Scholar
  36. 36.
    Kou, S.G.: A jump diffusion model for option pricing. Manag. Sci. 48, 1086–1101 (2002)CrossRefzbMATHGoogle Scholar
  37. 37.
    Maree, S.C., Ortiz-Gracia, L., Oosterlee, C.W.: Pricing early-exercise and discrete barrier options by Shannon wavelet expansions. Numer. Math. 136 (4), 1035–1070 (2017)Google Scholar
  38. 38.
    Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1976)CrossRefzbMATHGoogle Scholar
  39. 39.
    Neuenkirch, A., Szpruch, L.: First order strong approximations of scalar SDEs with values in a domain. Numer. Math. 128, 103–136 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ortiz-Gracia, L., Oosterlee, C.W.: A highly efficient Shannon wavelet inverse Fourier technique for pricing European options. SIAM J. Sci. Comput. 38(1), B118–B143 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pillay, E., O’Hara, J.: FFT based option pricing under a mean reverting process with stochastic volatility and jumps. J. Comput. Appl. Math. 235, 3378–3384 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Piterbarg, V.: Smiling hybrids. Risk Mag. 19(5), 66–70 (2006)Google Scholar
  43. 43.
    Rebonato, R.: Interest Rate Option Models, 2nd edn. Wiley, New York (1998)zbMATHGoogle Scholar
  44. 44.
    Sippel, J., Ohkoshi, S.: All power to PRDC notes. Risk Mag. 15(11), 1–3 (2002)Google Scholar
  45. 45.
    Stenger, F.: Handbook of Sinc Numerical Methods. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  46. 46.
    Zhang, S., Wang, L.: Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps. Appl. Math. Comput. 219, 10928–10933 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsThe University of QueenslandSt Lucia, BrisbaneAustralia
  2. 2.Universitat de Barcelona School of Economics, Faculty of Economics and BusinessUniversity of BarcelonaBarcelonaSpain

Personalised recommendations