Discrete Local Volatility for Large Time Steps (Short 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 a method to take small steps, fully consistent with the transition kernels of the large steps. The method's 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.

We demonstrate the power of our approach by showing its application to affine dividends calibrated to option prices given by proportional dividends, availability of Likelihood Greeks, and to mean-reverting assets such as VIX. We also comment on how to introduce jumps into our processes. This version of the paper is a rather concise summary; an extended version is available on SSRN at http://ssrn.com/abstract=2642630. The material discussed here was also presented at Global Derivatives 2016.

Keywords: Discrete Local Volatility, Discrete Martingale, Markov Margingale, Implied Volatility, Arbitrage-Free

JEL Classification: C60