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Calibration and hedging under jump diffusion

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Abstract

A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). By generating a set of option prices assuming a jump diffusion with known parameters, we investigate two crucial challenges intrinsic to this type of model: calibration of parameters and hedging of jump risk. Even though the estimation problem is ill-posed, our results suggest that the model can be calibrated with sufficient accuracy. Two different strategies are explored for hedging jump risk: a semi-static approach and a dynamic technique. Simulation experiments indicate that each of these methods can sharply reduce risk exposure.

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References

  • Andersen, L. and J. Andreasen. (1999). “Jumping Smiles,” Risk 12(11), 65–68.

    Google Scholar 

  • Andersen, L. and J. Andreasen. (2000). “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing,” Review of Derivatives Research 4, 231–262.

    Article  Google Scholar 

  • Bakshi, G. and C. Cao. (2003). “Risk-Neutral Kurtosis, Jumps, and Option Pricing: Evidence from 100 Most Actively Traded Firms on the CBOE,” Working paper, Smith School of Business, University of Maryland.

  • Bakshi, G., C. Cao, and Z. Chen. (1997). “Empirical Performance of Alternative Option Pricing Models,” Journal of Finance 52, 2003–2049.

    Article  Google Scholar 

  • Bates, D. S. (1988). “Pricing options under jump-diffusion processes,” Working paper 37–88, Rodney L. White Center for Financial Research, The Wharton School, University of Pennsylvania.

  • Bates, D. S. (1991). “The Crash of ’87: Was it Expected? The Evidence from Options Markets,” Journal of Finance 46, 1009–1044.

    Article  Google Scholar 

  • Bates, D. S. (2000). “Post-’87 Crash Fears in the S&P 500 Futures Option Market,” Journal of Econometrics 94, 181–238.

    Article  Google Scholar 

  • Broadie, M., M. Chernov, and M. Johannes. (2005). “Model Specification and Risk Premia: Evidence from Futures Options,” Working paper, Columbia Business School.

  • Carr, P. and A. Hirsa. (2003). “Why be Backward?,” Risk 16(1), 103–107.

    Google Scholar 

  • Carr, P. and L. Wu. (2004). “Static Hedging of Standard Options,” Working paper, Courant Institute, New York University.

  • Chernov, M. and E. Ghysels. (2000). “A Study Towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation,” Journal of Financial Economics 56, 407–458.

    Article  Google Scholar 

  • Coleman, T. F., Y. Kim, Y. Li, and A. Verma. (2001). “Dynamic Hedging With a Deterministic Volatility Function Model,” Journal of Risk 4(1), 63–90.

    Article  Google Scholar 

  • Coleman, T. F. and Y. Li. (1996). “An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds,” SIAM Journal on Optimization 6, 418–445.

    Article  Google Scholar 

  • Coleman, T. F., Y. Li, and A. Verma. (1999). “Reconstructing the Unknown Local Volatility Function,” Journal of Computational Finance 2(3), 77–102.

    Google Scholar 

  • Coleman, T. F., Y. Li, and A. Verma. (2002). “A Newton Method for American Option Pricing,” Journal of Computational Finance 5(3), 51–78.

    Google Scholar 

  • Cont, R. and P. Tankov. (2004a). Financial Modelling with Jump Processes. Boca Raton: Chapman & Hall/CRC.

    Google Scholar 

  • Cont, R. and P. Tankov. (2004b). “Non-Parametric Calibration of Jump-Diffusion Option Pricing Models,” Journal of Computational Finance 7(2), 1–49.

    Google Scholar 

  • Cox, J. C. and S. A. Ross. (1976). “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics 3, 145–166.

    Article  Google Scholar 

  • d’Halluin, Y., P. Forsyth, and K. Vetzal. (2005). “Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes,” IMA Journal of Numerical Analysis 25, 65–92.

    Article  Google Scholar 

  • d’Halluin, Y., P. A. Forsyth, and G. Labahn. (2004). “A Penalty Method for American Options with Jump Diffusion Processes,” Numerische Mathematik 97, 321–352.

    Article  Google Scholar 

  • Dierckx, P. (1993). Curve and Surface Fitting with Splines. Oxford: Oxford University Press.

    Google Scholar 

  • Dueker, M. and T. W. Miller. (2003). “Directly Measuring Early Exercise Premiums Using American and European S&P 500 Index Options,” Journal of Futures Markets 23, 287–313.

    Article  Google Scholar 

  • Duffie, D., J. Pan, and K. Singleton. (2000). “Transform Analysis and Asset Pricing for Affine Jump-Diffusions,” Econometrica 68, 1343–1376.

    Article  Google Scholar 

  • Dumas, B., J. Fleming, and R. E. Whaley. (1998). “Implied Volatility Functions: Empirical Tests,” Journal of Finance 53, 2059–2106.

    Article  Google Scholar 

  • Dupire, B. (1994). “Pricing With a Smile,” Risk 7(1), 18–20.

    Google Scholar 

  • Eraker, B., M. Johannes, and N. Polson. (2003). “The Impact of Jumps in Volatility and Returns,” Journal of Finance 58, 1269–1300.

    Article  Google Scholar 

  • Harvey, C. R. and R. E. Whaley. (1992). “Dividends and S&P 100 Index Option Valuation,” Journal of Futures Markets 12, 123–137.

    Google Scholar 

  • Heston, S. L. (1993). “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies 6, 327–343.

    Article  Google Scholar 

  • Hull, J. and A. White. (1987). “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance 42, 281–300.

    Article  Google Scholar 

  • Labahn, G. (2003). “Closed Form PDF for Merton’s Jump Diffusion Model,” Technical report, School of Computer Science, University of Waterloo.

  • Lagnado, R. and S. Osher. (1997). “Reconciling Differences,” Risk 10(4), 79–83.

    Google Scholar 

  • Liu, J., J. Pan, and T. Wang. (2005). “An Equilibrium Model of Rare-Event Premia and Its Implication for Option Smirks,” Review of Financial Studies 18, 131–164.

    Article  Google Scholar 

  • Merton, R. C. (1976). “Option Pricing when Underlying Stock Returns are Discontinuous,” Journal of Financial Economics 3, 125–144.

    Article  Google Scholar 

  • Naik, V. and M. Lee. (1990). “General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns,” Review of Financial Studies 3, 493–521.

    Article  Google Scholar 

  • Pan, J. (2002). “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study,” Journal of Financial Economics 63, 3–50.

    Article  Google Scholar 

  • Rubinstein, M. (1994). “Implied Binomial Trees,” Journal of Finance 49, 771–818.

    Article  Google Scholar 

  • Sung, H. M. (1995). “The Early Exercise Premia of American Put Options on Stocks,” Review of Quantitative Finance and Accounting 5, 365–373.

    Google Scholar 

  • Tikhonov, A. N. and V. Y. Arsenin. (1977). Solutions of Ill-Posed Problems. Washington: W. H. Winston.

    Google Scholar 

  • Vapnik, V. N. (1982). Estimation of Dependences Based on Empirical Data. Berlin: Springer-Verlag.

    Google Scholar 

  • Wahba, G. (1990). Spline Models for Observational Data. Philadelphia: SIAM.

    Google Scholar 

  • Wolcott, R. (2004). “Two of a kind?,” Risk 17(3), 24–26.

    Google Scholar 

  • Zivney, T. L. (1991). “The Value of Early Exercise in Option Prices: An Empirical Investigation,” Journal of Financial and Quantitative Analysis 26, 129–138.

    Article  Google Scholar 

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Authors and Affiliations

  1. J.P. Morgan Securities Inc., 270 Park Ave, Floor 6, New York, NY, 10017-2070, USA

    C. He

  2. Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    J. S. Kennedy

  3. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    T. F. Coleman

  4. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    P. A. Forsyth & Y. Li

  5. Centre for Advanced Studies in Finance, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    K. R. Vetzal

Authors
  1. C. He
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  2. J. S. Kennedy
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  3. T. F. Coleman
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  4. P. A. Forsyth
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  5. Y. Li
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  6. K. R. Vetzal
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Correspondence to K. R. Vetzal.

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JEL Classification G12 · G13

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He, C., Kennedy, J.S., Coleman, T.F. et al. Calibration and hedging under jump diffusion. Rev Deriv Res 9, 1–35 (2006). https://doi.org/10.1007/s11147-006-9003-1

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