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In this paper, we present a mixed risk-return optimization framework for selecting long put option positions for hedging the tail risk of investments in the S&amp;P 500 index. A tractable formulation is developed by constructing hypothetical portfolios that are constantly rolling put options. Variance and sample CVaR are used as risk measures. The models are tested against out-of-sample historical S&amp;P 500 index values as well as the values of the index paired with long put options of varying strike prices. The optimized hedged portfolio could provide sufficient protection in market downturns while not losing significant return the long horizons. This is achieved by dynamically adjusting the put option compositions to market trends in a timely manner. Allocations to different put options are analyzed in various market trends and investor risk aversion levels. The strategy overcomes the traditional drawbacks of protective put strategies and outperforms both directly investing in the underlying asset and holding a constant long position in a particular put option.

Optimization-based tail risk hedging of the S&P 500 index