Optimal Exit Time Research Articles

Conditional Beta Approximation: Two Applications

Suppose that X,Y are two independent positive continuous random variables. Let P=\frac{X}{X+Y} and Z=X+Y. If X, Y have gamma distributions with the same scale parameter, then P distribution will be beta and P,\ Z are independent. In the case that the distributions of these two variables are not gamma, the P distribution is well approximated by the beta distribution. However, P,\ Z are dependent. According to matching moment method, it is necessary to compute the moments of conditional distribution for beta fitting. In this paper, some new methods for computing moments of conditional distribution of P given Z are proposed. First of all, it is suggested to consider the regression method. Then Monte Carlo simulation is advised. The Bayesian posterior distribution of P is suggested. Applications of differential equations are also reviewed. These results are applied in two applications namely variance change point detection and winning percentage of gambling game are proposed. The probability of change in variance in a sequence of variables, as a leading indicator of possible change, is proposed. Similarly, the probability of winning in a sequential gambling framework is proposed. The optimal time to exit of gambling game is proposed. A game theoretic approach to problem of optimal exit time is proposed. In all cases, beta approximations are proposed. Finally, a conclusion section is also given.

THE OPTIMAL CONTROL PROBLEM FOR SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF-AXIS

This article is devoted to exploring the optimal control problem for a system of integro-differential equations on the infinite interval. Sufficient conditions for the existence of optimal controls and trajectories have been obtained in terms of right-hand sides and the quality criterion function. Integro-differential equation systems are the mathematical models for many natural science processes, such as those in fluid dynamics and kinetic chemistry, among others. Many of these equations have the control that minimizing specific functionals related to the dynamics of these processes. This work specifically focuses on deriving sufficient optimality conditions for integro-differential systems on the half-axis. The complexity of the research is in the following aspects: Firstly, the problem at hand involves optimal control with an infinite horizon, which makes the direct application of compactness criteria like the Arzela-Ascoli theorem impossible. Secondly, the problem is considered up to the moment $\tau$ when the solution reaches the boundary of the domain. This reach moment depends on the control $\tau = \tau(u)$. Hence, the solution to the problem is essentially represented by the triplet $(u^*, x^*, \tau^*)$ — the optimal control, the optimal trajectory, and the optimal exit time. We note that a particular case of this problem is the problem of optimal speed. The main idea of proving the existence of an optimal solution relies on a compactness approach and involves the following steps: identifying a weakly convergent minimizing sequence of admissible controls, extracting a strongly convergent subsequence of corresponding trajectories, and justifying boundary transitions in equations and the quality criterion. The work provides a problem statement, formulates, and proves the main result.

Approximative Policy Iteration for Exit Time Feedback Control Problems Driven by Stochastic Differential Equations using Tensor Train Format

We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space by a sequence of linear equations. As high degree multi-polynomials are needed, the corresponding equations suffer from the curse of dimensionality even in moderate dimensions. We employ tensor-train methods to account for this problem. The approximation process within the Policy Iteration is done via a Least-Squares ansatz and the integration is done via Monte-Carlo methods. Numerical evidences are given for the (multi dimensional) double well potential and a three-hole potential.

Closing a Bitcoin Trade Optimally under Partial Information: Performance Assessment of a Stochastic Disorder Model

The Bitcoin market exhibits characteristics of a market with pricing bubbles. The price is very volatile, and it inherits the risk of quickly increasing to a peak and decreasing from the peak even faster. In this context, it is vital for investors to close their long positions optimally. In this study, we investigate the performance of the partially observable digital-drift model of Ekström and Lindberg and the corresponding optimal exit strategy on a Bitcoin trade. In order to estimate the unknown intensity of the random drift change time, we refer to Bitcoin halving events, which are considered as pivotal events that push the price up. The out-of-sample performance analysis of the model yields returns values ranging between 9% and 1153%. We conclude that the return of the initiated Bitcoin momentum trades heavily depends on the entry date: the earlier we entered, the higher the expected return at the optimal exit time suggested by the model. Overall, to the extent of our analysis, the model provides a supporting framework for exit decisions, but is by far not the ultimate tool to succeed in every trade.

Influence of Venture Capital and Knowledge Transfer on Innovation Performance in the Big Data Environment

Technological innovation requires large investments. Venture capital (VC) is a prominent financial source for innovative start-ups. A venture capitalist will inevitably transfer knowledge to facilitate the innovation of a firm while monitoring and advising its portfolio companies. Only when a firm has its own valuable new knowledge and high growth potential would venture capitalists select it. At the same time, big data knowledge, such as customer demands and user preferences, is also important for the new product development of a firm in the big data environment. Therefore, private knowledge transferred from venture capitalists, new knowledge developed independently by a firm itself, and big data knowledge are the three main types of knowledge for venture-backed firms in the big data environment. To find the influences of VC and knowledge transfer on the innovative performance of venture-backed firms, a model of maximizing the present value of the expected profit of new product innovation performance of a venture-backed firm in the big data environment is presented. The model can help venture capitalists to determine the scale of investment and the optimal exit time and predict the internal rate of return (IRR). This model can also help innovative start-ups to illustrate the value and prospects of a project to attract investment in their business prospectus.

Order and exit decisions under non-increasing price curves for products with short life cycles

We consider a supplier selling a product with a relatively short life cycle and following a non-increasing price curve. Because of the short cycle, there is a single procurement opportunity at the beginning of the cycle. The objective of the supplier is to determine the initial order quantity and the time to remove the product from the market in order to maximize her profits. We study this problem in a continuous-time framework where the demand is modeled with a non-homogeneous Poisson process having a general intensity function and the pricing strategy is given by an arbitrary non-increasing function. We give a rigorous mathematical analysis for the problem and show how it can be solved in two stages. We also consider the special case with piecewise constant intensity and price functions. For this case, we show that the optimal exit time is included in the set of break points of these functions. This brings a fast method to obtain the optimal solution for this special case.

Optimal Exit Time from Casino Gambling: Strategies of Precommitted and Naive Gamblers

We consider a casino gambling model with an indefinite end date and gamblers endowed with cumulative prospect theory preferences. We study the optimal strategies of a precommitted gambler, who comm...

Price momentum and the premium for meeting or beating analysts' forecasts of earnings

This study provides a theoretical rationale and empirical support that relates the existence and magnitude of the premium for meeting/beating analysts' EPS forecasts to the existence of preannouncement price momentum. The study is based on the theoretical work that suggests that extreme levels of price momentum can cause security prices to deviate from fundamental values even in the presence of well-informed and well-financed rational arbitrageurs. Differences of opinion regarding the extent of mispricing and/or optimal exit time to exit the position allow this mispricing to persist (Abreu and Brunnermeir 2002, 2003). To correct mispricing, a news event, like an earnings announcement, is necessary to synchronize investors' exit strategy beliefs (Abreu and Brunnermeir 2002, 2003). In the case of an earnings announcement, this synchronization of beliefs triggers a price reaction of such magnitude that it cannot be explained by unexpected earnings. Instead, we hypothesize and show that the abnormal price reaction is largely captured in what empirical researchers have identified as the meet/beat market premium. Our findings provide a cohesive argument for the temporal variation in meet/beat premiums documented by Koh, Matsumoto and Rajgopal (2008).

Entry–Exit Decisions with Underlying Processes Following Geometric Lévy Processes

We study, from the perspective of optimal stopping theory, entry---exit decision problems of a project in the context that the log-price process follows a Levy process with exponential jumps. A closed-form solution to the problems is obtained. To be specific, we show explicitly an optimal entry time, an optimal exit time and an expression of the maximal expected present value of the project. Moreover, it is also anatomized how the jumping of the Levy process affects optimal entry and exit times. While the negative effect of jumping on prices grows, the optimal exit time gets earlier, and the optimal entry time, however, first moves up and then moves down. In addition, the optimal exit time decreases with the frequency increasing of jumps when the negative effect dominates the positive effect, and increases when the opposite situation holds; the optimal entry time first increases and then decreases as the frequency becomes higher if the negative effect dominates the positive effect, and increases if the contrary condition is satisfied.

Hold or sell? How capital gains taxation affects holding decisions

Investments with exit flexibility require decisions regarding both the investment and holding period. Because selling an investment often leads to taxable capital gains, which crucially depend on the duration of an investment, we investigate the impact of capital gains taxation on exit timing under different tax systems. We observed that capital gains taxation delays exit decisions but loses its decision relevance for very long holdings. Often the optimal exit time, which indicates the maximal present value of future cashflows, cannot be determined analytically. However, we identify the breakeven exit time that guarantees present values exceeding those of an immediate sale. While, after-taxes, an immediate sale is often optimal, long holding periods might also be attractive for investors depending on the degree of income and corporate tax integration. A classic corporate tax system often indicates holdings over more than 100 periods. By contrast, a shareholder relief system indicates the earliest breakeven exit time and thus the highest level of exit timing flexibility. Surprisingly, high retention rates are likely to accelerate sales under a classic corporate system. Additionally, the worst exit time, which should be avoided by investors, differs tremendously across tax systems. For an integrated tax system with full imputation, the worst time is reached earlier than under partial or non-integrated systems. These results could help to predict investors’ behavior regarding changes in capital gains taxation and thus are of interest for both investors and tax policymakers. Furthermore, the results emphasize the need to control for the underlying tax system in cross-country empirical studies.

Online Appendix to: 'Optimal Exit Time from Casino Gambling: Strategies of Pre-Committed and Naive Gamblers'

This document is an online appendix to Optimal Exit Time from Casino Gambling: Strategies of Pre-committed and Naive Gamblers by X.D. He, S. Hu, J. Obloj, and X.Y. Zhou.The paper to which these Appendices apply is available at the following URL: http://ssrn.com/abstract=2684043

Optimal Exit Time from Casino Gambling: Strategies of Pre-Committed and Naive Gamblers

We employ the casino gambling model in He et al (2015) (available at: http://ssrn.com/abstract=2534936) to study the strategies of a pre-committed gambler, who commits her future selves to the strategy she sets up today, and of a naive gambler, who fails to do so and thus keeps changing plans at every time. We identify conditions under which the pre-committed gambler, asymptotically, adopts a stop-loss strategy, exhibits the behavior of disposition effect, or does not exit. For a specific parameter setting when the utility function is piece-wise power and the probability weighting functions are concave power, we derive the optimal strategy of the pre-committed gambler in closed-form whenever it exists. Finally, we study the actual behavior of the naive gambler and highlight its marked differences from that of the pre-committed gambler.The appendices for this paper are available at the following URL: http://ssrn.com/abstract=2682637

A note on optimality conditions for optimal exit time problems

In this note, we obtain some optimality conditions for optimal control problems with exit time similar to those obtained in [Cannarsa, Pignotti and Sinestrari, Discrete Contin. Dynam. Systems 6 (2000), 975 - 997] without requiring an assumption on the Hamiltonian.

Impact of Capital Gains Taxation on the Holding Period of Investments Under Different Tax Systems

Investments with exit flexibility require decisions regarding both the Investment and holding period. Because selling an investment often leads to taxable capital gains, which crucially depend on the duration of an investment, we investigate the impact of capital gains taxation on exit timing under different tax systems. We observed that capital gains taxation delays exit decisions but loses its decision relevance for very long holdings. Often the optimal exit time, which indicates the maximal present value of future cashflows, cannot be determined analytically. However, we identify the breakeven exit time that guarantees present values exceeding those of an immediate sale. While, after-taxes, an immediate sale is often optimal, long holding periods might also be attractive for investors depending on the degree of income and corporate tax integration. A classic corporate tax system often indicates holdings over more than 100 periods. By contrast, a shareholder relief system indicates the earliest breakeven exit time and thus the highest level of exit timing flexibility. Surprisingly, high retention rates are likely to accelerate sales under a classic corporate system. Additionally, the worst exit time, which should be avoided by investors, differs tremendously across tax systems. For an integrated tax system with full imputation, the worst time is reached earlier than under partial or non-integrated systems. These results could help to predict investors’ behavior regarding changes in capital gains taxation and thus are of interest for both investors and tax policymakers. Furthermore, the results emphasize the need to control for the underlying tax system in cross-country empirical studies.

An ODE Comparison Theorem With Application in the Optimal Exit Time Control Problem

The optimal exit time control (OETC) problem tries to find the feedback control law with a reasonable cost that can keep the system state inside a certain subset of the state space, called the safe set, for the longest time under random perturbations. The symmetry property of its solutions has been proved previously when the state dimension is higher than one. In this note, a comparison theorem is established that compares the solutions to two ODEs arising in the 1-D OETC problem. The symmetry of solutions to the 1-D OETC problem is proved using this comparison theorem. An example is presented to show how the symmetry result can help to solve the OETC problem analytically in certain cases.

Symmetry of Solutions to the Optimal Exit Time Control Problem

In this paper, we study the solutions to the optimal exit time control problem. Such a problem tries to find the state feedback control law with a fixed cost that can keep the state of a randomly perturbed system inside a subset of the state space, called the safe set, for as long as possible on average. By formulating the problem as an optimization problem with PDE constraints and using symmetrization techniques, we show that, when the safe set is a ball, the optimal feedback control (if it exists) must be radially symmetric. Furthermore, we show that, among all safe sets with a fixed volume, the ball is the best in that it yields the most efficient optimal exit time control. The proofs make essential use of the general isoperimetric inequality.

Venture Capital and Timing of IPO

Our purpose is to find the optimal exit time of a venture capitalist (VC) under profit flow uncertainty. We consider that the VC sells his holding in two steps: at the offer price at the date of the initial public offering (IPO) taking into account the underpricing, then at the expiration of the lock-up period. We use a real options approach when the profit flow of the venture capital backed company follows an arithmetic Brownian motion. We find a closed-form solution of the threshold at which it is optimal for the VC to exit and also determine the expected first hitting time.

Rectifiability results for singular and conjugate points of optimal exit time problems

We consider the value function V of optimal control problems with exit time. Under suitable assumptions, through the study of the conjugate points, we prove that the closure of the singular set of V is rectifiable. Moreover, a sharper Hausdorff estimate is given on the set of the conjugate nonsingular points.

Semiconcavity for optimal control problems with exit time

In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of general form $\dot y(t)=f(y(t),u(t))$, $y(t)\in\mathbb R^n$, $u(t)\in U\subset \mathbb R^m$. However, suitable assumptions are needed relating $f$ with the running and exit costs. The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the Maximum Principle, and to study the singular set of the value function.