Enhancing the Black-Scholes Model for Option Valuation via Lévy Processes and Malliavin Calculus

Battle Count: 138

The paper provides a theoretically rigorous framework for option pricing that addresses key limitations of Black-Scholes (volatility smile, fat tails, jump risk). This is directly relevant to options trading desks, derivatives pricing, and risk management. However, the empirical validation is limited, the model is restricted to European options, and no trading strategy or backtesting is presented. The theoretical contribution is strong but practical implementation guidance is minimal.

The paper involves advanced stochastic calculus including multidimensional Itô-Lévy formulas, Malliavin calculus (Clark-Ocone formula for combined Gaussian and jump noise), compensated Poisson random measures, and Lévy subordinators. Implementing the full pricing formula (Theorem 2.4) and the exact ATM implied volatility expression (Theorems 3.1 and 3.2) requires deep expertise in stochastic analysis. Numerical implementation would involve Monte Carlo simulation of jump-diffusion processes, computation of Malliavin derivatives, and integration over Lévy measures. The OU calibration is straightforward, but the full Lévy-Malliavin framework is highly complex.

The paper provides complete mathematical derivations, explicit parameter estimates (Table 2), and mentions Python 3.14.0 for calculations. However, no source code or repository is provided. The VIX data source (Cboe) is publicly available. The theoretical framework is fully specified with all assumptions stated, but empirical validation is limited to a single time period and a single strike price (240).