随机价格影响下做市商的最优仓位管理
Optimal Position Management for a Market Maker with Stochastic Price Impacts
摘要：
研究了在非流动性市场中做市商面对不确定客户订单流的最优头寸管理问题，其中做市商的连续交易引起随机线性价格冲击.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进行大宗交易，而不会对价格产生直接影响。我们采用相当通用的随机过程的证券，订单流，价格的影响，二次罚款以及证券借贷利率。做市商最优仓位管理策略的解由随机哈密顿-雅可比-贝尔曼方程表示，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（BSDEs）。我们提供了一个单一的安全情况下，使用标准的BPENDIX技术的验证。对于多安全情形，我们利用非线性BSRDE与一种特殊类型的倒向随机Riccati微分方程（BSRDE）的联系，BSRDE的性质由Bismut（1976）研究。我们还提出了一个微扰近似方案所产生的BSRDE，它只需要一个系统的线性常微分方程求解在每个扩展阶。并给出了证明和收敛速度。
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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