Portfolio optimisation under nonlinear drawdown constraint in a general semimartingale market
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Thu, 17/11/2011 13:00 |
Vladimir Cherny |
Mathematical Finance Internal Seminar  |
DH 1st floor SR |
| We consider a portfolio optimisation problem on infinite horizon when the investment policy satisfies the drawdown constraint, which is the wealth process of an investor is always above a threshold given as a function of the past maximum of the wealth process. The preferences are given by a utility function and investor aims to maximise an asymptotic growth rate of her expected utility of wealth. This problem was firstly considered by Grossman and Zhou [3] and solved for a Black-Scholes market and linear drawdown constraint. The main contribution of the paper is an equivalence result: the constrained problem with utility U and drawdown function w has the same value function as the unconstrained problem with utility UoF, where function F is given explicitly in terms of w. This work was inspired by ideas from [2], whose results are a special case of our work. We show that the connection between constrained and unconstrained problems holds for a much more general setup than their paper, i.e. a general semimartingale market, larger class of utility functions and drawdown function which is not necessarily linear. The paper greatly simplifies previous approaches using the tools of Azema-Yor processes developed in [1]. In fact we show that the optimal wealth process for constrained problem can be found as an explicit Azema-Yor transformation of the optimal wealth process for the unconstrained problem. We further provide examples with explicit solution for complete and incomplete markets. [1] Carraro, L., Karoui, N. E., and Obloj, J. On Azema-Yor processes, their optimal properties and the Bachelier-Drawdown equation, to appear in Annals of Probability, 2011. [2] Cvitanic, J., and Karatzas, I. On portfolio optimization under drawdown constraints. IMA Volumes in Mathematics and Its Applications 65(3), 1994, 35-45 [3] Grossman, S. J., and Zhou, Z. Optimal investment strategies for controlling drawdowns. Mathematical Finance 3(3), 1993, 241-276 | |||