Optimal Consumption And Portfolio Choice Under Multi-factor Stochastic Volatility

It is well known that an optimal investment and consumption problem in continuous time is a classic problem in financial economics pioneered by Merton.There is strong empirical evidence that the conditional variance of asset returns,particularly stock market returns,is not constant over time.In recent years,stochastic covariance is a key feature in the modeling of financial asset returns.Numerous empirical studies have shown that modeling the stochastic variations in the covariance matrix of asset returns is very important for optimal portfolio choice is very important for optimal portfolio choice.Furthermore,it has also been well-documented in the option pricing literature that multifactor stochastic volatility models perform significantly better than their univariate counterparts.However,the optimal consumption and portfolio choice problem in a continuous-time multi-factor stochastic volatility setup has not yet been studied.This paper discusses a multivariate intertemporal consumption and portfolio choice problem in incomplete market with a multi-factor stochastic covariance matrix of asset returns.The main contents are summarized as follows:First,assuming that the investor make consumption and investment decisions based on the CRRA utility function,we study the optimization problem under the Principal Component Stochastic Volatility(PCSV)model.Using the stochastic dynamic programming approach,we obtain an Hamilton-Jacobi-Bellman equation related to the problem.In each of the power and logarithmic utility cases,we obtain expressions of the investment-consumption strategy and the value function.We also analyse the welfare losses due to suboptimal investment strategies and we find that if investors only consider single-factor stochastic volatility or ignore the random covariance between asset returns,it will cause significant wealth loss.Second,we study the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein-Zin type under the PCSV model.The explicit and semi-explicit expressions of optimal consumption and investment strategies in the case of unit and non-unit elasticity of inter-temporal substitution are derived respectively by using stochastic dynamic programming.We also analyze the influence of state variables on the optimal strategy.Moreover,the numerical simulation results show that the effect of risk aversion coefficient on the optimal investment strategy is much greater than that of elasticity of inter-temporal substitution,while for the optimal consumption-wealth ratio,this effect is different.In addition,we also find that when the inter-temporal substitution elasticity coefficient is greater than 1,the sensitivity of the optimal consumption-wealth ratio to the time discount rate is much greater than that when it is less than 1,and the convergence rate of the optimal consumption-wealth ratio when the inter-temporal substitution elasticity coefficient is less than 1 is faster than that when it is greater than 1.