Second-Order Esscher Pricing for L\'evy Models with Applications: Risk Management and Fear Quantification

This paper proposes the second-order Esscher transform as a tractable extension of the classical Esscher framework for option pricing and risk management in L\'evy-driven markets. For a general L\'evy process, we derive the associated densities and equivalent pricing measures, characterize the martingale condition in closed form, and obtain FFT-based valuation formulas for European call options. For jump-diffusion models, we establish explicit pricing formulas under the second-order Esscher measure and show that the resulting option prices lie in an interval bounded below by the Black--Scholes price and above by the underlying asset value. For the constant jump-diffusion model, we further prove monotonicity of option prices with respect to the second-order Esscher parameter. An empirical analysis based on market data shows that this additional parameter provides a tractable tool for stress testing, delta-hedging evaluation, and the construction of interval-valued risk measures in incomplete markets. We further document a strong association between the estimated second-order Esscher parameter and standard indicators of market stress, including the VIX, news sentiment, and crisis regimes. The proposed framework preserves analytical tractability while enlarging the class of admissible pricing measures, thereby supporting pricing, hedging, and stress-based risk assessment in incomplete markets with jump and general L\'evy dynamics.