Applications of the Second-Order Esscher Pricing in Risk Management

Paper: arXiv 2410.21649

Authors: Tahir Choulli, Ella Elazkany, Mich`ele Vanmaele

Abstract

This paper explores the application and significance of the second-order Esscher pricing model in option pricing and risk management. We split the study into two main parts. First, we focus on the constant jump diffusion (CJD) case, analyzing the behavior of option prices as a function of the second-order parameter and the resulting pricing intervals. Using real data, we perform a dynamic delta hedging strategy, illustrating how risk managers can determine an interval of value-at-risks (VaR) and expected shortfalls (ES), granting flexibility in pricing based on additional information. We compare our pricing interval to other jump-diffusion models, showing its comprehensive risk factor incorporation. The second part extends the second-order Esscher pricing to more complex models, including the Merton jump-diffusion, Kou’s Double Exponential jump-diffusion, and the Variance Gamma model. We derive option prices using the fast Fourier transform (FFT) method and provide practical formulas for European call and put options under these models.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical extension of continuous-time option pricing via second-order Esscher transforms, deriving closed-form and FFT-based solutions for jump-diffusion models. While strongly theoretical, it integrates real-data applications to demonstrate dynamic delta hedging and VaR/ES risk management intervals.

Research Flowchart

  flowchart TD
    A[Research Goal: Evaluate Second-Order Esscher Pricing in Option Pricing & Risk Management] --> B[Model Setups & Data Inputs: Real Market Data + Jump-Diffusion & Variance Gamma Models]
    B --> C[Part 1: Constant Jump Diffusion CJD Analysis]
    B --> D[Part 2: Complex Model Extensions Merton, Kou, Variance Gamma]
    
    C --> E[Dynamic Delta Hedging & VaR/ES Interval Computation]
    D --> F[Fast Fourier Transform FFT Pricing Derivations]
    
    E --> G[Outcomes: Flexible VaR/ES Risk Intervals & Comprehensive Factor Incorporation]
    F --> G