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- ItemSemitractability of optimal stopping problems via a weighted stochastic mesh algorithm(Oxford [u.a.] : Wiley-Blackwell, 2020) Belomestny, Denis; Kaledin, Maxim; Schoenmakers, JohnIn this paper, we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete- and continuous-time optimal stopping problems. In this context, we consider tractability of such problems via a useful notion of semitractability and the introduction of a tractability index for a particular numerical solution algorithm. It is shown that in the discrete-time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by (Formula presented.) with (Formula presented.) being the dimension of the underlying Markov chain. Furthermore, we study the WSM approach in the context of continuous-time optimal stopping problems and derive the corresponding complexity bounds. Although we cannot prove semitractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example. © 2020 Wiley Periodicals LLC
- ItemAn iteration procedure for solving integral equations related to the American put options(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestny, Denis; Gapeev, PavelA new algorithm for pricing American put option in the Black-Scholes model is presented. It is based on a time discretization of the corresponding integral equation. The proposed iterative procedure for solving the discretized integral equation converges in a finite number of steps and delivers in each step a lower or an upper bound for the price of discretized option on the whole time interval. The method developed can be easily implemented and carried over to the case of more general optimal stopping problems.
- ItemDynamic programming for optimal stopping via pseudo-regression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bayer, Christian; Redmann, Martin; Schoenmakers, John G.M.We introduce new variants of classical regression-based algorithms for optimal stopping problems based on computation of regression coefficients by Monte Carlo approximation of the corresponding L2 inner products instead of the least-squares error functional. Coupled with new proposals for simulation of the underlying samples, we call the approach pseudo regression. We show that the approach leads to asymptotically smaller errors, as well as less computational cost. The analysis is justified by numerical examples.
- ItemRepresentations for optimal stopping under dynamic monetary utility functionals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krätschmer, Volker; Schoenmakers, JohnIn this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
- ItemRegression on particle systems connected to mean-field SDEs with applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Belomestny, Denis; Schoenmakers, John G.M.In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.
- ItemOn non-asymptotic optimal stopping criteria in Monte Carlo simulations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Hoel, Hakon; von Schwerin, Erik; Tempone, RaúlWe consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.
- ItemOptimal stopping with signatures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Hager, Paul; Riedel, Sebastian; Schoenmakers, John G. M.We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes.
- ItemMinimum return guarantees with funds switching rights : an optimal stopping problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Mahayni, Antje; Schoenmakers, John G.M.Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
- ItemOn the rates of convergence of simulation based optimization algorithms for optimal stopping problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Belomestny, DenisIn this paper we study simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings