Abstract:
Optimal investment and consumption problem with proportional transaction costs known as Davis \& Norman problem is a challenging portfolio free-boundary problem to solve. According to the HJB equation, see Davis and Norman (1990), there are three optimal regions i.e. no-trade region, buy region, and sell region in which different actions are prescribed. However, the two boundaries separating the three regions are unknown and it is crucial to discover them in order to obtain the optimal policy. Nonetheless, under a special class of model in which the excess return of stock is assumed to be positively linear in variance reflecting the principle of risk-return tradeoff, the increase of two boundaries in variance suggests the investor to over-invest in a high-risk/high-return stock and to under-invest in a low-risk/low-return stock. This becomes contradictory to the principle eventually and reflects the incompleteness of the model setup. In this study, we aim to extend the model setup of the Davis \& Norman problem to include stochastic variance by assuming that it follows Heston stochastic volatility model. The problem becomes solving a two dimensional free-boundary partial differential equation problem which is different from the one in the Davis \& Norman problem, therefore, numerical methods for approximating boundaries of the Davis \& Norman problem are no longer applicable. As a result, we propose a neural network-based method inspired by the original Deep Galerkin Method (DGM) proposed by Sirignano and Spiliopoulos (2018). Unlike the solution of the Davis \& Norman problem, the neural network approximated solution implies that the stochastic variance can reduce the domination of excess return on the buy boundary resulting the boundary to be less steep i.e. less positively sensitive to initial variance compared to that of the Davis \& Norman problem. Moreover, stochastic variance completely gains the dominance over excess mean return on the sell boundary making it downward-sloping i.e. decreases when the initial variance level is increased for the set of parameters that we used. However, in the presence of a negative correlation between the stochastic variance and stock returns, the buy boundary barely transforms while the sell boundary is no longer downward-sloping and becomes almost insensitive to initial variance. Therefore, under the Heston model with this set of parameters, the investor can implement roughly the same policy for every level of initial variance.
