Mathematical Modelling of an Investor’s Wealth with Different Stochastic Volatility Models

Abstract

This paper investigates the application of various stochastic volatility models in determining optimal investment strategies in the stock market. The study explores the geometric Brownian motion (GBM), constant elasticity of variance (CEV), modified CEV (M-CEV), and Heston volatility models. Each model offers a unique perspective on volatility dynamics and option pricing. The research formulates the Hamilton-Jacobi-Bellman (HJB) equations for each model and employs the Legendre transformation method to convert them into linear partial differential equations (PDEs). The quadratic utility function is utilized to derive optimal portfolio distributions under each model. Numerical simulations are conducted to analyze the impact of market parameters such as appreciation rate, volatility, interest rate, elasticity parameter, tax, and investor's wealth on the optimal portfolio distribution. The results indicate that optimal investment strategies vary significantly based on market conditions and investor preferences. Overall, this study provides valuable insights into the dynamic nature of financial markets and offers practical guidance for portfolio optimization and risk management strategies