Discrete Local Volatility for Large Time Steps (Extended Version)

Abstract

We construct a state-and-time discrete martingale which is calibrated globally to a set of given input option prices which may exhibit arbitrage. We also provide an method to take small steps, fully consistent with the transition kernels of the large steps

Its robustness vs. arbitrage violations in the input surface makes our approach particularly suited for computations in stressed scenarios. Indeed, our method of finding a globally closest arbitrage-free surface under constraints on implied and local volatility is useful in its own right.

While we do not aim at approximating a diffusion in our approach, our method may still be interpreted as a discrete version of Dupire's LocalVolatility. We will make this link-explicit with the introduction of our 'Backward Local Volatility.'

This version of the paper is very extensive and provides much detail on how to construct our discrete martingale. A second, much more concise version is also available on SSRN at http://ssrn.com/abstract=2783409. The material discussed here was also presented at Global Derivatives~2016.

Keywords: Implied Volatility, Arbitrage-Free Fitting, Expensive Martingales, Large Step Monte-Carlo, Discrete Pricing

JEL Classification: C60