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.