Nonparametric and arbitrage-free construction of call surfaces using l1-recovery

TL;DR

This paper introduces a nonparametric, arbitrage-free method for constructing call option surfaces using $l_1$-minimization, inspired by compressed sensing, to fit sparse market data accurately.

Contribution

It presents a novel $l_1$-minimization approach for arbitrage-free call surface construction that is model-free and consistent with market quotes.

Findings

Successfully constructs arbitrage-free call surfaces for S&P 500 options.

Extends methodology to FX options, specifically HKD/USD.

Produces surfaces that match market data and eliminate static arbitrage.

Abstract

This paper is devoted to the application of an $l_1$ -minimisation technique to construct an arbitrage-free call-option surface. We propose a nononparametric approach to obtaining model-free call option surfaces that are perfectly consistent with market quotes and free of static arbitrage. The approach is inspired from the compressed-sensing framework that is used in signal processing to deal with under-sampled signals. We address the problem of fitting the call-option surface to sparse option data. To illustrate the methodology, we proceed to the construction of the whole call-price surface of the S\&P500 options, taking into account the arbitrage possibilities in the time direction. The resulting object is a surface free of both butterfly and calendar-spread arbitrage that matches the original market points. We then move on to an FX application, namely the HKD/USD call-option surface.