Optimal Stopping Value Research Articles

Risk sensitive optimal stopping

In this paper we consider continuous time risk sensitive optimal stopping problem. Using the probabilistic approach and dyadic discrete time approximations we prove continuity of the generic optimal stopping value function for a large class of Feller-Markov processes. Also, we provide formulas for the corresponding optimal stopping policies and study regularity of approximating functions.

Long-Run Risk-Sensitive Impulse Control

In this paper we consider long-run risk-sensitive average cost impulse control applied to a continuous-time Feller--Markov process. Using the probabilistic approach, we show how to get the solution to a suitable continuous-time Bellman equation and link it with the impulse control problem. The optimal strategy for the underlying problem is constructed as a limit of dyadic impulse strategies by exploiting regularity properties of the linked risk-sensitive optimal stopping value functions. In particular, this shows that the discretized setting could be used to approximate near-optimal strategies for the underlying continuous-time control problem, which facilitates the usage of the standard approximation tools. For completeness, we present examples of processes that could be embedded into our framework.

Executive Stock Option Exercise with Full and Partial Information on a Drift Change Point

We analyze the optimal exercise of an American call executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point and seek to optimally exercise the option by maximizing its discounted payoff under the physical measure. The first agent has full information and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterize the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.

Cost-based optimization of the stopping threshold for local disease surveillance during progressive eradication of tuberculosis from New Zealand wildlife.

Bovine tuberculosis (TB) is managed in New Zealand largely via population reduction of the major wildlife disease reservoir of Mycobacterium bovis, the introduced brushtail possum Trichosurus vulpecula. New Zealand aims to eradicate M.bovis infection from its livestock and wildlife within 40years, as the culmination of progressive regional eradication programmes. Declarations of regional eradication are decided after extensive possum population control and post-control surveillance; hence, we developed a modelling framework, based on eco-epidemiological simulation data, to provide cost-evaluated options for deciding when to make these declarations. A decision-support framework evaluated potential costs of wildlife surveillance (and recontrol, if required) with respect to the calculated probability of successful eradication of M.bovis from wildlife. This enabled expected costs to be predicted in terms of stopping thresholds, allowing selection of optimal stopping rules based on minimizing costs. We identified factors that could influence optimal stopping values applied during regional eradication. Where vector/disease surveillance was inexpensive (for example, using low-cost detection devices or sentinel wildlife hosts) optimization involved setting a higher rather than lower stopping value, as it would be cheaper to minimize the risk of making a false declaration of eradication than to remedy any such failure. In addition, any cost of recontrol would largely depend on the time to rediscovery of residual M.bovis infection in wildlife, which would in turn be linked to the level of ongoing passive surveillance (with more rapid detection of re-emergent infection among wildlife in farmland situations than in remote forested regions). These two scenarios would favour different optimal stopping rules, as would the consideration of stakeholder confidence and socio-political issues, which are discussed. The framework presented here provides guidance to assess the economics underlying eradication of bovine TB from New Zealand farming; this eliminates reliance upon a pre-determined and uniform stopping rule for ceasing active management.

Optimal Multiple Stopping with Sum-Payoff

We consider the optimal stopping of independent, discrete time sequences $X_1,\dots, X_n$, where $m$ stops are allowed. The payoff is the sum of the stopped values. Under the assumption of convergence of related imbedded point processes to a Poisson process in the plane, we derive approximatively optimal stopping times and stopping values. The solutions are obtained via a system of $m$ differential equations of first order. As an application we consider the case that $X_i=c_iZ_i+d_i$, with $(Z_i)$ independent identically distributed in the domain of attraction of an extreme value distribution. We obtain explicit results for stopping values and approximative optimal stopping rules.

On general optimal stopping problems using penalty method

Abstract In the paper we use penalty method to approximate a number of general stopping problems over finite horizon. We consider optimal stopping of discrete time or right continuous stochastic processes, and show that suitable version of Snell’s envelope can by approximated by solutions to penalty equations. Then we study optimal stopping problem for Markov processes on a general Polish space, and again show that the optimal stopping value function can be approximated by a solution to a Markov version of the penalty equation.

Constructing time-homogeneous generalized diffusions consistent with optimal stopping values

Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting. In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions, we can characterize the existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalized convexity theory.

Penalty Method for Finite Horizon Stopping Problems

In this paper we use a penalty method to approximate a number of stopping problems over a finite horizon. In particular we prove existence and continuity of the value function corresponding to Dynkin games over a finite horizon. Since stopping problems can be studied in the context of Dynkin games, as a by-product we obtain continuity of the optimal stopping value. We then study Dynkin games with delayed stopping and finally impulse control. In each case the value function is approximated by a solution to a suitable penalty equation.

Continuity Properties of Optimal Multiple Stopping Value

This article is concerned with the optimal multiple stopping problem for discrete time finite stage stochastic processes. We study lower semicontinuity and continuity properties of optimal stopping values with respect to the topology of convergence in distribution. Also, we formulate the multiple stopping version of the prophet inequality for the optimal stopping problem and apply the lower semicontinuity property of optimal stopping values to the prophet inequality for the optimal multiple stopping problem.

Semi-parametric forecasts of the implied volatility surface using regression trees

We present a new semi-parametric model for the prediction of implied volatility surfaces that can be estimated using machine learning algorithms. Given a reasonable starting model, a boosting algorithm based on regression trees sequentially minimizes generalized residuals computed as differences between observed and estimated implied volatilities. To overcome the poor predictive power of existing models, we include a grid in the region of interest, and implement a cross-validation strategy to find an optimal stopping value for the boosting procedure. Back testing the out-of-sample performance on a large data set of implied volatilities from S&P 500 options, we provide empirical evidence of the strong predictive power of our model.

Lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set N d . The optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable, is lower semicontinuous for the topology of convergence in distribution. The multiparameter version of prophet inequality for the one-parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality.

Forecasting Implied Volatility Surfaces

This paper introduces a new semi-parametric methodology for the implied volatility surface, which incorporates machine learning algorithms. Given a starting model, a tree boosting algorithm sequentially minimizes the residuals of observed and estimated implied volatility. To overcome the poor predicting power of existing models, we include a grid in the region of interest, and implement a cross-validation strategy to find an optimal stopping value for the tree boosting. Back testing the out-of-sample performance on a large data set of implied volatilities on S&P 500 options, we provide empirical evidence of the strong predictive potential of our methodology.

Prophet Inequalities for I.I.D. Random Variables with Random Arrival Times

Suppose X 1, X 2,… are independent and identically distributed nonnegative random variables with finite expectation, and for each k, X k is observed at the kth arrival time S k of a Poisson process with a unit rate independent of the sequence {X k }. For t > 0, comparisons are made between the expected maximum M(t) := E[max k≥1 X k I(S k ≤ t)] and the optimal stopping value V(t) := supτ∈𝒯E[X τ I(S τ ≤ t)], where 𝒯 is the set of all ℕ-valued random variables τ such that {τ = i} is measurable with respect to the σ-algebra generated by (X 1, S 1),…, (X i , S i ). For instance, it is shown that M(t)/V(t) ≤ 1 + α0, where α0 ≐ 0.34149 satisfies , and this bound is asymptotically sharp as t → ∞. Another result is that M(t)/V(t) < 2 − (1 − e −t )/t, and this bound is asymptotically sharp as t ↓ 0. Upper bounds for the difference M(t) − V(t) are also given, under the additional assumption that the X k 's are bounded.

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Let X 1, X 2… and B 1, B 2… be mutually independent [0, 1]-valued random variables, with EB j = β > 0 for all j. Let Y j = B 1 … sB j−1 X j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y 1,…,Y n ):=sup{EY τ:τ is a stopping rule for Y 1,…,Y n } and E(max 1≤j≤n Y j ). It is shown that the set of ordered pairs {(x, y):x = V(Y 1,…,Y n ), y = E(max 1≤j≤n Y j ) for some sequence Y 1,…,Y n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ n, β(x)}, where Ψ n, β(x) = [(1 − β)n + 2β]x − β−(n−2) x 2 if x ≤ β n−1, and Ψ n, β(x) = min j≥1{(1 − β)jx + β j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.

Triangular function and continuity property of multiparameter optimal stopping value

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. A generalization of Elton's results for one-parameter optimal stopping problems to the case of multiparameter optimal stopping problems is given. The optimal stopping value of a discrete time multiparameter independent uniformly integrable stochastic process is continuous for the topology of convergence in distribution.

An application of prophet regions to optimal stopping with a random number of observations

Let X 1,X 2 , … be any sequence of nonnegative integrable random variables, and let N∈{1,2 , …} be a random variable with known distribution, independent of X 1,X 2 , …. The optimal stopping value sup t E(Xt I(N≥ t)) is considered for two players: one who has advance knowledge of the value of N, and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz [T.P. Hill and R.P. Kertz (1983). Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc., 278, 197–207].

On the optimal stopping values induced by general dependence structures

The optimal stopping value of random variables X 1,…,X n depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F 1,…,F n . They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

On the optimal stopping values induced by general dependence structures

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

Optimal stopping with discount and observation costs

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.

Optimal stopping with discount and observation costs

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.