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A Spectral Element Approximation to Price European Options with One Asset and Stochastic Volatility

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Abstract

We develop a Legendre quadrilateral spectral element approximation for the Black-Scholes equation to price European options with one underlying asset and stochastic volatility. A weak formulation of the equations imposes the boundary conditions naturally along the boundaries where the equation becomes singular, and in particular, we use an energy method to derive boundary conditions at outer boundaries for which the problem is well-posed on a finite domain. Using Heston’s analytical solution as a benchmark, we show that the spectral element approximation along with the proposed boundary conditions gives exponential convergence in the solution and the Greeks to the level of time and boundary errors in a domain of financial interest.

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References

  1. Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. SIAM, Philadelphia (2005)

    MATH  Google Scholar 

  2. Andersen, L.B.G., Piterbarg, V.V.: Moment explosions in stochastic volatility models. Finance Stoch. 11(1), 29–50 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B.: An empirical comparison of alternative models of the short-term interest rate. J. Finance 47(3), 1209–1227 (1992)

    Article  Google Scholar 

  5. Clarke, N., Parrott, K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Financ. 6(3), 177–195 (1999)

    Article  MATH  Google Scholar 

  6. Cox, J., Ingersoll Jr., J., Ross, S.: A theory of the term structure of interest rates. Econometrica 53(2), 385–407 (1985)

    Article  MathSciNet  Google Scholar 

  7. Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and their applications to mesh generation. Int. J. Numer. Methods Eng. 7, 461–477 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  8. Heston, S.L.: A closed-form solution for option with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)

    Article  Google Scholar 

  9. Hilber, N., Matache, A., Schwab, C.: Sparse wavelet methods for option pricing under stochastic volatility. J. Comput. Financ. 8(4), 1–42 (2005)

    Google Scholar 

  10. Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42(2), 281–300 (1987)

    Article  Google Scholar 

  11. Hull, J.C.: Options, Futures, and Other Derivatives, 4th edn. Oxford University Press, London (2000)

    Google Scholar 

  12. Ikonen, S., Toivanen, J.: Efficient numerical methods for pricing American options under stochastic volatility. Numer. Methods Partial Differ. Equ. 24(1), 104–126 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jackwerth, J.C., Rubinstein, M.: Recovering probability distributions from option prices. J. Finance 51, 1611–1631 (1996)

    Article  Google Scholar 

  14. Johnson, H., Shanno, D.: Option pricing when the variance is changing. J. Financ. Quant. Anal. 22(2), 143–151 (1987)

    Article  Google Scholar 

  15. macBeth, J.D., Merville, L.J.: An empirical examination of the Black-Scholes call option pricing model. J. Finance 34, 1173–1186 (1979)

    Article  Google Scholar 

  16. Patera, A.T.: A spectral element method for fluid dynamics—Laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ronquist, E.M., Patera, A.T.: A Lagrange spectral element method for the Stefan problem. Int. J. Numer. Methods Eng. 24(12), 2273–2299 (1987)

    Article  MathSciNet  Google Scholar 

  18. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    MATH  Google Scholar 

  19. Scott, L.O.: Option pricing when the variance changes randomly: Theory, estimation, and an application. J. Financ. Quant. Anal. 22(4), 419–438 (1987)

    Article  Google Scholar 

  20. Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: An analytic approach. Rev. Financ. Stud. 4(4), 727–752 (1991)

    Article  Google Scholar 

  21. Vasicek, O.: An equilibrium characterisation of the term structure. J. Financ. Econ. 5, 177–188 (1977)

    Article  Google Scholar 

  22. Van Der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13(2), 631–644 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wiggens, J.B.: Option values under stochastic volatility: Theory and empirical estimates. J. Financ. Econ. 19, 351–372 (1987)

    Article  Google Scholar 

  24. 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(2), 222–243 (2009)

    Article  MathSciNet  Google Scholar 

  25. Zhu, W., Kopriva, D.A.: A spectral element approximation to price European options. II. The Black-Scholes model with two underlying assets. J. Sci. Comput. 39(3), 323–339 (2009)

    Article  MathSciNet  Google Scholar 

  26. Zvan, R., Forsyth, P., Vetzal, K.: A penalty method for American options with stochastic volatility. J. Comput. Appl. Math. 91, 199–218 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. NextEra Energy Power Marketing, LLC, Juno Beach, FL, USA

    Wuming Zhu

  2. Department of Mathematics, The Florida State University, Tallahassee, FL, USA

    David A. Kopriva

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  1. Wuming Zhu
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  2. David A. Kopriva
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Correspondence to David A. Kopriva.

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Zhu, W., Kopriva, D.A. A Spectral Element Approximation to Price European Options with One Asset and Stochastic Volatility. J Sci Comput 42, 426–446 (2010). https://doi.org/10.1007/s10915-009-9333-x

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  • DOI: https://doi.org/10.1007/s10915-009-9333-x

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