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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 823-834.doi: 10.3969/j.issn.1005-3085.2015.06.004

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奈特不确定下带高水印的对冲基金最优投资组合

费为银,    朱涛涛,    费   晨   

  1. 安徽工程大学金融工程系，安徽 芜湖 241000
  • 收稿日期:2014-06-03 接受日期:2014-12-11 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金 (71171003; 71271003)；安徽省自然科学基金 (1208085MG116)；安徽省高校自然科学基金 (KJ2012B019; KJ2013B023).

Optimal Portfolio of Hedge Funds with High Water Marks under Knightian Uncertainty

FEI Wei-yin,  ZHU Tao-tao,   FEI Chen   

  1. Department of Financial Engineering, Anhui Polytechnic University, Wuhu, Anhui 241000
  • Received:2014-06-03 Accepted:2014-12-11 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (71171003; 71271003); the Natural Science Foundation of Anhui Province (1208085MG116); the Natural Science Foundation of Universities of Anhui Province (KJ2012B019; KJ2013B023).

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摘要： 对冲基金管理者往往面临投资资产价格的不确定，这里的不确定包含通常意义下的概率不确定和奈特不确定．资产价格在经典意义下可用布朗运动扰动的随机微分方程予以刻画，然而由于金融市场环境的复杂性，资产价格的干扰源用彭实戈提出的G-布朗运动来刻画更为合理．本文研究了风险资产价格具有奈特不确定下对冲基金管理者最优投资策略．首先建立了基金管理者在风险资产和无风险资产上投资的动态模型，此时风险资产受G-布朗运动干扰，对冲基金带有高水印激励合约，基金管理者的目标是最大化预期累积激励费的净现值．然后通过非线性期望下的随机分析和随机动态规划方法推导出了带有特定边界条件的值函数的G-哈密尔顿-雅可比-贝尔曼(G-HJB)方程，得出相应的基金管理者最优投资组合策略．最后进行了静态经济分析．

关键词: 奈特不确定, 高水印, 对冲基金, 最优投资组合, G-HJB方程

Abstract:

The hedge fund manager is often faced with the uncertainty of asset prices, where uncertainty includes both the classical probabilistic uncertainty and the Knightian uncertainty. We know that asset prices can be addressed by stochastic differential equations disturbed by a Brownian motion in the classical sense. However, due to the complexity of financial markets, it might be more reasonable that the interference sources of asset prices are characterized through Peng's G-Brownian motion. This paper investigates the optimal strategy of the fund manager's portfolio as the volatility of the asset price has Knightian uncertainty. First, we establish a dynamic model in which a fund manager invests in a riskless asset and a risky asset under the framework of G-Brownian motion. On the other hand, for the hedge fund with the contract of high water marks, the fund manager wants to maximize the expected net present value of the cumulative incentive fees. Then we deduce the corresponding G-Hamilton-Jacobi-Bellman equation of the value function with specific boundary conditions through the stochastic calculus and the stochastic dynamic programming method  under nonlinear expectations, and the corresponding optimal portfolio strategy of the fund manager is obtained. Finally, we give the static economic analyses for our results.

Key words: Knightian uncertainty, high water marks, hedge fund, optimal portfolio, G-HJB equation

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