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

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.