Revisiting semidefinite programming approaches to options pricing: complexity and computational perspectives

TL;DR

This paper explores semidefinite programming methods, specifically the Moment-SOS hierarchy, to compute bounds on multi-asset option prices without relying on specific price models, emphasizing computational feasibility and data incorporation.

Contribution

It applies the Moment-SOS hierarchy to the options pricing problem, providing a model-free approach to derive bounds and compare computational methods.

Findings

The approach yields viable bounds on option prices in numerical examples.

Incorporates observable data like moments and option prices into the bounds.

Demonstrates the computational viability of semidefinite programming for complex options pricing.

Abstract

In this paper we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics, but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known Moment-Sum-of-Squares (SOS) hierarchy of Lasserre to obtain bounds on the range of the possible prices. A complementary approach (also due to Lasserre) is employed for comparison. We present several numerical examples to demonstrate the viability of our approach. The framework we consider makes it possible to incorporate different kinds of observable data, such as moment information, as well as observable prices of options on the assets of interest.