, Volume 44, Issue 1, pp 33–57 | Cite as

Implicit–explicit numerical schemes for jump–diffusion processes

  • Maya Briani
  • Roberto Natalini
  • Giovanni Russo


We study the numerical approximation of solutions for parabolic integro-differential equations (PIDE). Similar models arise in option pricing, to generalize the Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term, unconditionally stable implicit difference schemes are not practically feasible. Here we propose using implicit-explicit (IMEX) Runge-Kutta methods for the time integration to solve the integral term explicitly, giving higher-order accuracy schemes under weak stability time-step restrictions. Numerical tests are presented to show the computational efficiency of the approximation.

Mathematics Subject Classification (1991): Primary: 65M12; Secondary: 35K55, 49L25


Stability Region Option Price Stochastic Volatility Integral Term Explicit Numerical Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1. Amadori, A.L.: Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. Differential Integral Equations 16, 787–811 (2003)zbMATHMathSciNetGoogle Scholar
  2. 2. Andersen, L., Andreasen, J.: Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev. Derivatives Research 4, 231–262 (2000)CrossRefGoogle Scholar
  3. 3. Ascher, U., Ruuth, S., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependant partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4. Barndorff-Nielsen, O., Mikosch T., Resnick S.: Lévy processes. Theory and applications. Boston: Birkhäuser, Boston 2001Google Scholar
  5. 5. Barndorff-Nielsen, O., Shephard, N.: Modeling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E. et al. (eds.): Lévy processes. Theory and applications. Boston: Birkhäuser, Boston 2001, pp. 283–318Google Scholar
  6. 6. Björk, T., Kabanov, Y., Runggaldier, W.: Bond market structure in the presence of marked point processes. Math. Finance 7, 211–239 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7. Black, F., Scholes, M.: The pricing of option and corporate liabilities. J. Political Economy 81, 637–659 (1973)CrossRefGoogle Scholar
  8. 8. Briani, M., La Chioma, C., Natalini, R.: Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Numer. Math. 98, 607–646 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. Berlin: Springer 1988Google Scholar
  10. 10. Cont, R., Tankov, P.: Financial modelling with jump processes. Boca Raton, FL: Chapman & Hall/CRC 2004Google Scholar
  11. 11. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12. Das, S., Foresi, S.: Exact solutions for bond and option prices with systematic jump risk. Rev. Derivatives Research 1, 7–24 (1996)CrossRefGoogle Scholar
  13. 13. d'Halluin, Y., Forsyth, P.A., Labahn, G.: A penalty method for American options with jump diffusion processes. Numer. Math. 97, 321–352 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15. Eberlein, E., Raible, S.: Term structure models driven by general Lévy processes. Math. Finance 9, 31–53 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  17. 17. Hull, J., White, A.: The pricing of options with stochastic volatilities. J. Finance 42, 281–300 (1987)CrossRefGoogle Scholar
  18. 18. Karniadakis, G.E., Israeli, M., Orszag, S.A.: High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19. Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44, 139–181 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20. Lapeyre, B., Lamberton, D.: An introduction to stochastic calculus applied to finance. London: Chapman & Hall 1996Google Scholar
  21. 21. Madan, D.B., Seneta, E.: The variance gamma model for share market returns. J. Business 63, 511–524 (1990)CrossRefGoogle Scholar
  22. 22. Madan, D., Milne, F.: Option pricing with variance gamma martingale components. Math. Finance 1, 39–56 (1991)zbMATHCrossRefGoogle Scholar
  23. 23. Matache, A.M., Schwab, C., Wihler, T.P.: Fast numerical solution of parabolic integrodifferential equations with applications in finance. SIAM J. Sci. Comput. 27, 369–393 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24. Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3, 125–144 (1976)CrossRefGoogle Scholar
  25. 25. Page, F.H., Sanders, A.B.: A general derivation of the jump process option pricing formula. J. Financial Quant. Anal. 21, 437–446 (1986)CrossRefGoogle Scholar
  26. 26. Pareschi, L., Russo, G.: High order asymptotically strong-stability-preserving methods for hyperbolic systems with stiff relaxation. In: Hou, T.Y., Tadmor, E. (eds.): Hyperbolic problems: theory, numerics, applications. Berlin: Springer 2003, pp. 241–251Google Scholar
  27. 27. Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic system with relaxation. J. Sci. Comput. 25, 129–155 (2005)CrossRefMathSciNetGoogle Scholar
  28. 28. Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes for stiff system of differential equations. In: Trigiante, D. (ed.): Recent trends in numerical analysis. Dedicated to the 65th birthday of Professor I. Galligani. (Advances in the Theory of Computational Mathematics 3), Huntington, NY: Nova Science Publishers 2001, pp. 269–288Google Scholar
  29. 29. Randall, C., Tavella, D.: Pricing financial instruments: the finite difference method. New York: Wiley 2000Google Scholar
  30. 30. Strikwerda, J.C.: Finite difference schemes and partial differential equations. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software 1989Google Scholar
  31. 31. Verwer, J.G., Blom, J.G., Hundsdorfer, W.: An implicit-explicit approach for atmospheric transport-chemistry problems. Appl. Numer. Math. 20, 191–209 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32. Yong, J., Zhou, X.Y.: Stochastic controls. Hamiltonian systems and HJB equations. (Applications of Mathematics 43) New York: Springer 1999Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Maya Briani
    • 1
  • Roberto Natalini
    • 2
  • Giovanni Russo
    • 3
  1. 1.Istituto per le Applicazioni del Calcolo “Mauro Picone” – CNR, 00161 Rome, Italy, & LUISS Guido Carli, 00198 RomeItaly
  2. 2.Istituto per le Applicazioni del Calcolo “Mauro Picone” – CNR, 00161 RomeItaly
  3. 3.Dipartimento di Matematica Pura e Informatica, Università di Catania, 95129 CataniaItaly

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