Implied Volatility Surface Simulation with Tangent Levy Models

Abstract

With the recent developments of a liquid derivative market, as well as the demands for an improved risk management framework post the financial crisis, it is becoming increasingly important to consistently model the implied volatility dynamics of an asset. Many attempts have been made on this front, but few manage to exclude arbitrage opportunities with reasonable tractability.

In this thesis, we present two approaches based on tangent Levy models to achieve the task. One of the biggest advantages of tangent Levy models is that, by using the tangent process' jump density as the codebook to describe the option price dynamics, it enables an explicit expression of the no-arbitrage conditions, hence allows for tractable implementation.

Our first approach is based on the tangent Levy model with tangent processes being derived from the double exponential process. This approach is easy to implement given the small number of parameters and the availability of an analytical pricing formula. In the second approach, the tangent process takes only finitely many jump sizes. With this specification, the no-arbitrage conditions are simplified and the model offers more flexibility since a non-parametric calibration procedure is allowed. For both approaches, we illustrate in detail the calibration and simulation procedure.

We stress the benefits of our approaches in a portfolio optimization problem by comparing the performance against the SABR model. Our approaches produce scenarios that are more consistent with the real-world dynamics, and lead to more stationary and stable portfolio returns, hence are more desirable in practical risk management.