Model-free bounds for multi-asset options using option-implied information and their exact computation

TL;DR

This paper develops a method to compute tight, model-free bounds for multi-asset option prices using market-implied information, with algorithms that are efficient even in high-dimensional settings and applicable for arbitrage detection.

Contribution

It introduces a new framework combining superhedging duality and linear semi-infinite optimization to accurately bound multi-asset option prices from market data, including bid-ask spreads.

Findings

Algorithms efficiently compute $ ext{ε}$-optimal bounds in high dimensions.

The approach detects arbitrage opportunities and strategies.

Including additional market information reduces model risk.

Abstract

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives, and even allow for the presence of bid-ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem, and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are $\varepsilon$-optimal, as well as a characterization of the optimal pricing measures. These algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g. when $d=60$). Moreover, these algorithms can be used to detect arbitrage opportunities and identify the corresponding arbitrage strategies. Numerical experiments using both synthetic and real market data showcase the efficiency of these algorithms, while they also allow to understand the reduction of model risk by including additional information, in the form of known derivative prices.