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Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models

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Option prices in exponential Lévy models solve certain partial integro-differential equations. This work focuses on developing novel, computable error approximations for a finite difference scheme that is suitable for solving such PIDEs. The scheme was introduced in (Cont and Voltchkova, SIAM J. Numer. Anal. 43(4):1596–1626, 2005). The main results of this work are new estimates of the dominating error terms, namely the time and space discretisation errors. In addition, the leading order terms of the error estimates are determined in a form that is more amenable to computations. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschitz continuous as in previous works. If the underlying Lévy process has infinite jump activity, then the jumps smaller than some \(\epsilon > 0\) are approximated by diffusion. The resulting diffusion approximation error is also estimated, with leading order term in computable form, as well as the dependence of the time and space discretisation errors on this approximation. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the cut off parameter \(\epsilon \).

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  1. Andersen, L., Andreasen, J.: Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res. 4(4), 231–262 (2000)

    Article  MATH  Google Scholar 

  2. Bayer, C., Szepessy, A., Tempone, R.: Adaptive weak approximation of reflected and stopped diffusions. Monte Carlo Methods Appl. 16(1), 1–67 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertoin, J.: Lévy processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)

  4. Bangerth, W., Rannacher, R.: Adaptive finite element methods for differential equations. Birkhauser, Basel (2003)

    Book  MATH  Google Scholar 

  5. Bingham, N.H., Kiesel, R.: Risk-neutral valuation, 2nd edn. Springer Finance. Springer-Verlag London Ltd., London (2004)

  6. Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Financ. 13(3), 345–382 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cont, R., Tankov, P.: Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton (2004)

  8. Cont, R. Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43(4), 1596–1626 (electronic) (2005)

  9. Cont, R., Voltchkova, E.: Integro-differential equations for option prices in exponential Lévy models. Financ. Stoch. 9(3), 299–325 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dzougoutov, A. Moon, K.-S., von Schwerin, E., Szepessy, A., Tempone, R. Adaptive Monte Carlo algorithms for stopped diffusion. In: Multiscale Methods in Science and Engineering, vol. 44 of Lecture Notes Computer Science Engineering, pp. 59–88. Springer, Berlin (2005)

  11. Eriksson, K., Johnson, C.: Error estimates and automatic time step control for nonlinear parabolic problems. I. SIAM J. Numer. Anal. 24(1), 12–23 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hilber, N., Reich, N., Schwab, C., Winter, C.: Numerical methods for Lévy processes. Financ. Stoch. 13(4), 471–500 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Johnson, C.: Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 25(4), 908–926 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Katsoulakis, M.A., Szepessy, A.: Stochastic hydrodynamical limits of particle systems. Commun. Math. Sci. 4(3), 513–549 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kiessling, J., Tempone, R.: Diffusion approximation of Lévy processes with a view towards finance. Monte Carlo Methods Appl. 17(1), 11–45 (2010)

    MathSciNet  Google Scholar 

  16. Koponen, I.: Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52(1), 1197–1199 (Jul 1995)

  17. Steven, G.: Kou. A Jump Diffusion Model For Option Pricing, SSRN eLibrary (2001)

  18. Madan, D.B., Carr, P.P., Chang, E.C.: The variance gamma process and option pricing. Eur. Financ. Rev. 2(1), 79–105 (1998)

    Article  MATH  Google Scholar 

  19. Matache, A.-M., von Petersdorff, T., Schwab, C.: Fast deterministic pricing of options on Lévy driven assets. M2AN. Math. Model. Numer. Anal. 38(1), 37–71 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1976)

    Article  MATH  Google Scholar 

  21. Mordecki, E., Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of diffusions with jumps. SIAM J. Numer. Anal. 46(4), 1732–1768 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Reich, N., Schwab, C., Winter, C.: On Kolmogorov equations for anisotropic multivariate Lévy processes. Financ. Stoch. 14(4), 527–567 (2010)

  23. Rydberg, T.H.: The normal inverse Gaussian Lévy process: simulation and approximation. Commun. Stat. Stoch. Models 13(4), 887–910 (1997) Heavy tails and highly volatile phenomena

  24. Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, volu. 68. Cambridge University Press, Cambridge (1999). Translated from the 1990 Japanese original, Revised by the author

  25. Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54(10), 1169–1214 (2001)

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The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) is also acknowledged. The research has also been supported by the Swedish Foundation for Strategic Research (SSF) via the Center for Industrial and Applied Mathematics (CIAM) at KTH. The second author is a member of the KAUST SRI center for Uncertainty Quantification.

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Correspondence to Jonas Kiessling.

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Communicated by Desmond Higham.

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Kiessling, J., Tempone, R. Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models. Bit Numer Math 54, 1023–1065 (2014).

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