scholarly journals A Threshold Type Policy for Trading a Mean-Reverting Asset with Fixed Transaction Costs

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 107
Author(s):  
Phong Luu ◽  
Jingzhi Tie ◽  
Qing Zhang
Keyword(s):  
Asset Price ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Stopping Times ◽  
Programming Approach ◽  
Value Functions ◽  
Algebraic Equations ◽  
Hjb Equations ◽  
Optimal Trading ◽  
Optimal Stopping Times

A mean-reverting model is often used to capture asset price movements fluctuating around its equilibrium. A common strategy trading such mean-reverting asset is to buy low and sell high. However, determining these key levels in practice is extremely challenging. In this paper, we study the optimal trading of such mean-reverting asset with a fixed transaction (commission and slippage) cost. In particular, we focus on a threshold type policy and develop a method that is easy to implement in practice. We formulate the optimal trading problem in terms of a sequence of optimal stopping times. We follow a dynamic programming approach and obtain the value functions by solving the associated HJB equations. The optimal threshold levels can be found by solving a set of quasi-algebraic equations. In addition, a verification theorem is provided together with sufficient conditions. Finally, a numerical example is given to illustrate our results. We note that a complete treatment of this problem was done recently by Leung and associates. Nevertheless, our work was done independently and focuses more on developing necessary optimality conditions.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (03) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (3) ◽  
pp. 799-817 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

Discounted optimal stopping problems for the maximum process

2000 ◽  
Vol 37 (4) ◽  
pp. 972-983 ◽  
Author(s):  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Stopping Times ◽  
Value Functions ◽  
Black Scholes Model ◽  
Lookback Options ◽  
Black Scholes ◽  
Maximum Process ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Discounted optimal stopping problems for the maximum process

2000 ◽  
Vol 37 (04) ◽  
pp. 972-983 ◽  
Author(s):  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Stopping Times ◽  
Value Functions ◽  
Black Scholes Model ◽  
Lookback Options ◽  
Black Scholes ◽  
Maximum Process ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

scholarly journals Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

2014 ◽  
Vol 51 (03) ◽  
pp. 799-817
Author(s):  
Pavel V. Gapeev ◽  
Neofytos Rodosthenous
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Stopping Times ◽  
Value Functions ◽  
Diffusion Type ◽  
Closed Form Solutions ◽  
Optimal Stopping Problems ◽  
Optimal Stopping Times ◽  
Maximum Drawdown

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

scholarly journals Pricing Perpetual American Put Options with Asset-Dependent Discounting

2021 ◽  
Vol 14 (3) ◽  
pp. 130
Author(s):  
Jonas Al-Hadad ◽  
Zbigniew Palmowski
Keyword(s):  
Value Function ◽  
Asset Price ◽  
Stopping Times ◽  
Martingale Measure ◽  
Put Options ◽  
American Put Options ◽  
Exact Calculations ◽  
Negative Exponential ◽  
American Put ◽  
The Value Function

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

Some diffusion models for the mabinogion sheep problem of williams

1996 ◽  
Vol 28 (03) ◽  
pp. 763-783 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Local Time ◽  
Discrete Time ◽  
Continuous Time ◽  
Programming Approach ◽  
Value Functions ◽  
Stochastic Version ◽  
Optimal Boundary ◽  
Dynamic Programming Approach ◽  
Time Problem ◽  
A Free Boundary Problem

The ‘Mabinogion sheep’ problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.

P-D Feedback for Decoupling and Pole Assignment of Singular Systems

1998 ◽  
Vol 120 (3) ◽  
pp. 378-388 ◽  
Author(s):  
F. N. Koumboulis ◽  
B. G. Mertzios
Keyword(s):  
Closed Loop ◽  
Singular Systems ◽  
Sufficient Conditions ◽  
Pole Assignment ◽  
Loop System ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Equations ◽  
Input Output ◽  
Closed Loop System ◽  
Necessary And Sufficient

The problem of reducing a multi input-multi output system to many single input-single output systems, namely the problem of input-output decoupling, is studied for the case of singular systems i.e., for systems described by dynamic and algebraic equations. The problem of input-output decoupling with simultaneous arbitrary pole assignment, via proportional plus derivative (P-D) state feedback, is extensively solved. The general explicit expression of all P-D controllers solving the decoupling problem is determined. The general form of the diagonal elements of the decoupled closed-loop system is proven to be in a form having a fixed numerator polynomial and an arbitrary denominator polynomial. The necessary and sufficient conditions for the solvability of the problem of decoupling with simultaneous asymptotic stabilizability or arbitrary pole assignment are established. Furthermore, the necessary and sufficient conditions for decoupling with simultaneous impulse elimination, as well as the necessary and sufficient conditions for decoupling with arbitrary assignment of the finite and infinite poles of the closed-loop system, are established.

Optimal stopping with discount and observation costs

2000 ◽  
Vol 37 (01) ◽  
pp. 64-72 ◽  
Author(s):  
Robert Kühne ◽  
Ludger Rüschendorf
Keyword(s):  
Differential Equations ◽  
Optimal Stopping ◽  
Point Processes ◽  
Numerical Solutions ◽  
Domain Of Attraction ◽  
Approximate Solutions ◽  
Poisson Approximation ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Times

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

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