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Multiscale exponential L�vy-type models

Author

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  • Matthew Lorig
  • Oriol Lozano-Carbass�

Abstract

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L�vy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors-one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of Fouque et al. [ Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives , 2011] to models of the exponential L�vy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility, as demonstrated in a calibration to S&P500 options data.

Suggested Citation

  • Matthew Lorig & Oriol Lozano-Carbass�, 2015. "Multiscale exponential L�vy-type models," Quantitative Finance, Taylor & Francis Journals, vol. 15(1), pages 91-100, January.
    • Handle: RePEc:taf:quantf:v:15:y:2015:i:1:p:91-100
    DOI: 10.1080/14697688.2014.934712
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    References listed on IDEAS

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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    4. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    5. repec:bla:jfinan:v:59:y:2004:i:3:p:1367-1404 is not listed on IDEAS
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    9. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2009. "Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options," CIRANO Working Papers 2009s-34, CIRANO.
    10. repec:dau:papers:123456789/1392 is not listed on IDEAS
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