scholarly journals Optimal Execution considering Trading Signal and Execution Risk Simultaneously

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yuan Cheng ◽  
Lan Wu
Keyword(s):  
Kernel Function ◽  
Price Impact ◽  
Value Functions ◽  
Feedback Form ◽  
Bellman Equations ◽  
Impact Function ◽  
Optimal Execution ◽  
Dirac Function ◽  
Hamilton Jacobi Bellman ◽  
Optimal Execution Problem

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

scholarly journals Optimal execution of limit and market orders with trade director, speed limiter, and fill uncertainty

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750020 ◽  
Author(s):  
Brian Bulthuis ◽  
Julio Concha ◽  
Tim Leung ◽  
Brian Ward
Keyword(s):  
Model Parameters ◽  
Value Functions ◽  
Continuous Time Model ◽  
Time Model ◽  
Limit Orders ◽  
Price Impacts ◽  
Dynamic Strategy ◽  
Optimal Execution ◽  
Optimal Dynamic ◽  
Optimal Execution Problem

We study the optimal execution of market and limit orders with permanent and temporary price impacts as well as uncertainty in the filling of limit orders. Our continuous-time model incorporates a trade speed limiter and a trade director to provide better control on the trading rates. We formulate a stochastic control problem to determine the optimal dynamic strategy for trade execution, with a quadratic terminal penalty to ensure complete liquidation. In addition, we identify conditions on the model parameters to ensure optimality of the controls and finiteness of the associated value functions. For comparison, we also solve the schedule-following optimal execution problem that penalizes deviations from an order schedule. Numerical results are provided to illustrate the optimal market and limit orders over time.

STOCHASTIC DIFFERENTIAL GAMES BETWEEN TWO INSURERS WITH GENERALIZED MEAN-VARIANCE PREMIUM PRINCIPLE

2018 ◽  
Vol 48 (1) ◽  
pp. 413-434 ◽  
Author(s):  
Shumin Chen ◽  
Hailiang Yang ◽  
Yan Zeng
Keyword(s):  
Differential Game ◽  
Game Problem ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Programming Technique ◽  
Bellman Equations ◽  
Equilibrium Strategies ◽  
Generalized Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

AbstractWe study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

scholarly journals Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

2021 ◽  
Vol 14 (9) ◽  
pp. 399
Author(s):  
Pedro Pólvora ◽  
Daniel Ševčovič
Keyword(s):  
Risk Aversion ◽  
Nonlinear Partial Differential Equation ◽  
Option Price ◽  
Transformation Method ◽  
Value Functions ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Option Pricing Model ◽  
Utility Indifference ◽  
Hamilton Jacobi Bellman

Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.

scholarly journals Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a General Risk Model with Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Lidong Zhang ◽  
Ximin Rong ◽  
Ziping Du
Keyword(s):  
Risk Model ◽  
Theoretical Perspective ◽  
Optimal Strategies ◽  
Investment Strategy ◽  
Value Functions ◽  
Lagrange Method ◽  
Bellman Equations ◽  
Strategy Versus ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We mainly study a general risk model and investigate the precommitted strategy and the time-consistent strategy under mean-variance criterion, respectively. A lagrange method is proposed to derive the precommitted investment strategy. Meanwhile from the game theoretical perspective, we find the time-consistent investment strategy by solving the extended Hamilton-Jacobi-Bellman equations. By comparing the precommitted strategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Furthermore, we theoretically and numerically provide the effect of the parameters on these two optimal strategies and the corresponding value functions.

scholarly journals A dynamic optimal execution strategy under stochastic price recovery

2015 ◽  
Vol 02 (04) ◽  
pp. 1550025 ◽  
Author(s):  
Masashi Ieda
Keyword(s):  
Order Book ◽  
Market Impact ◽  
Time Step ◽  
Limit Order ◽  
Optimal Execution ◽  
Execution Strategy ◽  
Quasi Variational Inequality ◽  
Finite Time Horizon ◽  
Hamilton Jacobi Bellman ◽  
Optimal Execution Problem

In the present paper, we study the optimal execution problem under stochastic price recovery based on limit order book dynamics. We model price recovery after execution of a large order by accelerating the arrival of the refilling order, which is defined as a Cox process whose intensity increases by the degree of the market impact. We include not only the market order, but also the limit order in our strategy in a restricted fashion. We formulate the problem as a combined stochastic control problem over a finite time horizon. The corresponding Hamilton–Jacobi–Bellman quasi-variational inequality is solved numerically. The optimal strategy obtained consists of three components: (i) the initial large trade; (ii) the unscheduled small trades during the period; (iii) the terminal large trade. The size and timing of the trade is governed by the tolerance for market impact depending on the state at each time step, and hence the strategy behaves dynamically. We also provide competitive results due to inclusion of the limit order, even though a limit order is allowed under conservative evaluation of the execution price.

On Optimal Options Book Execution Strategies with Market Impact

2016 ◽  
Vol 02 (03n04) ◽  
pp. 1750002
Author(s):  
Aymeric Kalife ◽  
Saad Mouti
Keyword(s):  
Implied Volatility ◽  
Option Price ◽  
Spot Price ◽  
Market Impact ◽  
Impact Function ◽  
Optimal Execution ◽  
Execution Strategy ◽  
Variance Risk ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We consider the optimal execution of a book of options when market impact is a driver of the option price. We aim at minimizing the mean-variance risk criterion for a given market impact function. First, we develop a framework to justify the choice of our market impact function. Our model is inspired from Leland’s option replication with transaction costs where the market impact is directly part of the implied volatility function. The option price is then expressed through a Black– Scholes-like PDE with a modified implied volatility directly dependent on the market impact. We set up a stochastic control framework and solve an Hamilton–Jacobi–Bellman equation using finite differences methods. The expected cost problem suggests that the optimal execution strategy is characterized by a convex increasing trading speed, in contrast to the equity case where the optimal execution strategy results in a rather constant trading speed. However, in such mean valuation framework, the underlying spot price does not seem to affect the agent’s decision. By taking the agent risk aversion into account through a mean-variance approach, the strategy becomes more sensitive to the underlying price evolution, urging the agent to trade faster at the beginning of the strategy.

scholarly journals Policy iteration for Hamilton–Jacobi–Bellman equations with control constraints

2021 ◽  
Author(s):  
Sudeep Kundu ◽  
Karl Kunisch
Keyword(s):  
Linear Equations ◽  
Hjb Equation ◽  
Policy Iteration ◽  
Optimal Feedback Control ◽  
Iteration Step ◽  
Control Constraints ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Hamilton Jacobi Bellman ◽  
Feedback Control Theory

AbstractPolicy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

Sign in / Sign up

Export Citation Format

Share Document