随机价格下做市商的最优仓位管理

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英文标题：
《Optimal Position Management for a Market Maker with Stochastic Price
  Impacts》
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作者：
Masaaki Fujii
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最新提交年份：
2015
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英文摘要：
  This paper deals with an optimal position management problem for a market maker who has to face uncertain customer order flows in an illiquid market, where the market maker\'s continuous trading incurs a stochastic linear price impact. Although the execution timing is uncertain, the market maker can also ask its OTC counterparties to transact a block trade without causing a direct price impact. We adopt quite generic stochastic processes of the securities, order flows, price impacts, quadratic penalties as well as security borrowing/lending rates. The solution of the market maker\'s optimal position-management strategy is represented by a stochastic Hamilton-Jacobi-Bellman equation, which can be decomposed into three (one non-linear and two linear) backward stochastic differential equations (BSDEs). We provide the verification using the standard BSDE techniques for a single security case. For a multiple-security case, we make use of the connection of the non-linear BSDE to a special type of backward stochastic Riccati differential equation (BSRDE) whose properties were studied by Bismut(1976). We also propose a perturbative approximation scheme for the resultant BSRDE, which only requires a system of linear ODEs to be solved at each expansion order. Its justification and the convergence rate are also given.
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中文摘要：
本文研究了一个做市商在非流动市场中面临不确定客户订单流的最优仓位管理问题，其中做市商的连续交易会产生随机线性价格影响。虽然执行时间不确定，但做市商也可以要求其场外交易对手在不造成直接价格影响的情况下进行大宗交易。我们采用了证券、订单流、价格影响、二次惩罚以及证券借贷利率的非常一般的随机过程。做市商最优仓位管理策略的解由随机Hamilton-Jacobi-Bellman方程表示，该方程可分解为三个（一个非线性和两个线性）倒向随机微分方程（BSDE）。我们使用标准BSDE技术为单个安全案例提供验证。对于多重安全情形，我们利用非线性BSDE与一类特殊的倒向随机Riccati微分方程（BSRDE）的联系，Bimit（1976）研究了该方程的性质。我们还提出了一个扰动近似方案，只需要在每个展开阶上求解线性常微分方程组。文中还给出了算法的合理性和收敛速度。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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关键词：仓位管理 做市商 Mathematical Differential Quantitative