Analysis of optimal liquidation in limit order books for portfolios of correlated assets with stochastic volatility

In this paper we study optimal liquidation under two settings: the first being for a basket of correlated assets, the second being for a portfolio of a single asset but under a stochastic volatility model. Under both frameworks we use a combined approach of accurate numerical methods and asymptotic analysis to investigate and gain insight into the solution, with each approach informing and confirming the other. We are able to make a significant improvement in efficiency in both problems, reducing the resulting Hamiliton-Jacobi-Bellman (HJB) partial differential equations (PDEs) to classical non-linear PDEs, as well as reducing the number of variables and input parameters, the latter through non-dimensionalisation. We present numerical solutions to both problems, before further investigating the solution topology through the use of asymptotic analysis in various limits. In some cases we are able to find analytic approximations for both the value function and indeed the optimal liquidation strategies. Furthermore, the solutions we present are comparable with those of Markowitz Portfolio Theory (MPT) for the multiple-asset case, and to those of option pricing theory under stochastic volatility for the stochastic volatility model. For the former we find the trader trades in a way that results in a diversified portfolio, while for the latter we find that more noise in the volatility can be beneficial for the trader in certain regimes, despite being risk-averse