An Arbitrage-Free Interpolation of Class C² for Option Prices

Abstract

This article presents simple formulae for the local variance gamma model of Carr and Nadtochiy (2017), extended with a piecewise-linear local variance function. The new formulae allow us to calibrate the model efficiently to market option quotes. On a small set of quotes, exact calibration is achieved under one millisecond. This effectively results in an arbitrage-free interpolation of class . The article proposes a good regularization when the quotes are noisy. Finally, it puts in evidence an issue of the model at-the-money, which is also present in the related one-step finite difference technique of Andreasen and Huge (2011), and gives two solutions for it.

TOPICS: Options, statistical methods

Key Findings

• ▪ The local variance gamma model, extended with piecewise-linear local variance function, leads to simple formulae for vanilla option prices.

• ▪ This model leads to a fast, exact arbitrage-free interpolation of market quotes.

• ▪ A specific regularization is required to overcome an artificial spike in the implied probability density, when fitting the model to noisy quotes.