Paper: arXiv 2507.20338

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a highly rigorous theoretical extension of the Lindquist-Rachev framework by incorporating pure-jump Lévy processes (NIG and CGMY) via Itô-Lévy calculus. It validates the theoretical results empirically by calibrating to S&P 500 options data using FFT/COS methods and demonstrating clear performance improvements over Black-Scholes while extracting meaningful shadow risk signals.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Lindquist-Rachev Framework to Lévy Jumps without Risk-Free Asset] --> B[Model Dynamics: Introduce Common NIG & CGMY Jump Processes for Two Risky Assets]
    B --> C[Derive Endogenous Shadow Short Rate & LR-PIDE via Itô-Lévy Calculus]
    C --> D[Data & Calibration: S&P 500 Index Options & Paired Asset Data]
    D --> E[Numerical Pricing: Compute Option Values via FFT and COS Algorithms]
    E --> F[Key Findings: Superior Smile Fit, Reduced Pricing Errors CGMY Best, & Shadow Rate Reveals Liquidity Stress]

Cited References (14)

  1. The pricing of options and corporate liabilities (1973) — link
  2. Alternatives to Classical Option Pricing (2025) — link
  3. Financial Markets with No Riskless (Safe) Asset (2017) — link
  4. Option Pricing When Underlying Stock Returns Are Discontinuous (1976) — link
  5. The Variance Gamma (V.G.) Model for Share Market Returns (1990) — link
  6. The Variance Gamma Process and Option Pricing (1998) — link
  7. Processes of Normal Inverse Gaussian Type (1998) — link
  8. The Fine Structure of Asset Returns: An Empirical Investigation (2002) — link
and 6 more
  1. Financial Modelling with Jump Processes (2004)
  2. Option Valuation Using the Fast Fourier Transform (1999)
  3. A Novel Pricing Method for European Options Based on Fourier–Cosine Series Expansions (2009) — link
  4. On a Non-Classical Invariance Principle (2008) — link
  5. Limit Theorems for Stochastic Processes (2003) — link
  6. A General Version of the Fundamental Theorem of Asset Pricing (1994) — link

0 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.