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L\'evy-Driven Option Pricing without a Riskless Asset

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  • Ziyao Wang

Abstract

We extend the Lindquist-Rachev (LR) option-pricing framework--which values derivatives in markets lacking a traded risk-free bond--by introducing common Levy jump dynamics across two risky assets. The resulting endogenous "shadow" short rate replaces the usual risk-free yield and governs discounting and risk-neutral drifts. We focus on two widely used pure-jump specifications: the Normal Inverse Gaussian (NIG) process and the Carr-Geman-Madan-Yor (CGMY) tempered-stable process. Using Ito-Levy calculus we derive an LR partial integro-differential equation (LR-PIDE) and obtain European option values through characteristic-function methods implemented with the Fast Fourier Transform (FFT) and Fourier-cosine (COS) algorithms. Calibrations to S and P 500 index options show that both jump models materially reduce pricing errors and fit the observed volatility smile far better than the Black-Scholes benchmark; CGMY delivers the largest improvement. We also extract time-varying shadow short rates from paired asset data and show that sharp declines coincide with liquidity-stress episodes, highlighting risk signals not visible in Treasury yields. The framework links jump risk, relative asset pricing, and funding conditions in a tractable form for practitioners.

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  • Ziyao Wang, 2025. "L\'evy-Driven Option Pricing without a Riskless Asset," Papers 2507.20338, arXiv.org.
    • Handle: RePEc:arx:papers:2507.20338
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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. W. Brent Lindquist & Svetlozar T. Rachev, 2025. "Alternatives to classical option pricing," Annals of Operations Research, Springer, vol. 346(1), pages 489-509, March.
    3. 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.
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