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A note on exact likelihoods of the Carr-Wu models for leverage effects and volatility in financial economics

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  • Lancelot F. James

Abstract

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed L\'evy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and the instantaneous volatility. Naturally similar to a recent work of Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a likelihood based framework. These likelihoods are based on the unobserved integrated volatility, rather than the instantaneous volatility. James (2005) establishes general results for the likelihood and estimation of a large class of such models which include possible leverage effects. In this note we show that exact expressions for likelihood models based on generalizations of Carr and Wu (2005) and Huang and Wu (2005), follow essentially from the arguments in Theorem 5.1 in James (2005) with some slight modification. This serves to formally verify a claim made by James (2005).

Suggested Citation

  • Lancelot F. James, 2005. "A note on exact likelihoods of the Carr-Wu models for leverage effects and volatility in financial economics," Papers math/0503314, arXiv.org, revised Mar 2005.
    • Handle: RePEc:arx:papers:math/0503314
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    1. Lancelot F. James, 2005. "Analysis of a Class of Likelihood Based Continuous Time Stochastic Volatility Models including Ornstein-Uhlenbeck Models in Financial Economics," Papers math/0503055, arXiv.org, revised Aug 2005.
    2. Jing-zhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time-Changed Lévy Processes," Journal of Finance, American Finance Association, vol. 59(3), pages 1405-1440, June.
    3. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    4. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
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