Local Variance Gamma and Explicit Calibration to Option Prices

TL;DR

This paper introduces an efficient algorithm for calibrating jump Markov models to match European option prices across multiple strikes and maturities, applicable to various markets with different listing conventions.

Contribution

It provides a novel calibration method using simple root-search problems for constructing time-homogeneous and piecewise processes matching market smiles.

Findings

Algorithm efficiently calibrates models to market prices.

Constructs time-homogeneous processes for single smiles.

Builds piecewise processes for multiple smiles.

Abstract

In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we provide an algorithm for calibrating a pure jump Markov martingale model to match the market prices of European options of multiple strikes and maturities. This algorithm only requires solutions of several one-dimensional root-search problems, as well as application of elementary functions. We show how to construct a time-homogeneous process which meets a single smile, and a piecewise time-homogeneous process which can meet multiple smiles.