Valdo Durrleman, Rama Cont
We propose a market-based approach to the modeling 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 factors. Our modeling approach is based on empirical studies on statistical behavior of implied volatility time series .
We illustrate how this approach extends and improves the accuracy of the well-known ``sticky moneyness'' rule used by option traders for updating implied volatilities. Our approach gives a justification for use of ``Vega''s for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors.
We examine the existence of arbitrage free realizations of such stochastic implied volatility models and show that they lead to simple Delta-Vega hedging strategies for portfolios of options.