Simulation of Implied Volatility Surfaces via Tangent Lévy Models

Abstract

In this paper, we implement and test a market-based model for European-type options, based on the tangent Lévy models proposed in [R. Carmona and S. Nadtochiy, Int. J. Theoret. Appl. Finance, 14 (2001), pp. 107--135; Finance Stoch., 16 (2012), pp. 63--104]. 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 market data to estimate the parameters of this model, and conduct an empirical study to compare the performance of the chosen market-based model with the classical SABR (stochatic alpha beta rho) model and with the method based on direct simulation of implied volatility, described in [R. Cont and J. da Fonseca, Quant. Finance, 2 (2002), pp. 45--60]. We focus on the problem of minimal-variance portfolio choice as the main measure of model performance and compare the three models. Our study demonstrates that the tangent Lévy model does a better job at finding a portfolio with the smallest variance than does the SABR model. In addition, the prediction of return variance provided by the tangent Lévy model is more reliable and the portfolio weights are more stable. We also find that the performance of the direct simulation method on the portfolio choice problem is not much worse than that of the tangent Lévy model. However, the direct simulation method of Cont and da Fonseca is not arbitrage-free. We illustrate this shortcoming by comparing the direct simulation method and the tangent Lévy model on a different problem---estimation of Value at Risk of an options' portfolio. To the best of our knowledge, this paper is the first example of empirical analysis, based on real market data, which provides convincing evidence of the superior performance of market-based models for European options, as compared to the classical spot models.