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On the calibration of local jump-diffusion asset price models

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Abstract

We consider the inverse problem of calibrating a localized jump-diffusion process to given option price data. It is shown that applying Tikhonov regularization to the originally ill-posed problem yields a well-posed optimization problem. For the solution of the latter, i.e., the calibrated (infinite-dimensional) parameter of the process, we prove the stability and furthermore obtain convergence results. The work-horse for these proofs is the forward partial integro-differential equation associated to the European call price. Moreover, by providing a precise link between the parameters and the corresponding asset price models, we are able to carry over the stability and convergence results to the associated asset price models and hence to the model prices of exotic derivatives. Finally we indicate some possible applications.

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References

  1. Achdou, Y., Pironneau, O.: Numerical procedure for calibration of volatility with American options. Appl. Math. Finance 12, 201–241 (2005)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  3. Avellaneda, M., Buff, R., Friedman, C., Grandchamp, N., Kruk, L., Newman, J.: Weighted Monte Carlo: a new technique for calibrating asset-pricing models. Int. J. Theor. Appl. Finance 4, 91–119 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Belomestny, D., Reiss, M.: Spectral calibration for exponential Lévy models. Finance Stoch. 10, 449–474 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Belomestny, D., Reiss, M.: Implementing the spectral calibration of exponential Lévy models (2006). http://math.uni-heidelberg.de/studinfo/reiss/Publikationen/LevyCalImpl.pdf

  6. Bentata, A., Cont, R.: Mimicking the marginal distributions of a semimartingale (2009). Working paper, available at http://arxiv.org/abs/0910.3992v2

  7. Bentata, A., Cont, R.: Forward equations for option prices in semimartingale models (2010). Working paper, available at http://arxiv.org/abs/1001.1380v2

  8. Breeden, D., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978)

    Article  Google Scholar 

  9. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20, 1411–1421 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carmona, R., Nadtochiy, S.: An infinite dimensional stochastic analysis approach to local volatility dynamic models. Commun. Stoch. Anal. 2, 109–123 (2008)

    MathSciNet  Google Scholar 

  11. Carmona, R., Nadtochiy, S.: Local volatility dynamic models. Finance Stoch. 13, 1–48 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Carmona, R., Nadtochiy, S.: Tangent Lévy market models. Finance Stoch., in press (2011)

  13. Carr, P., Geman, H., Madan, D., Yor, M.: From local volatility to local Lévy models. Quant. Finance 4, 581–588 (2004)

    Article  MathSciNet  Google Scholar 

  14. Carr, P., Javaheri, A.: The forward PDE for European options on stocks with fixed fractional jumps. Int. J. Theor. Appl. Finance 8, 239–253 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cont, R., Tankov, P.: Nonparametric calibration of jump-diffusion option pricing models. J. Comput. Finance 7, 1–49 (2004)

    Google Scholar 

  16. Cont, R., Tankov, P.: Recovering exponential Lévy models from option prices: regularization of an ill-posed inverse problem. SIAM J. Control Optim. 43, 1–25 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Crépey, S.: Calibration of the local volatility in a generalized Black–Scholes model using Tikhonov regularization. SIAM J. Math. Anal. 34, 1183–1206 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Derman, E., Kani, I.: Riding on a smile. Risk 7, 32–39 (1994)

    Google Scholar 

  20. Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience Publishing, New York (1958)

    Google Scholar 

  21. Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)

    Google Scholar 

  22. Egger, H., Engl, H.W.: Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Inverse Probl. 21, 1027–1045 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Egger, H., Hein, T., Hofmann, B.: On decoupling of volatility smile and term structure in inverse option pricing. Inverse Probl. 22, 1247–1259 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Elstrodt, J.: Maß- und Integrationstheorie, 3rd edn. Grundwissen Mathematik. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  25. Engl, H.W.: Calibration problems—an inverse problems view. Wilmott, July, 16–20 (2007)

  26. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  27. Engl, H.W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl. 5, 523–540 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10, 39–52 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Goll, T., Rüschendorf, L.: Minimal distance martingale measures and the valuation problem in incomplete markets. In: Buckdahn, R., Engelbert, H.J., Yor, M. (eds.) Stochastic Processes and Related Topics. Stochastic Monographs, vol. 12, pp. 141–154. Taylor & Francis, London (2002)

    Google Scholar 

  30. Gyöngy, I.: Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Relat. Fields 71, 501–516 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hein, T.: Some analysis of Tikhonov regularization for the inverse problem of option pricing in the price-dependent case. Numer. Funct. Anal. Optim. 21, 439–464 (2000)

    Article  MathSciNet  Google Scholar 

  32. Hofmann, B., Kaltenbacher, B., Pöschl, C., Scherzer, O.: A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Probl. 23, 987–1010 (2007)

    Article  MATH  Google Scholar 

  33. Jacod, J., Protter, P.: Risk-neutral compatibility with option prices. Finance Stoch. 14, 285–315 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  35. Jiang, L., Chen, Q., Wang, L., Zhang, J.E.: A new well posed algorithm to recover local volatility. Quant. Finance 3, 451–457 (2003)

    Article  MathSciNet  Google Scholar 

  36. Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  37. Kallsen, J., Shiryaev, A.N.: The cumulant process and Esscher’s change of measure. Finance Stoch. 6, 397–428 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kindermann, S., Mayer, P., Albrecher, H., Engl, H.W.: Identification of the local speed function in a Lévy model for option pricing. J. Integral Equ. Appl. 12, 161–200 (2008)

    Article  MathSciNet  Google Scholar 

  39. Klebaner, F.: Option price when the stock is a semimartingale. Electron. Commun. Probab. 7, 79–83 (2002)

    MathSciNet  Google Scholar 

  40. Lagnado, R., Osher, S.: A technique for calibrating derivative security pricing models: numerical solution of the inverse problem. J. Comput. Finance 1, 13–25 (1997)

    Google Scholar 

  41. Matache, A.M., von Petersdorff, T., Schwab, C.: Fast deterministic pricing of options on Lévy driven assets. Mathematical Modelling and Numerical. Analysis 38, 37–71 (2004)

    MATH  Google Scholar 

  42. Matache, A.M., Schwab, C., Wihler, T.: Fast numerical solution of parabolic integrodifferential equations with applications in finance. SIAM J. Sci. Comput. 27, 369–393 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  43. Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  44. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  45. Schock, E.: On the asymptotic order of accuracy of Tikhonov regularization. J. Optim. Theory Appl. 44, 95–104 (1985)

    Article  MathSciNet  Google Scholar 

  46. Schoutens, W.: Lévy Processes in Finance. Wiley, Chichester (2003)

    Book  Google Scholar 

  47. Schweizer, M., Wissel, J.: Arbitrage-free market models for option prices: the multi-strike case. Finance Stoch. 12, 469–505 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  48. Schweizer, M., Wissel, J.: Term structures of implied volatilities: absence of arbitrage and existence results. Math. Finance 18, 77–114 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  49. Seidmann, T.I.: Vogel, C.R.: Well posedness and convergence of some regularization methods for non-linear ill posed problems. Inverse Probl. 5, 227–238 (1989)

    Article  Google Scholar 

  50. Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 209–244 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  51. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040, Linz, Austria

    S. Kindermann

  2. Department of Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    P. A. Mayer

Authors
  1. S. Kindermann
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  2. P. A. Mayer
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Correspondence to P. A. Mayer.

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The second author was supported by the Austrian Science Fund Project P-18392.

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Kindermann, S., Mayer, P.A. On the calibration of local jump-diffusion asset price models. Finance Stoch 15, 685–724 (2011). https://doi.org/10.1007/s00780-011-0159-7

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