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Portfolio optimization of stochastic volatility models through the dynamic programming equations
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In this work we study the problem of portfolio optimization in markets with stochastic volatility.The optimization criteria considered consists in the maximization of the utility of terminal wealth.The most usual method to solve this type of problem passes by the solution of an equation with partial derivatives,deterministic and nonlinear, named the Hamilton-Jacobi-Bellman equation (HJB) or the dynamic programming equation. One of the biggest challenges consists in verifying that the solution to the HJB equation coincides with the payoof the optimal portfolio.These results are known as verication theorems.In this sense,we follow the approach by Kraft[13],generalizing the verication theorems for more general utility functions.
The most significant contribution of this work consists in the resolution of the optimal portfolio problem for the 2-hypergeometric stochastic volatility model considering power utilities. Specifically we obtain a Feynman-Kac formula for the solution of the HJ Bequation.Based on this stochastic representation weapply the Monte Carlo method to approximate the solution to the HJB equation,which if it sufifciently regular it coincides with the payoff function of the optimal portfolio.
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Optimal portfolios stochastic volatility verification theorems 2-hypergeometric model