The collocating local volatility framework – a fresh look at efficient pricing with smile

Abstract

ABSTRACTIt is a market practice to price exotic derivatives, like callable basket options, with the local volatility model [B. Dupire, Pricing with a smile, Risk 7 (1994), pp. 18–20; E. Derman and I. Kani, Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility, Int. J. Theor. Appl. Finance 1 (1998), pp. 61–110.] which can, contrary to stochastic volatility frameworks, handle multi-dimensionality easily. On the other hand, a well-known limitation of the nonparametric local volatility model is the necessity of a short-stepping simulation, which, in high dimensions, is computationally expensive. In this article, we propose a new local volatility framework called the collocating local volatility (CLV) model which allows for large Monte Carlo steps and therefore it is computationally efficient. The CLV model is by its construction guaranteed to be almost perfectly calibrated to implied volatility smiles/skews at a given set of expiries. Additionally, the framework allows to control forward volatilities without affecting the fit to plain vanillas. The model requires only a fraction of a second for complete calibration to simple vanilla products.