Optimal investment in S&P 500 using SDDP and the implied-calibrated ARMA–GARCH model

Abstract

We study a dynamic portfolio optimization problem in which probabilistic forecasts of the S&P 500 index are derived from option market prices. To overcome the limitations of classical approaches to reconstructing implied density from option prices which fail to generate conditional multi-period return distributions, we propose a method that involves calibrating the discrete-time ARMA–GARCH model from the observed call option prices in a risk-neutral measure with subsequent transition to a physical measure using a representative-agent framework. The calibrated model provides conditional multi-period distributions of asset returns, which are used to construct scenario lattices in multi-stage stochastic optimization. The resulting portfolio optimization problem is formulated as a multi-stage stochastic programming problem. At each stage, a weighted combination of the expected negative objective value for the next stage and the Conditional Value-at-Risk (CVaR) is minimized. The optimization is performed by means of the Stochastic Dual Dynamic Programming (SDDP) method. Historical simulations over the period 2019–2023 demonstrate that the proposed option-calibrated ARMA–GARCH–SDDP method consistently outperforms benchmark approaches based on static implied densities, equal-probability scenarios, and buy-and-hold investment. The results underscore the economic value of using option-implied information in portfolio management.