PCA for Implied Volatility Surfaces

Dynamics of implied volatility surfaces

The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to ‘Vega’ risk in o...

Stochastic Models of Implied Volatility Surfaces

We propose a market-based approach to the modelling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random...

Normalization for Implied Volatility

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the...

Simulation of Implied Volatility Surfaces via Tangent Lévy Models

In this paper, we implement and test a market-based model for European-type options, based on the tangent Lévy models proposed in [4] and [3]. 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 va...

Deformation of Implied Volatility Surfaces: An Empirical Analysis