Solution Of Optimal Stopping Problem Research Articles

Deep neural network expressivity for optimal stopping problems

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document} by a deep ReLU neural network of size at most κdqε−r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa d^{\\mathfrak{q}} \\varepsilon ^{-\\mathfrak{r}}$\\end{document}. The constants κ,q,r≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa ,\\mathfrak{q},\\mathfrak{r} \\geq 0$\\end{document} do not depend on the dimension d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} of the state space or the approximation accuracy ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}. This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of optimal stopping problems. The framework covers for example exponential Lévy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions.

Pricing Multidimensional American Options

A new explicit form is provided for the solution of optimal stopping problems involving a multidimensional geometric Brownian motion. A free-boundary value approach is adopted and the value function is obtained via fundamental solution methods. There are many applications for the valuation of perpetual options of American style, which are of interest for finance and managerial decisions.

Kriging metamodels and experimental design for Bermudan option pricing

We investigate two new strategies for the numerical solution of optimal stopping problems in the regression Monte Carlo (RMC) framework proposed by Longstaff and Schwartz in 2001. First, we propose using stochastic kriging (Gaussian process) metamodels for fitting the continuation value. Kriging offers a flexible, non-parametric regression approach that quantifies approximation quality. Second, we connect the choice of stochastic grids used in RMC with the design of experiments (DoE) paradigm. We examine space-filling and adaptive experimental designs; we also investigate the use of batching with replicated simulations at design sites to improve the signal-to-noise ratio. Numerical case studies for valuing Bermudan puts and max calls under a variety of asset dynamics illustrate that our methods offer a significant reduction in simulation budgets compared with existing approaches.

THE FORMULATION AND METHOD OF SOLUTION OF OPTIMAL STOPPING PROBLEM

THE FORMULATION AND METHOD OF SOLUTION OF OPTIMAL STOPPING PROBLEM

On the existence of solutions of unbounded optimal stopping problems

Known conditions of existence of solutions of optimal stopping problems for Markov processes assume that payoff functions are bounded in some sense. In this paper we prove weaker conditions which are applicable to unbounded payoff functions. The results obtained are applied to the optimal stopping problem for a Brownian motion with the payoff function G(τ, Bτ )= |Bτ |− c/(1 − τ ).

Viscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory

We consider a finite time horizon optimal stopping problem for a system of stochastic functional differential equations with a bounded memory. Under some sufficiently smooth conditions, a Hamilton-Jacobi-Bellman (HJB) variational inequality for the value function is derived via dynamical programming principle. It is shown that the value function is the unique viscosity solution of the HJB variational inequality.

On approximative solutions of optimal stopping problems

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

A new approach to the solution of optimal stopping problem in a discrete time

An optimal stopping problem of Markov chain with infinite horizon is considered. For the case of finite number m of states, Sonin proposed an algorithm, which allows to find the value function and the stopping set in not more than steps. The algorithm is based on a modification of Markov chain on each step, related with the elimination of the states which certainly belong to the continuation set. To solve the problem with arbitrary state space and to have a possibility of generalization to the continuous time, one needs to modify the procedure. We propose a procedure which is based on a sequential modification of the pay-off function for the same chain in such a way, that the value function is the same for both problems and the modified pay-off function is greater than the initial one on some set and is equal to it on the complement. We show the efficiency of this procedure.

Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options

Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time. Our approach is to recursively compute the so-called continuation values. They are defined as regression functions of the cash flow, which would occur over a series of subsequent time periods, if the approximated optimal exercise strategy is applied. We use nonparametric least squares regression estimates to approximate the continuation values from a set of sample paths which we simulate from the underlying stochastic process. The parameters of the regression estimates and the regression problems are chosen in a data-dependent manner. We present results concerning the consistency and rate of convergence of the new algorithm. Finally, we illustrate its performance by pricing high-dimensional Bermudan basket options with strangle-spread payoff based on the average of the underlying assets.

Viscosity solutions of optimal stopping problems

We prove that the value function of an optimal stopping problem is the unique viscosity solution of the associated variational inequalities. We illustrate by an example how this can be used to solve optimal stopping problems where the high contact (smooth fit) principle does not necessarily hold

092059 (E50) Viscosity solutions of optimal stopping problems : Reikvam K.,Presented at the International Workshop on The Interplay between Insurance Finance and Control, organized by the Mathematical Research Centre at Aarhus University, also supported by the Danish Science Research Council and the Centre for Analytical Finance

092059 (E50) Viscosity solutions of optimal stopping problems : Reikvam K.,Presented at the International Workshop on The Interplay between Insurance Finance and Control, organized by the Mathematical Research Centre at Aarhus University, also supported by the Danish Science Research Council and the Centre for Analytical Finance

An Optimal Stopping Problem for Sums of Dichotomous Random Variables

Let $Y_t$ be a stochastic process starting at $y$ which changes by i.i.d. dichotomous increments $X_t$ with mean 0 and variance 1. The cost of proceeding one step is one and the payoff is zero unless $n$ steps are taken and the final value $\hat{Y}$ of $Y_t$ is negative in which case the payoff is $\hat{Y}^2$. The optimal procedure consists of stopping as soon as $Y_t \geq \tilde{y}_m$ where $m$ is the number of steps left to be taken. The limit of $\tilde{y}_m$ as $m \rightarrow \infty$ is desired as a function of $p = P(X_t < 0)$. This limit $\tilde{y}$ is evaluated for $p$ rational and proved to be continuous in $p$. One can use $\tilde{y}$ to relate the solution of optimal stopping problems involving a Wiener process to those involving certain discrete-time discrete-process stopping problems. Thus $\tilde{y}$ is useful in calculating simple numerical approximations to solutions of various stopping problems.