ScholarWorks@성균관대학교

초록

We propose a stable numerical method for constructing a bias-reduced nonparametric local volatility surface without relying on a pre-constructed continuous implied volatility surface. Applying the Feynman-Kac formula to Dupire's equation, we introduce the concept of average local volatility. We construct a natural cubic spline to approximate continuous call option values in the strike direction and evaluate the Feynman-Kac expectation. Subsequently, we estimate the expectation by analytically integrating the call-option spline with the stochastic process governed by the average local volatility. Average local volatility values at grid points are then determined using the bisection method. Finally, we employ an interpolation method on these values to achieve a continuous local volatility surface across the entire domain. Our numerical analysis and experiments demonstrate that the proposed method enhances numerical stability across various time and space discretizations, outperforming the traditional finite difference method. Moreover, we establish that our method maintains stability under data sparsity and uncertainty conditions, adhering to Lipschitz continuity. An empirical application using the DAX option chain, together with recent snapshots from the S&P500 and EURO STOXX50, confirms the efficacy of our method in reducing bias in the local volatility surface, thereby enabling the detection of market distortions and arbitrage opportunities.

키워드