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Asymptotic approximation of optimal portfolio for small time horizons

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  • Rohini Kumar
  • Hussein Nasralah

Abstract

We consider the problem of portfolio optimization in a simple incomplete market and under a general utility function. By working with the associated Hamilton-Jacobi-Bellman partial differential equation (HJB PDE), we obtain a closed-form formula for a trading strategy which approximates the optimal trading strategy when the time horizon is small. This strategy is generated by a first order approximation to the value function. The approximate value function is obtained by constructing classical sub- and super-solutions to the HJB PDE using a formal expansion in powers of horizon time. Martingale inequalities are used to sandwich the true value function between the constructed sub- and super-solutions. A rigorous proof of the accuracy of the approximation formulas is given. We end with a heuristic scheme for extending our small-time approximating formulas to approximating formulas in a finite time horizon.

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  • Rohini Kumar & Hussein Nasralah, 2016. "Asymptotic approximation of optimal portfolio for small time horizons," Papers 1611.09300, arXiv.org, revised Feb 2018.
    • Handle: RePEc:arx:papers:1611.09300
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    References listed on IDEAS

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    Cited by:

    1. Mrudul Y. Jani & Manish R. Betheja & Amrita Bhadoriya & Urmila Chaudhari & Mohamed Abbas & Malak S. Alqahtani, 2022. "Optimal Pricing Policies with an Allowable Discount for Perishable Items under Time-Dependent Sales Price and Trade Credit," Mathematics, MDPI, vol. 10(11), pages 1-19, June.
    2. Minglian Lin & Indranil SenGupta, 2021. "Analysis of optimal portfolio on finite and small time horizons for a stochastic volatility market model," Papers 2104.06293, arXiv.org.
    3. Minglian Lin & Indranil SenGupta, 2023. "Analysis of optimal portfolio on finite and small-time horizons for a stochastic volatility model with multiple correlated assets," Papers 2302.06778, arXiv.org, revised Dec 2024.

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