Análise Comparativa entre Exposição Direta a Índices e Estratégias com Opções sob Diferentes Regimes de Mercado

Abstract

This work proposes a comparative analysis between direct exposure to market indices and the use of options strategies, combining a theoretical approach based on simplified scenarios with an empirical study on real S&P 500 options data. In financial markets, the decision between maintaining a position in the underlying asset or using derivative instruments involves relevant considerations related to potential return, capital protection, payoff asymmetry, hedging cost, and sensitivity to variables such as volatility and the passage of time. The study begins with a historical contextualization of the evolution of financial markets and the emergence of derivatives, followed by the theoretical foundation necessary to understand options, their Greeks, and the arbitrage pricing principles underlying the Black-Scholes-Merton model. Building on this foundation, three valuation methodologies are compared for the same position in out-of-the-money calls: full repricing via Black-Scholes- Merton (Full Valuation), the traditional local Delta-Gamma approximation, and an extension proposed in this work, in which gamma is adjusted as a function of the underlying asset’s proximity to the strike. These three methodologies are then confronted with real daily closing data of SPX options throughout 2023, organized into three 90-day windows with distinct market regimes, under a risk equalization criterion that calibrates the size of the options position by the expected loss equivalent to that of a direct position in the index. The results show that the advantage of the real option over the spot, when it exists, is concentrated in specific time windows and is not sustained throughout the analyzed horizon, outperforming the index return in only between 7% and 27% of the tested vertices, depending on the market regime. The Delta-Gamma approximations, by freezing sensitivities at the initial instant, overestimate the real result in more than 80% of cases, as they fail to incorporate the time decay or the implied volatility variation observed in the market. It was also found that, during periods of index decline, the options structure acted more consistently as a loss-limiting instrument than as a gain amplifier. It is concluded that the convexity of OTM options is an economically real mechanism, but its conversion into a net advantage over direct exposure depends critically on the timing of evaluation and on the simplifications adopted by each methodology.