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

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