Limites superiores e inferiores para opções de compra europeia: uma formulação via optimização convexa

Abstract

The options market offers flexible investment strategies, among which European call options stand out by allowing the buyer to acquire an underlying asset at a fixed price on the expiration date. However, pricing these options becomes challenging when the distribution of the underlying asset is unknown, making traditional models potenti- ally inapplicable in structurally uncertain environments. This study proposes a robust approach to pricing European call options based solely on empirical moments of the un- derlying asset, without relying on parametric assumptions about its distribution. Framed within the context of incomplete markets, a robust formulation based on the Generalized Moment Problem (GMP) is adopted. The study uses real data from the S&P 500 in- dex, covering the period from February 1 to 15, 2024, to compute the required empirical moments. Numerical implementation was performed in MATLAB Online using the Se- quential Quadratic Programming algorithm via the ‘fmincon‘ function, with the domain discretized into 300 points. The results yielded an upper bound of 1159.32 USD and a lower bound of 563.54 USD for an option with a strike price of k = 5000. Sensitivity analyses for k = 4900 and k = 5100 confirmed the model’s consistency with option the- ory, highlighting the monotonicity of the option’s value with respect to the strike price. It is concluded that the moment-based formulation via GMP provides a mathematically rigorous, computationally feasible, and financially interpretable alternative for pricing European options under uncertainty. The methodology yields conservative and theoreti- cally sound estimates while preserving the convexity and strong duality properties of the problem