Review of Derivatives Research

, Volume 4, Issue 3, pp 231–262

Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing

  • Leif Andersen
  • Jesper Andreasen
Article

Abstract

This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (PartialIntegro-Differential Equation) and demonstrates how this equationcan be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples fromthe S&P500 market.

jump-diffusion process local time forward equation volatility smile ADI finite difference method fast Fourier transform 

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References

  1. Ait-Sahalia, Y., Yubo Wang, and Francis Yared. (1998). “Do Option Markets Correctly Asses the Probabilities of Movements of the Underlying Asset?” Forthcoming, Journal of Econometrics.Google Scholar
  2. Amin, Kaushuk. (1996). “Jump Diffusion OptionValuation in Discrete Time,” Journal of Finance 48, 1833–1863.CrossRefGoogle Scholar
  3. Andersen, Leif, and Rupert Brotherton-Ratcliffe. (1998). “The Equity OptionVolatility Smile: AFinite Difference Approach,” Journal of Computational Finance 1, 2, 5–38.Google Scholar
  4. Andersen, Torben, Luca Benzoni, and Jesper Lund. (1999). “Estimating Jump-Diffusions for Equity Returns,” Working Paper, Northwestern University and Aarhus School of Business.Google Scholar
  5. Andreasen, Jesper. (1997). “Implied Modelling: Stable Implementation, Hedging, and Duality,” Working Paper, University of Aarhus.Google Scholar
  6. Andreasen, Jesper. (1998). “The Pricing of Discretely Sampled Asian and Lookback Options: A Change of Numeraire Approach,” Journal of Computational Finance 2, 1, 5–30.Google Scholar
  7. Andreasen, Jesper, and Barbara Gruenewald. (1996). “American Option Pricing in the Jump-Diffusion Model,” Working Paper, Aarhus University and University of Mainz.Google Scholar
  8. Avallaneda Marco, Craig Friedman, Richard Holmes, and Dominick Samperi. (1997). “Calibrating Volatility Surfaces via Relative-Entropy Minimization,” Applied Mathematical Finance 4, 1.CrossRefGoogle Scholar
  9. Bakshi, Gurdip, Charles Cao, and Zhiwu Chen. (1997). “Empirical Performance of Alternative Option Pricing Models,” Journal of Finance 52, 2003–2049.CrossRefGoogle Scholar
  10. Bates, David. (1996). “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Review of Financial Studies 9, 1, 69–107.CrossRefGoogle Scholar
  11. Bjork, Tomas, Yuri Kabanov, and Wolfgang Runggaldier. (1997). “Bond Market Structure in the Presence of Marked Point Processes,” Mathematical Finance 7, 211–239.CrossRefGoogle Scholar
  12. Black, Fischer, and Myron Scholes. (1973). “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81, 637–654.CrossRefGoogle Scholar
  13. Boyle, Phelim, Mark Broadie, and Paul Glasserman. (1997). “Monte Carlo Methods for Security Pricing,” Journal of Economic Dynamics and Control 21, 8–9, 1267–1321.CrossRefGoogle Scholar
  14. Breeden, Douglas, and Robert Litzenberger. (1978). “Prices of State-Contingent Claims Implicit in Options Prices,” Journal of Business 51, October, 621–651.Google Scholar
  15. Brown, Gregory, and Klaus Toft. (1999). “Constructing Implied Binomial Trees from Multiple Probability Distributions,” Journal of Derivatives 7, 2, 83–100.CrossRefGoogle Scholar
  16. Buraschi, Andrea, and Jens Jackwerth. (1998). “Explaining Option Prices: Deterministic vs. Stochastic Models,” Working Paper, London Business School.Google Scholar
  17. Coleman, Thomas, Yuying Li, and Arun Verma. (1999). “Reconstructing the Optimal Volatility Surface,” Journal of Computational Finance 2, 3, 77–102.Google Scholar
  18. Chriss, Neill. (1996). “Transatlantic Trees,” RISK July, 45–48.Google Scholar
  19. Das, Sanjiv, and Silverio Foresi. (1996). “Exact Solutions for Bond and Option Prices with Systematic Jump Risk,” Review of Derivatives Research 1, 7–24.CrossRefGoogle Scholar
  20. Derman, Emanuel and Iraj Kani. (1994). “Riding on a Smile,” RISK Magazine February, 32–39.Google Scholar
  21. Dierckx, Paul. (1995). Curve and Surface Fitting with Splines. Oxford Science Publications.Google Scholar
  22. Dumas, Bernard, Jeff Fleming, and Robert E. Whaley. (1997). “Implied Volatility Functions: Empirical Tests,” Journal of Finance 53, 2059–2106.CrossRefGoogle Scholar
  23. Duffie, Darrell, Jun Pan, and Kenneth Singleton. (1999). “Transform Analysis and Option Pricing for Affine Jump-Diffusions,” Working Paper, Stanford University.Google Scholar
  24. Dupire, Bruno. (1994). “Pricing with a Smile,” RISK Magazine January, 18–20.Google Scholar
  25. Heston, Steven. (1993). “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies 6, 2, 327–343.CrossRefGoogle Scholar
  26. Hull, John, and Allan White. (1987). “The Pricing of Options with Stochastic Volatilities,” Journal of Finance 42, 281–300.CrossRefGoogle Scholar
  27. Jackwerth, Jens. (1996). “Generalized Binomial Trees,” Working Paper, University of California at Berkeley.Google Scholar
  28. Karatzas, Ioannis, and Steven Shreve. (1991). Brownian Motion and Stochastic Calculus. Springer Verlag.Google Scholar
  29. Kloeden, Peter, and Eckhardt Platen. (1992). Numerical Solution of Stochastic Differential Equations. Springer Verlag.Google Scholar
  30. Krishnan, Venkatarama. (1984). Nonlinear Filtering and Smoothing. John Wiley and Sons.Google Scholar
  31. Lagnado, Ronald, and Stanley Osher. (1997). “Reconciling Differences,” RISK Magazine April, 79–83.Google Scholar
  32. Merton, Robert. (1976). “Option Pricing when Underlying Stock Returns are Discontinuous,” Journal ofFinancial Economics May, 125–144.Google Scholar
  33. Mitchell, Andrew, and D. F. Griffiths. (1980). The Finite Difference Method in Partial Differential Equations. John Wiley & Sons.Google Scholar
  34. Naik, Vasant, and Moon Lee. (1990). “General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns,” The Review of Financial Studies 3, 493–521.CrossRefGoogle Scholar
  35. Papparlardo, Luca. (1996), “Option Pricing and Smile Effect when Underlying Stock Prices are Driven by a Jump Process,” Working Paper, University of Warwick.Google Scholar
  36. Press, William H., Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. (1992). Numerical Recipes in C. Cambridge University Press.Google Scholar
  37. Rubinstein, Mark. (1994). “Implied Binomial Trees,” Journal of Finance 49, 771–818.CrossRefGoogle Scholar
  38. Stein, Elias, and Jeremy Stein. (1991). “Stock Price Distributions with Stochastic Volatility: An Analytic Approach,” Review of Financial Studies 4, 4, 727–752.CrossRefGoogle Scholar
  39. Zhang, X. (1993). “Options Americaines et Modeles de Diffusion avec Sauts,” C. R. Acad. Sci. Paris, Serie I, 857–862.Google Scholar
  40. Zvan, Robert, Peter Forsyth, and Kenneth Vetzal. (1998). “Robust Numerical Methods for PDE Models of Asian Options,” Journal of Computational Finance 1, 39–78.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Leif Andersen
    • 1
  • Jesper Andreasen
    • 2
  1. 1.Gen Re SecuritiesUSA
  2. 2.Bank of AmericaUSA

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