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Valuation of asset and volatility derivatives using decoupled time-changed L\'evy processes

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  • Lorenzo Torricelli

Abstract

In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) L\'evy processes to model the underlying asset of contingent claims. A DTC L\'evy process is a generalized time-changed L\'evy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC L\'evy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying L\'evy decomposition allows to introduce asset price models consistent with the assumption of a correlated pair of continuous and jump market activities; we study one illustrative DTC model having this property by assuming that the instantaneous activity rates follow the the so-called Wishart process. The theory developed is applied to the problem of pricing claims depending not only on the price or the volatility of an underlying asset, but also to more sophisticated derivatives that pay-off on the joint performance of these two financial variables, like the target volatility option (TVO). We solve the pricing problem through a Fourier-inversion method; numerical computations validating our technique are provided.

Suggested Citation

  • Lorenzo Torricelli, 2012. "Valuation of asset and volatility derivatives using decoupled time-changed L\'evy processes," Papers 1210.5479, arXiv.org, revised Jan 2015.
    • Handle: RePEc:arx:papers:1210.5479
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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. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    3. Giuseppe Di Graziano & Lorenzo Torricelli, 2012. "Target Volatility Option Pricing," World Scientific Book Chapters, in: Matheus R Grasselli & Lane P Hughston (ed.), Finance at Fields, chapter 8, pages 207-223, World Scientific Publishing Co. Pte. Ltd..
    4. Lorenzo Torricelli, 2012. "Pricing joint claims on an asset and its realized variance under stochastic volatility models," Papers 1206.2112, arXiv.org.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    7. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    9. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    10. Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
    11. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
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