Optimal investment strategy in the family of 4/2 stochastic volatility models

This paper derives optimal investment strategies for the 4/2 stochastic volatility model proposed in [Grasselli, M., The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math. Finance, 2017, 27(4), 1013-1034] and the embedded 3/2 model [Heston, S.L., A simple new formula for options with stochastic volatility. 1997]. We maximize the expected utility of terminal wealth for a constant relative risk aversion (CRRA) investor, solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations in closed form for both complete and incomplete markets. Conditions for the verification theorems are provided. Interestingly, the optimal investment strategy displays a very intuitive dependence on current volatility levels, a trend which has not been previously reported in the literature of stochastic volatility models. A full empirical analysis comparing four popular embedded models-i.e. the Merton (geometric Brownian motion), Heston (1/2), 3/2 and 4/2 models-is conducted using S&P 500 and VIX data. We find that the 1/2 model carries the larger weight in explaining the 4/2 behaviour, and optimal investments in the 1/2 and 4/2 models are similar, while investments in the 3/2 model are the most conservative in high-variance settings (20% of Merton's solution). Despite the similarities between the 1/2 and 4/2 models, wealth-equivalent losses due to deviations from the 4/2 model are largest for the1/2 and GBM models (40% over 10 years). Meanwhile, the wealth losses due to market incompleteness are harsher for the 1/2 model than for the 4/2 and 3/2 models (60% versus 40% and 30% respectively), highlighting the benefits of choosing the 4/2 or the 3/2 over the 1/2 model.