Enhancing the Black-Scholes Model for Option Valuation via L\'evy Processes and Malliavin Calculus
The Black-Scholes model has been extensively used for option pricing, but exhibits limitations in its reliance on geometric Brownian motion and fixed volatility assumptions. This paper proposes an enhanced model incorporating stochastic volatility with jumps modeled by a L\'evy process. Leveraging multidimensional It\^o calculus, we derive a pricing formula for European call options under the new framework. Additionally, Malliavin calculus enables the derivation of an exact expression for at-the-money implied volatility. The proposed model is shown to better capture empirical features like volatility smiles. Analysis of VIX data demonstrates the model's ability to match observed market volatility. The integration of L\'evy processes and Malliavin calculus represents a valuable advancement in addressing deficiencies in the classic Black-Scholes model. Further empirical testing is warranted to validate the approach across varying market conditions and option types.