Local Implied Volatility: A Market Model Approach

Abstract

Option valuation models specifying the dynamics of the stock price directly always face the problem of calibration to the volatility surface, which shows to be a troublesome procedure. Market models take the option price surface as an input of the model and are hence perfectly calibrated to the market. Market models try to specify the dynamics of the option prices, rather than the stock price itself. Due to no-arbitrage constraints, specifying the dynamics of the option prices becomes unworkable quickly. Therefore, the idea is to find a one-to-one parameterization of the option price surface. We then specify dynamics for this parameterization to equivalently have the evolution of the option prices. The dynamics of the parameterization are subject to no-arbitrage conditions, which are either complicated or restricted to a single strike or maturity. Johannes Wissel recently came with a model that works for option prices on a fixed grid of strikes and maturities, without invoking complicated restraints. Based on this option valuation model, we analyze the way such a model works in practice. We find ways to calibrate the dynamics of the parameterization and apply the model to the valuation of exotic derivatives that potentially benefit from this approach, such as forward start options, forward start variance swaps and structured products.