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.

In this article, we explore a method introduced by Buehler and Ryskin in the paper titled “Discrete Local Volatility for Large Time Steps” which explores the removal of arbitrage from a lattice of option prices. The method presented does not only remove arbitrage from the price lattice, but it also finds the globally closest arbitrage-free lattice using simple linear programming. Inspired by the Dupire’s local volatility Buehler introduces a discrete state-and-time Markov chain to model the dynamics of the underlying asset prices. This approach discretizes the continuous-time stochastic process typically assumed in option pricing into a series of discrete time steps and price levels. By doing so, Buehler’s method allows for a more robust computation of local volatilities that are consistent with the observed market prices of options, even when large time steps are involved.

The innovation in this approach lies in the transformation of the continuous problem into a discrete one…