Multiple Stopping Research Articles

A Study on Optimal Multiple Stopping and Swing Options Pricing

In this paper, we have studied the optimal stopping of random process as well as the costing of Swing options, specially the valuation of electricity market which is considered to an American style option having multiple practicing rights. Since this type of options are widely used in investing, so it requires some methods for valuation and that should be as precise as possible. So, we discuss two numerical methods for getting swing options prices in the field of electricity market, namely Monte Carlo and Finite difference. Finally, we compare our obtained results numerically and graphically with the help of MATLAB. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 145-155

Two-Stage Bayesian Sequential Change Diagnosis

In this paper, we formulate and solve a two-stage Bayesian sequential change diagnosis (SCD) problem. Different from the one-stage sequential change diagnosis problem considered in the existing work, after a change has been detected, we can continue to collect low-cost samples so that the post-change distribution can be identified more accurately. The goal of a two-stage SCD rule is to minimize the total cost including delay, false alarm probability, and misdiagnosis probability. To solve the two-stage SCD problem, we first convert the problem into a two-ordered optimal stopping time problem. Using tools from optimal multiple stopping time theory, we obtain the optimal SCD rule. Moreover, to address the high computational complexity issue of the optimal SCD rule, we further propose a computationally efficient threshold-based two-stage SCD rule. By analyzing the asymptotic behaviors of the delay, false alarm, and misdiagnosis costs, we show that the proposed threshold SCD rule is asymptotically optimal as the per-unit delay costs go to zero.

Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects

<p style='text-indent:20px;'>So far, the optimal investment timing to maximize the total profit of multi-stage capacity expansion infrastructure projects is not clear. In the case of uncertain demands, the optimal multiple stopping time theory is adopted to model the optimal decision-making of investment timing for multi-stage expansion infrastructure projects in a finite time horizon. In this context, the first-stage of the project involves a dedicated asset investment for later expansion, and the capacity of the project at each stage is constrained, which makes the cash flow of the project exhibit the characteristic of bull call spread. The upwind finite difference method and multi-least squares Monte Carlo simulation are combined to solve the project value and determine the optimal exercise boundaries at all stages described by a sequence of demand thresholds. A multi-stage power plant project is taken as an example to validate the model. Through the example, the optimal investment strategies and the value of the multi-stage project are provided; the effects of the dedicated asset and capacity constraint are illustrated. This study novelly reveals the effect of the capacity constraints on the project value using the bull call spread theory.</p>

Partial liquidation under reference-dependent preferences

We propose a multiple optimal stopping model where an investor can sell a divisible asset position at times of her choosing. Investors have S-shaped reference-dependent preferences, whereby utility is defined over gains and losses relative to a reference level and is concave over gains and convex over losses. For a price process following a time-homogeneous diffusion, we employ the constructive potential-theoretic solution method developed by Dayanik and Karatzas (Stoch. Process. Appl. 107:173–212, 2003). As an example, we revisit the single optimal stopping model of Kyle et al. (J. Econ. Theory 129:273–288, 2006) to allow partial liquidation. In contrast to the extant literature, we find that the investor may partially liquidate the asset at distinct price thresholds above the reference level. Under other parameter combinations, the investor sells the asset in a block, either at or above the reference level.

Belief Propagation Decoder With Multiple Bit-Flipping Sets and Stopping Criteria for Polar Codes

Compared with successive cancellation list (SCL) decoders, belief propagation (BP)-based decoders suffer performance loss in middle-and high-signal-to-noise ratio (SNR) regions. By analyzing the behavior of the incorrect decoding results of the bit-flipping BP decoder with a critical set of order ω (BFBP-CS ω ), we found that undetected errors mainly contribute to the error floor. Based on this observation, we proposed a belief propagation decoder with multiple bit-flipping sets (BFSs) and stopping criteria (BP-MF-MC) in this work. We use multiple stopping criteria to identify undetected errors and a small BFS to find an additional estimated codeword given by the bit-flipping BP (BFBP) function. For uncorrected errors, we use multiple BFSs to find estimated codewords with the BFBP function. Furthermore, we propose a method to dynamically generate a BFS based on the submatrix check. This method can remove unnecessary bit-flipping positions and increase the order of the critical set. Then, the best codeword is selected from all estimated codewords according to the maximum likelihood principle. Numerical results show that BP-MF-MC performs similarly to the cyclic redundancy check-aided SCL (CA-SCL) decoder with list size 16 and is slightly worse than CA-SCL with list size 32.

Optimal multiple stopping under catastrophic event

Optimal multiple stopping under catastrophic event

Optimal multiple stopping under catastrophic event

In this paper, we introduce a new optimal multiple stopping times problem, where we assume each exercise right happens before the date of release of a catastrophic event modelled by a random variable and this catastrophe can be natural (e.g., earthquake, tsunami) or technological (e.g., nuclear event). Since a sudden catastrophe will have a direct influence on prices variation, especially those of underlying as well as option's prime, eventual catastrophic event will be modelled by the first time the underlying's price exceeds some large barrier. The originality of this paper comes from a mathematical model taking account of a nonlinear criteria of sum of the underlying stopped at stopping times of the holder's filtration information as well as a random number of exercise rights at sopping times involving prior to a catastrophic event. This will generalises the concept of swing contracts, where the exercise rights number is only deterministic and finite.

New class of optimal multiple stopping times problems

New class of optimal multiple stopping times problems

A Convex Optimization Approach to Multiple Stopping: Pricing Chooser Caps and Swing Options

In this current work, we generalize the recent Pathwise Optimization approach of Desai et al to Multiple stopping problems. The approach also minimizes the dual bound as in Desai et al to find the best approximation architecture for the Multiple stopping problem. Though, we establish the convexity of the dual operator, in this setting as well, we cannot directly take advantage of this property because of the computational issues that arise due to the combinatorial nature of the problem. Hence, we deviate from the pure martingale dual approach to marginal dual approach of Meinshausen and Hambly and solve each such optimal stopping problem in the framework of Desai et al. Though, this Pathwise Optimization approach as generalized to the Multiple stopping problem is computationally intensive, we highlight that it can produce superior dual and primal bounds in certain settings.

Non-Nested Monte Carlo Dual Bounds for Multi-Exercisable Options

We study the optimal marginal value of discrete-time optimal multiple stopping problems and find that it can be formulated as a single optimal stopping optimization as well. Based on this result propose a marginal-value-based lower bound method to achieve a small bound on the iterative error. We further introduce a non-nested upper bound method. The convergence of both methods is analysed. The implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency and the tightness in dual bounds.

Conflict-free routing of multi-stop warehouse trucks

The recent interest in greater vehicular autonomy for factory and warehouse automation has stimulated research in conflict-free routing: a challenging network routing problem in which vehicles may not pass each other. Motivated by a real-world case study, we consider one such application: truck movements in a tightly constrained warehouse. We propose an extension of an existing conflict-free routing algorithm to consider multiple stopping points per route. A high level metaheuristic is applied to determine the route construction and assignment of vehicles to routes.

Double continuation regions for American and Swing options with negative discount rate in Lévy models

AbstractIn this paper, we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate that arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black–Scholes model and the jump‐diffusion model with exponentially distributed jumps.

Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies

The topic of investment timing in multi-stage public-private partnership (PPP) projects has not been received much attention so far. This study investigates optimal expansion timing decisions in multi-stage PPP projects under an uncertain demand and where the first-stage greenfield project involving a dedicated asset is immediately implemented as the PPP contract is closed, whereas the timing of the later expansion is flexibly decided according to the demand. In this setting, the optimal multiple stopping timing theory is adopted to model the expansion framework. Furthermore, we integrate a government subsidy, including an investment subsidy and revenue subsidy, into the expansion timing decisions. Through a hypothetical three-stage investment plan for a sanitary sewerage project, the optimal expansion strategies and the value of the multi-stage project before and after the subsidy are provided using a least squares Monte Carlo simulation. Also, the influences of a dedicated asset on the expansion strategies and project value are illustrated. In addition, we compare the incremental value before and after the subsidy and earlier expansion derived from two types of subsidies. The comparisons show that there is more incremental value for the revenue subsidy, and that the investment subsidy brings an earlier expansion.

The viscosity solutions approach to swing options pricing under a regime-switching mean-reverting model

In this paper, we study the valuation of swing options on electricity markets with local volume and refraction time constraints, under the setting that the dynamic of the underlying spot price is a 2-state regime-switching mean-reverting process. We derive the corresponding optimal multiple stopping problem, reduce it to a sequence of optimal single stopping problems, and further find that those value functions satisfy HJB variational inequalities subject to suitable conditions. Then after a prior estimation for the value functions, the viscosity solutions approach is adopted to get existence and uniqueness results in viscosity sense.

Online Resource Allocation Under Partially Predictable Demand

For online resource allocation problems, we propose a new demand arrival model where the sequence of arrivals contains both an adversarial component and a stochastic one. Our model requires no demand forecasting; however, due to the presence of the stochastic component, we can partially predict future demand as the sequence of arrivals unfolds. Under the proposed model, we study the problem of the online allocation of a single resource to two types of customers, and design online algorithms that outperform existing ones. Our algorithms are adjustable to the relative size of the stochastic component, and our analysis reveals that as the portion of the stochastic component grows, the loss due to making online decisions decreases. This highlights the value of (even partial) predictability in online resource allocation. We impose no conditions on how the resource capacity scales with the maximum number of customers. However, we show that using an adaptive algorithm — which makes online decisions based on observed data — is particularly beneficial when capacity scales linearly with the number of customers. Our work serves as a first step in bridging the long-standing gap between the two well-studied approaches to the design and analysis of online algorithms based on (1) adversarial models and (2) stochastic ones. Using novel algorithm design, we demonstrate that even if the arrival sequence contains an adversarial component, we can take advantage of the limited information that the data reveals to improve allocation decisions. We also study the classical secretary problem under our proposed arrival model, and we show that randomizing over multiple stopping rules may increase the probability of success.

Multiple stopping time POMDPs: Structural results & application in interactive advertising on social media

This paper considers a multiple stopping time problem for a Markov chain observed in noise, where a decision maker chooses at most L stopping times to maximize a cumulative objective. We formulate the problem as a Partially Observed Markov Decision Process (POMDP) and derive structural results for the optimal multiple stopping policy. The main results are as follows: (i) The optimal multiple stopping policy is shown to be characterized by threshold curves Γl, for l=1,…,L, in the unit simplex of Bayesian Posteriors. (ii) The stopping sets Sl (defined by the threshold curves Γl) are shown to exhibit the following nested structure Sl−1⊂Sl. (iii) The optimal cumulative reward is shown to be monotone with respect to the copositive ordering of the transition matrix. (iv) A stochastic gradient algorithm is provided for estimating linear threshold policies by exploiting the structural results. These linear threshold policies approximate the threshold curves Γl, and share the monotone structure of the optimal multiple stopping policy. (v) Application of the multiple stopping framework to interactively schedule advertisements in live online social media. It is shown that advertisement scheduling using multiple stopping performs significantly better than currently used methods.

An exactly solvable multiple stochastic optimal stopping problem

A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.

On the Optimal Exercise Boundaries of Swing Put Options

We use probabilistic methods to characterise time-dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon. Motivated by financial applications, we consider a payoff of immediate stopping of “put” type, and the underlying dynamics follows a geometric Brownian motion. The optimal stopping region relative to each optimal stopping time is described in terms of two boundaries, which are continuous, monotonic functions of time and uniquely solve a system of coupled integral equations of Volterra-type. Finally, we provide a formula for the value function of the problem.

Optimal Liquidation of Child Limit Orders

The present paper studies the optimal placement problem of a child order. In practice, short-term momentum indicators inferred from order book data play important roles in order placement decisions. In the present work, we first propose to explicitly model the short-term momentum indicator and then formulate the order placement problem as an optimal multiple stopping problem. In contrast to the common formulation in the existing literature, we allow zero refracting period between consecutive stopping times to take into account the common practice of submitting multiple orders at the same time. This optimal multiple stopping problem will be explored over both infinite and finite horizons. It is shown that the optimal placement of a child order and the optimal replacement of outstanding limit orders by market ones are determined by first passage times of the short-term momentum indicator across a sequence of time-dependent boundaries. The aggressiveness of the optimal order replacement strategy is also examined via several numerical examples. In particular, our work illustrates that the optimal order replacement strategy is more aggressive when the bid–ask spread is smaller, when the impact from the momentum indicator is larger, or when the remaining time becomes shorter. All of these decision-making behaviors predicted by our model are natural and agree with empirical studies in the existing literature.

Crowd Foraging: A QoS-Oriented Self-Organized Mobile Crowdsourcing Framework Over Opportunistic Networks

Recent years have witnessed the proliferation of mobile crowdsourcing that brings a new opportunity to leverage human intelligence and movement behaviors to wider application areas. In parallel with the development of online centralized platforms, we look into the realization of self-organized mobile crowdsourcing drawing on opportunistic networks, and propose the Crowd Foraging framework, in which a mobile task requester can proactively recruit a massive crowd of opportunistic encountered mobile workers in real time for quick and high-quality results. We present a comprehensive framework model that fully integrates human behavior factors for modeling task profile, worker arrival, and work ability, and then introduce a service quality concept to indicate the expected service gain that a requester can enjoy when she recruits an arrival worker by jointly considering the work ability of workers as well as timeliness and reward of tasks. Furthermore, we formulate a sequential worker recruitment problem as an online multiple stopping problem to maximize the expected sum of service quality, and accordingly derive an optimal worker recruitment policy through the dynamic programming principle, which exhibits a nice threshold-based structure. We provide data-driven case studies to validate the assumptions used in the policy design, and conduct extensive trace-driven numerical evaluations, which demonstrate that our policy can achieve superior performance (e.g., improve more than 30% performance over classic policies). Besides, our Android prototype shows that the Crowd Foraging framework is cost-efficient, such as requiring less than 7 s and 6 J in terms of time and energy consumption for the optimal threshold calculation in our policy in most cases.