scholarly journals OPTIMAL LIQUIDATION UNDER STOCHASTIC PRICE IMPACT

2019 ◽  
Vol 22 (02) ◽  
pp. 1850059 ◽  
Author(s):  
WESTON BARGER ◽  
MATTHEW LORIG
Keyword(s):  
Continuous Time ◽  
Value Function ◽  
Price Impact ◽  
Trading Strategies ◽  
Impact Model ◽  
Optimal Liquidation ◽  
Special Cases ◽  
Time Price ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

We assume a continuous-time price impact model similar to that of Almgren–Chriss but with the added assumption that the price impact parameters are stochastic processes modeled as correlated scalar Markov diffusions. In this setting, we develop trading strategies for a trader who desires to liquidate his inventory but faces price impact as a result of his trading. For a fixed trading horizon, we perform coefficient expansion on the Hamilton–Jacobi–Bellman (HJB) equation associated with the trader’s value function. The coefficient expansion yields a sequence of partial differential equations that we solve to give closed-form approximations to the value function and optimal liquidation strategy. We examine some special cases of the optimal liquidation problem and give financial interpretations of the approximate liquidation strategies in these cases. Finally, we provide numerical examples to demonstrate the effectiveness of the approximations.

scholarly journals Optimal Portfolios for Financial Markets with Wishart Volatility

2013 ◽  
Vol 50 (4) ◽  
pp. 1025-1043 ◽  
Author(s):  
Nicole Bäuerle ◽  
Zejing Li
Keyword(s):  
Value Function ◽  
Numerical Study ◽  
Heston Model ◽  
Hard Part ◽  
Control Approach ◽  
Portfolio Strategy ◽  
Hamilton Jacobi Bellman ◽  
The One ◽  
Hedging Demand ◽  
The Value Function

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals Dynamic Programming and Hamilton–Jacobi–Bellman Equations on Time Scales

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yingjun Zhu ◽  
Guangyan Jia
Keyword(s):  
Dynamic Programming ◽  
Dynamic System ◽  
Time Scales ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Dynamic ◽  
Optimality Principle ◽  
Bellman Equations ◽  
Special Cases ◽  
Hamilton Jacobi Bellman

Bellman optimality principle for the stochastic dynamic system on time scales is derived, which includes the continuous time and discrete time as special cases. At the same time, the Hamilton–Jacobi–Bellman (HJB) equation on time scales is obtained. Finally, an example is employed to illustrate our main results.

Solving the Hamilton-Jacobi-Bellman equation of stochastic control by a semigroup perturbation method

1984 ◽  
Vol 16 (1) ◽  
pp. 16-16
Author(s):  
Domokos Vermes
Keyword(s):  
Bellman Equation ◽  
Value Function ◽  
Sufficient Optimality Condition ◽  
Hamilton Jacobi Bellman Equation ◽  
Deterministic Processes ◽  
Random Jumps ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
The Value Function ◽  
Deterministic Trajectories

We consider the optimal control of deterministic processes with countably many (non-accumulating) random iumps. A necessary and sufficient optimality condition can be given in the form of a Hamilton-jacobi-Bellman equation which is a functionaldifferential equation with boundary conditions in the case considered. Its solution, the value function, is continuously differentiable along the deterministic trajectories if. the random jumps only are controllable and it can be represented as a supremum of smooth subsolutions in the general case, i.e. when both the deterministic motion and the random jumps are controlled (cf. the survey by M. H. A. Davis (p.14)).

scholarly journals Basel III and the Net Stable Funding Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-20 ◽  
Author(s):  
F. Gideon ◽  
Mark A. Petersen ◽  
Janine Mukuddem-Petersen ◽  
LNP Hlatshwayo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Continuous Time ◽  
Value Function ◽  
Analytic Solution ◽  
Market Liquidity ◽  
Basel Iii ◽  
Quantitative Manner ◽  
The Value Function

We validate the new Basel liquidity standards as encapsulated by the net stable funding ratio in a quantitative manner. In this regard, we consider the dynamics of inverse net stable funding ratio as a measure to quantify the bank’s prospects for a stable funding over a period of a year. In essence, this justifies how Basel III liquidity standards can be effectively implemented in mitigating liquidity problems. We also discuss various classes of available stable funding and required stable funding. Furthermore, we discuss an optimal control problem for a continuous-time inverse net stable funding ratio. In particular, we make optimal choices for the inverse net stable funding targets in order to formulate its cost. This is normally done by obtaining analytic solution of the value function. Finally, we provide a numerical example for the dynamics of the inverse net stable funding ratio to identify trends in which banks behavior convey forward looking information on long-term market liquidity developments.

A quasilinear elliptic equation in ℝN

1996 ◽  
Vol 126 (5) ◽  
pp. 911-921 ◽  
Author(s):  
O. Alvarez
Keyword(s):  
Lower Bound ◽  
Elliptic Equation ◽  
Control Problem ◽  
Stochastic Control ◽  
Quasilinear Elliptic Equation ◽  
Value Function ◽  
Optimal Criterion ◽  
Hamilton Jacobi Bellman ◽  
Quasilinear Elliptic ◽  
The Value Function

A quasilinear elliptic equation in ℝN of Hamilton-Jacobi-Bellman type is studied. An optimal criterion for uniqueness which involves only a lower bound on the functions is given. The unique solution in this class is identified as the value function of the associated stochastic control problem.

Optimal Insurance-Package and Investment Problem for an Insurer

2018 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
Delei Sheng ◽  
Linfang Xing
Keyword(s):  
Value Function ◽  
Investment Strategy ◽  
Optimal Combination ◽  
Hjb Equations ◽  
Terminal Wealth ◽  
Optimal Insurance ◽  
Yield Rate ◽  
Investment Problem ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

AbstractAn insurance-package is a combination being tie-in at least two different categories of insurances with different underwriting-yield-rate. In this paper, the optimal insurance-package and investment problem is investigated by maximizing the insurer’s exponential utility of terminal wealth to find the optimal combination-share and investment strategy. Using the methods of stochastic analysis and stochastic optimal control, the Hamilton-Jacobi-Bellman (HJB) equations are established, the optimal strategy and the value function are obtained in closed form. By comparing with classical results, it shows that the insurance-package can enhance the utility of terminal wealth, meanwhile, reduce the insurer’s claim risk.

Optimal stopping with random intervention times

2002 ◽  
Vol 34 (01) ◽  
pp. 141-157 ◽  
Author(s):  
Paul Dupuis ◽  
Hui Wang
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Value Function ◽  
Value Functions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Time Optimal ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient ◽  
The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞 0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞 2 otherwise, both contradicting the usual 𝒞 1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

OPTIMAL PORTFOLIOS WITH STOCHASTIC SHORT RATE: PITFALLS WHEN THE SHORT RATE IS NON-GAUSSIAN OR THE MARKET PRICE OF RISK IS UNBOUNDED

2009 ◽  
Vol 12 (06) ◽  
pp. 767-796 ◽  
Author(s):  
HOLGER KRAFT
Keyword(s):  
Interest Rates ◽  
Continuous Time ◽  
Value Function ◽  
Market Price ◽  
Gaussian Model ◽  
Power Utility ◽  
Market Price Of Risk ◽  
Price Of Risk ◽  
Non Gaussian ◽  
The Value Function

The aim of this paper is to provide a survey of some of the problems occurring in portfolio problems with power utility, Non-Gaussian interest rates, and/or unbounded market price of risk. Using stochastic control theory, we solve several portfolio problems for different specifications of the short rate and the market price of risk. In particular, we consider a Gaussian model, the Cox-Ingersoll-Ross model, and squared Gaussian as well as lognormal specifications of the short rate. We find that even in a Gaussian framework the canonical candidate for the value function may not be finite if the market price of risk is unbounded. It is thus not straightforward to generalize results on continuous-time portfolio problems with power utility, Gaussian interest rates, and bounded market price of risk to situations where the short rate is Non-Gaussian or the market price of risk is unbounded.

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