Using Discrete Local Volatilities for Arbitrage Repair with Gleam

Co-written by Magnus Wiese

TL;DR

• Arbitrageable prices do not allow for a proper evaluation / modeling of risks. Even more, cleaning / repairing option prices from arbitrage is a tedious task.
• The DLV algorithm is a linear program that repairs arbitrage and solves for the globally closest price lattice.
• The option price parametrization reduces the complex no-arbitrage conditions: if the (discrete) backward local volatilities are positive, then the call prices will satisfy no static arbitrage conditions.
• Method allows to interpolate for any maturity, i.e. intermediate maturities that lie between tenors.

Introduction

The world of financial markets is driven by complex data and calculations, where even the smallest discrepancies can lead to significant consequences. One such challenge that traders, quants, and investors face is the existence of arbitrage opportunities in option price data when pricing derivatives. In derivatives pricing it does not make any economical sense for prices to inherit arbitrage: any arbitrage would directly be exploited and removed by traders. It is therefore crucial when calibrating pricing models to remove any arbitrage from option prices. To this end, we made a framework called gleam that contains methods for working with call price and implied volatility surfaces.