Sequential Design for Optimal Stopping Problems

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.Comment: 24 page

Processing second-order stochastic dominance models using cutting-plane representations

This is the post-print version of the Article. The official published version can be accessed from the links below. Copyright @ 2011 Springer-VerlagSecond-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).This study was funded by OTKA, Hungarian National Fund for Scientific Research, project 47340; by Mobile Innovation Centre, Budapest University of Technology, project 2.2; Optirisk Systems, Uxbridge, UK and by BRIEF (Brunel University Research Innovation and Enterprise Fund)

Sequential change-point detection when unknown parameters are present in the pre-change distribution

In the sequential change-point detection literature, most research specifies a required frequency of false alarms at a given pre-change distribution $f_{\theta}$ and tries to minimize the detection delay for every possible post-change distribution $g_{\lambda}$. In this paper, motivated by a number of practical examples, we first consider the reverse question by specifying a required detection delay at a given post-change distribution and trying to minimize the frequency of false alarms for every possible pre-change distribution $f_{\theta}$. We present asymptotically optimal procedures for one-parameter exponential families. Next, we develop a general theory for change-point problems when both the pre-change distribution $f_{\theta}$ and the post-change distribution $g_{\lambda}$ involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations.Comment: Published at http://dx.doi.org/10.1214/009053605000000859 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dual representations for general multiple stopping problems

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b), Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cashflow structures than the additive structure in the above references. For example some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for the price of multiple exercise options and exemplify it by a numerical study on the pricing of a swing option in an electricity market.Comment: This is an updated version of WIAS preprint 1665, 23 November 201