Simulation of Implied Volatility Surfaces via Tangent Lévy Models

In this paper, we implement and test two types of market-based models for European-type options, based on the tangent Levy models proposed recently by R. Carmona and S. Nadtochiy. As a result, we obtain a method for generating Monte Carlo samples of future paths of implied volatility surfaces. These paths and the surfaces themselves are free of arbitrage, and are constructed in a way that is consistent with the past and present values of implied volatility. We use a real market data to estimate the parameters of these models and conduct an empirical study, to compare the performance of market-based models with the performance of classical stochastic volatility models. We choose the problem of minimal-variance portfolio choice as a measure of model performance and compare the two tangent Levy models to SABR model. Our study demonstrates that the tangent Levy models do a much better job at finding a portfolio with smallest variance, their predictions for the variance are more reliable, and the portfolio weights are more stable. To the best of our knowledge, this is the first example of empirical analysis that provides a convincing evidence of the superior performance of the market-based models for European options using real market data.