Valuation of complex structured instruments in financial and energy markets

D. Belomestny, J. Schoenmakers, Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance, Macmillan Publishers Ltd., London, 2018, 364 pages, (Monograph Published), DOI 10.1057/978-1-137-03351-2 .

J.G.M. Schoenmakers, Robust Libor Modelling and Pricing of Derivative Products, Chapman & Hall CRC Press, 2005, 202 pages, (Monograph Published).

D. Belomestny, M. Kaledin, J.G.M. Schoenmakers, Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, published online on 27.05.2020, DOI 10.1111/mafi.12271 .
Abstract
In this article we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete and continuous time optimal stopping problems. It is shown that in the discrete time case the WSM algorithm leads to semi-tractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by $varepsilon^-4log^d+2(1/varepsilon)$ with $d$ 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 can not prove semi-tractability 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.

D. Belomestny, J.G.M. Schoenmakers, V. Spokoiny, B. Zharkynbay, Optimal stopping via reinforced regression, Communications in Mathematical Sciences, 18 (2020), pp. 109--121, DOI 10.4310/CMS.2020.v18.n1.a5 .
Abstract
In this note we propose a new approach towards solving numerically optimal stopping problems via boosted regression based Monte Carlo algorithms. The main idea of the method is to boost standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from finance.

CH. Bayer, R.F. Tempone , S. Wolfers, Pricing American options by exercise rate optimization, Quantitative Finance, published online on 07.07.2020, DOI 10.1080/14697688.2020.1750678 .
Abstract
We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.

D. Belomestny, R. Hildebrand, J.G.M. Schoenmakers, Optimal stopping via pathwise dual empirical maximisation, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 79 (2019), pp. 715--741, DOI 10.1007/s00245-017-9454-9 .
Abstract
The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the path-wise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

CH. Bayer, J. Häppölä, R. Tempone, Implied stopping rules for American basket options from Markovian projection, Quantitative Finance, 19 (2019), pp. 371--390.
Abstract
This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.

D. Belomestny, R. Hildebrand, J.G.M. Schoenmakers, Optimal stopping via pathwise dual empirical maximisation, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, pp. published online on 08.11.2017, urlhttps://doi.org/10.1007/s00245-017-9454-9, DOI 10.1007/s00245-017-9454-9 .
Abstract
The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the path-wise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

CH. Bayer, M. Siebenmorgen, R. Tempone, Smoothing the payoff for efficient computation of basket option prices, Quantitative Finance, 18 (2018), pp. 491--505 (published online on 20.07.2017), DOI 10.1080/14697688.2017.1308003 .
Abstract
We consider the problem of pricing basket options in a multivariate Black Sc- holes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non- smooth integrands. Due to this lack of regularity, higher order numerical integration tech- niques may not be directly available, requiring the use of methods like Monte Carlo specif- ically designed to work for non-regular problems. We propose to use the inherent smooth- ing property of the density of the underlying in the above models to mollify the payo ff function by means of an exact conditional expectation. The resulting conditional expec- tation is unbiased and yields a smooth integrand, which is amenable to the e ffi cient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 25.

F. Dickmann, N. Schweizer, Faster comparison of stopping times by nested conditional Monte Carlo, Journal of Computational Finance, 20 (2016), (24).

D. Belomestny, F. Dickmann, T. Nagapetyan, Pricing Bermudan options via multilevel approximation methods, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 6 (2015), pp. 448--466.
Abstract
In this article we propose a novel approach to reducing the computational complexity of various approximation methods for pricing discrete time American or Bermudan options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation, we propose a multilevel low biased estimate for the price of the option. It turns out that the resulting complexity gain can be of order ? ?1 with ? denoting the desired precision. The performance of the proposed multilevel algorithms is illustrated by a numerical example.

D. Belomestny, M. Ladkau, J.G.M. Schoenmakers, Simulation based policy iteration for American style derivatives -- A multilevel approach, SIAM ASA J. Uncertainty Quantification, 3 (2015), pp. 460--483.
Abstract
This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration.

CH. Bender, J.G.M. Schoenmakers, J. Zhang, Dual representations for general multiple stopping problems, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 25 (2015), pp. 339--370.
Abstract
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.

S. Balder, A. Mahayni, J.G.M. Schoenmakers, Primal-dual linear Monte Carlo algorithm for multiple stopping --- An application to flexible caps, Quantitative Finance, 13 (2013), pp. 1003--1013.
Abstract
In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal-dual linear Monte Carlo algorithm that allows for efficient simulation of lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of a primal-dual Monte Carlo algorithm for standard Bermudan options proposed in Schoenmakers et al (2011), to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers (2010). At each level the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. At the same time, the algorithm also provides approximate continuation functions which may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull White (1990) model setup. The simple model choice allows for comparison of the computed price bounds with the exact price which is obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.

D. Belomestny, J.G.M. Schoenmakers, F. Dickmann, Multilevel dual approach for pricing American style derivatives, Finance and Stochastics, 17 (2013), pp. 717-742.
Abstract
In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.

J.G.M. Schoenmakers, J. Zhang, J. Huang, Optimal dual martingales, their analysis and application to new algorithms for Bermudan products, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 4 (2013), pp. 86--116.
Abstract
In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options. We provide a theorem which give conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these theorems we develop a regression based backward construction of such a martingale in a Wiener environment. In turn this martingale may be utilized for computing upper bounds by non-nested Monte Carlo. As a by-product, the algorithm also provides approximations to continuation values of the product, which in turn determine a stopping policy. Hence, we obtain lower bounds at the same time. The proposed algorithm is pure dual in the sense that it doesn't require an (input) approximation to the Snell envelope, is quite easy to implement, and in a numerical study we show that, regarding the computed upper bounds, it is comparable with the method of Belomestny, et. al. (2009).

A. Mahayni, J.G.M. Schoenmakers, Minimum return guarantees with funds switching rights --- An optimal stopping problem, Journal of Economic Dynamics & Control, 35 (2012), pp. 1880--1897.
Abstract
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.

J.G.M. Schoenmakers, A pure martingale dual for multiple stopping, Finance and Stochastics, 16 (2012), pp. 319--334.
Abstract
In this paper we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such it is a natural generalization of the method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping problem for American options. We consider this representation as the real dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple dual representation we present three Monte Carlo simulation algorithms which require only one degree of nesting.

V. Krätschmer, J.G.M. Schoenmakers, Representations for optimal stopping under dynamic monetary utility functionals, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 1 (2010), pp. 811--832.
Abstract
In 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.

D. Belomestny, L. Rüschendorf, M. Urusov, Optimal stopping of integral functionals and a ``no-loss'' free boundary formulation, SIAM Journal of Theory of Probability and Its Applications, 54 (2010), pp. 14--28.

D. Belomestny, A. Kolodko, J.G.M. Schoenmakers, Regression methods for stochastic control problems and their convergence analysis, SIAM Journal on Control and Optimization, 48 (2010), pp. 3562--3588.
Abstract
In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.

D. Belomestny, G.N. Milstein, J.G.M. Schoenmakers, Sensitivities for Bermudan options by regression methods, Decisions in Economics and Finance. A Journal of Applied Mathematics, 33 (2010), pp. 117--138.
Abstract
In this article we propose several pathwise and finite difference based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations which allows, in combination with a regression approach, an efficient simultaneous computation of sensitivities at all initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods.

D. Belomestny, J. Kampen, J.G.M. Schoenmakers, Holomorphic transforms with application to affine processes, Journal of Functional Analysis, 257 (2009), pp. 1222--1250.
Abstract
In a rather general setting of Itô?Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Itô?Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform

D. Belomestny, Ch. Bender, J.G.M. Schoenmakers, True upper bounds for Bermudan products via non-nested Monte Carlo, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 19 (2009), pp. 53--71.
Abstract
We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic “true” stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.

D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quantitative Finance, 9 (2009), pp. 315--327.
Abstract
Here we develop methods for efficient pricing multidimensional discrete-time American and Bermudan options by using regression based algorithms together with a new approach towards constructing upper bounds for the price of the option. Applying sample space with payoffs at the optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach admits constructing both low and upper bounds for the price by Monte Carlo simulations. The methods are illustrated by pricing Bermudan swaptions and snowballs in the Libor market model.

J. Kampen, A. Kolodko, J.G.M. Schoenmakers, Monte Carlo Greeks for financial products via approximative transition densities, SIAM Journal on Scientific Computing, 31 (2008), pp. 1--22.

CH. Bender, A. Kolodko, J.G.M. Schoenmakers, Enhanced policy iteration for American options via scenario selection, Quantitative Finance, 8 (2008), pp. 135--146.
Abstract
In Kolodko & Schoenmakers (2004) and Bender & Schoenmakers (2004) a policy iteration was introduced which allows to achieve tight lower approximations of the price for early exercise options via a nested Monte-Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of actually performed inner simulations, and thus can heavily speed up the method (up to factor 10 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original one such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.

CH. Fries, J. Kampen, Proxy simulation schemes for generic robust Monte Carlo sensitivities, process-oriented importance sampling and high-accuracy drift approximation, Journal of Computational Finance, 10 (2007), pp. 97--128.
Abstract
We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme (target scheme) by adjusting measure (via likelihood ratio) to match the distribution. This makes the approach independent of the product (the function f) and even of the model, it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method and Malliavin's Calculus reconsidered on the level of the discrete numerical simulation scheme. Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin's Calculus). The framework is completely generic and may be used for high accuracy drift approximations, process oriented importance sampling and the robust calculation of partial derivatives of expectations w.r.t. model parameters (i.e. sensitivities, aka. Greeks) by applying finite differences by reevaluating the expectation with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking.

G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Forward and reverse representations for Markov chains, Stochastic Processes and their Applications, 117 (2007), pp. 1052--1075.
Abstract
In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.

A. Kolodko, J.G.M. Schoenmakers, Iterative construction of the optimal Bermudan stopping time, Finance and Stochastics, 10 (2006), pp. 27--49.
Abstract
We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by duality, the method induces a convergent sequence of upper bounds as well. In a Markovian setting the presented procedure allows to calculate approximative solutions with only a few nestings of conditional expectations and is therefore tailor-made for a plain Monte Carlo implementation. The method may be considered generic for all discrete optimal stopping problems. The power of the procedure is demonstrated for Bermudan swaptions in a full factor LIBOR market model.

D. Belomestny, G.N. Milstein, Monte Carlo evaluation of American options using consumption processes, International Journal of Theoretical and Applied Finance, 9 (2006), pp. 455--481.
Abstract
We develop a new approach for pricing both continuous-time and discrete-time American options which is based on the fact that any American option is equivalent to a European one with a consumption process involved. This approach admits the construction of an upper bound (a lower bound) on the true price using some lower bound (an upper bound) by Monte Carlo simulation. A number of effective estimators of upper and lower bounds with the reduced variance are proposed. The method is supported by numerical experiments which look promising.

CH. Bender, A. Kolodko, J.G.M. Schoenmakers, Iterating cancelable snowballs and related exotics, Risk Magazine, 9 (2006), pp. 126--130.
Abstract
Effective valuation procedures for callable exotics are a thorny problem. Standard methods reveal limitations in pricing many-dimensional and path-dependent products, such as cancellable snowballs. Christian Bender, Anastasia Kolodko and John Schoenmakers ally these methods with their recent iterative methodology to fill the final gap.

CH. Bender, A. Kolodko, J.G.M. Schoenmakers, Policy iteration for American options: Overview, Monte Carlo Methods and Applications, 12 (2006), pp. 347--362.
Abstract
This paper is an overview of recent results by Kolodko and Schoenmakers (2006), Bender and Schoenmakers (2006) on the evaluation of options with early exercise opportunities via policy improvement. Stability is discussed and simulation results based on plain Monte Carlo estimators for conditional expectations are presented.

CH. Bender, J.G.M. Schoenmakers, An iterative method for multiple stopping: Convergence and stability, Advances in Applied Probability, 38 (2006), pp. 729--749, DOI 10.1239/aap/1158684999 .
Abstract
We present a new iterative procedure for solving the multiple stopping problem in discrete time and discuss the stability of the algorithm. The algorithm produces monotonically increasing approximations of the Snell envelope, which coincide with the Snell envelope after finitely many steps. Contrary to backward dynamic programming, the algorithm allows to calculate approximative solutions with only a few nestings of conditional expectations and is, therefore, tailor-made for a plain Monte-Carlo implementation.

A. Kolodko, J.G.M. Schoenmakers, Upper bounds for Bermudan style derivatives, Monte Carlo Methods and Applications, 10 (2004), pp. 331-343.
Abstract
Based on a duality approach for Monte Carlo construction of upper bounds for American/Bermudan derivatives (Rogers, Haugh & Kogan), we present a new algorithm for computing dual upper bounds in a more e?cient way. The method is applied to Bermudan swaptions in the context of a LIBOR market model, where the dual upper bound is constructed from the maximum of still alive swaptions. We give a numerical comparison with Andersen's lower bound method.

G.N. Milstein, O. Reiss, J.G.M. Schoenmakers, A new Monte Carlo method for American options, International Journal of Theoretical and Applied Finance, 7 (2004), pp. 591--614, DOI 10.1142/S0219024904002554 .
Abstract
We introduce a new Monte Carlo method for constructing the exercise boundary of an American option in a generalized Black-Scholes framework. Based on a known exercise boundary, it is shown how to price and hedge the American option by Monte Carlo simulation of suitable probabilistic representations in connection with the respective parabolic boundary value problem. The method presented is supported by numerical experiments.

G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Transition density estimation for stochastic differential equations via forward-reverse representations, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 10 (2004), pp. 281--312.
Abstract
The general reverse diffusion equations are derived and applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem we propose density estimation based on forward?reverse representations and show that this method allows essentially better results to be achieved than the usual kernel or projection estimation based on forward representations only.

O. Kurbanmuradov, K. Sabelfeld, J.G.M. Schoenmakers, Lognormal approximations to LIBOR market models, Journal of Computational Finance, 6 (2002), pp. 69--100.
Abstract
We study several lognormal approximations for LIBOR market models, where special attention is paid to their simulation by direct methods and lognormal random fields. In contrast to conventional numerical solution of SDE's this approach simulates the solution directly at a desired point in time and therefore may be more efficient. As such the proposed approximations provide valuable alternatives to the Euler method, in particular for long dated instruments. We carry out a path-wise comparison of the different lognormal approximations with the 'exact' SDE solution obtained by the Euler scheme using sufficiently small time steps. Also we test approximations obtained via numerical solution of the SDE by the Euler method, using larger time steps. It turns out that for typical volatilities observed in practice, improved versions of the lognormal approximation proposed by Brace, Gatarek and Musiela, citeBrace, appear to have excellent path-wise accuracy. We found out that this accuracy can also be achieved by Euler stepping the SDE using larger time steps, however, from a comparative cost analysis it follows that, particularly for long maturity options, the latter method is more time consuming than the lognormal approximation. We conclude with applications to some example LIBOR derivatives.

G.N. Milstein, J.G.M. Schoenmakers, Numerical construction of hedging strategies against multi-asset European claims, Stochastics and Stochastics Reports, 73 (2002), pp. 125--157.
Abstract
For evaluating a hedging strategy we have to know at every moment the solution of the Cauchy problem for a corresponding parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these quantities by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. As illustrations we consider a Markovian multi-asset model with an instantaneously riskless saving bond and also some applications to the LIBOR rate model of Brace, Gatarek, Musiela and Jamshidian.

D. Becherer, J.G.M. Schoenmakers, E3 -- Stochastic simulation methods for optimal stopping and control -- Towards multilevel approaches, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 317--331.

J.G.M. Schoenmakers, Optionsbewertung, in: Besser als Mathe --- Moderne angewandte Mathematik aus dem MATHEON zum Mitmachen, K. Biermann, M. Grötschel, B. Lutz-Westphal, eds., Reihe: Populär, Vieweg+Teubner, Wiesbaden, 2010, pp. 187--192.

J. Kampen, On optimal strategies of multivariate passport options, in: Progress in Industrial Mathematics at ECMI 2006, L.L. Bonilla, M. Moscoso, G. Platero, J.M. Vega, eds., 12 of Mathematics in Industry, Springer, Berlin, Heidelberg, 2008, pp. 666--670.

C. Croitoru, Ch. Fries, W. Jäger, J. Kampen, D.-J. Nonnenmacher, On the dynamics of the forward interest rate curve and the evaluation of interest rate derivatives and their sensitivities, in: Mathematics --- Key Technology for the Future, W. Jäger, H.-J. Krebs, eds., Springer, Heidelberg, 2008, pp. 343--357.

CH. Bender, A. Kolodko, J.G.M. Schoenmakers, Policy iteration method for American options, in: Proceedings of 4th Actuarial and Financial Mathematics Day, M. Vanmaele, A. De Schepper, J. Dhaene, H. Reynaerts, W. Schoutens, P. Van Goethem, eds., Royal Flemish Academy of Belgium for Sciences and Arts, Brussels, 2006, pp. 31--45.

R.J.A. Laeven, J.G.M. Schoenmakers, N.F.F. Schweizer, M. Stadje, Robust multiple stopping -- A path-wise duality approach, Preprint no. 2728, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2728 .
Abstract, PDF (576 kByte)
In this paper we develop a solution method for general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem which satisfy appealing path-wise optimality (almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds which, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine upper and lower bounds. We illustrate the applicability of our general approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies.

CH. Bayer, D. Belomestny, P. Hager, P. Pigato, J.G.M. Schoenmakers, Randomized optimal stopping algorithms and their convergence analysis, Preprint no. 2697, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2697 .
Abstract, PDF (367 kByte)
In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimization algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.

CH. Bayer, M. Redmann, J.G.M. Schoenmakers, Dynamic programming for optimal stopping via pseudo-regression, Preprint no. 2532, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2532 .
Abstract, PDF (337 kByte)
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.

CH. Bayer, Pricing american options by exercise rate optimization, Research Seminar on Insurance Mathematics and Stochastic Finance, Eidgenössische Technische Hochschule Zürich, Switzerland, January 9, 2020.

CH. Bayer, Pricing american options by exercise rate optimization, Mathrisk- INRIA/ LPSM Paris-Diderot Seminar, Inria Paris Research Centre, Mathrisk Research Team, France, February 6, 2020.

CH. Bayer, Pricing american options by exercise rate optimization, Lunch at the Lab, University of Calgary, Department of Mathematics and Statistics, Canada, March 3, 2020.

CH. Bayer, Pricing American options by exercise rate optimization, Workshop on Financial Risks and Their Management, February 19 - 20, 2019, Ryukoku University, Wagenkan, Kyoto, Japan, February 19, 2019.

CH. Bayer, Pricing American options by exercise rate optimization, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Stochastic Optimization and Its Applications (Part III)'', August 5 - 8, 2019, Berlin, August 7, 2019.

J.G.M. Schoenmakers, Tractability of continuous time optimal stopping problems, DynStoch 2019, June 12 - 15, 2019, Delft University of Technology, Institute of Applied Mathematics, Netherlands, June 14, 2019.

J.G.M. Schoenmakers, Tractability of continuous time optimal stopping problems, Séminaire du Groupe de Travail ``Finance Mathématique, Probabilités Numériques et Statistique des Processus'', Université Paris Diderot, LPSM-Equipe Mathématiques Financières et Actuarielles, Probabilités Numériques, France, June 27, 2019.

CH. Bayer, Smoothing the payoff for computation of basket options, Berlin-Paris Young Researchers Workshop Stochastic Analysis with Applications in Biology and Finance, May 2 - 4, 2018, Institut des Systèmes Complexes de Paris Ile-de-France (ISC-PIF), National Center for Scientific Research, Paris, France, May 3, 2018.

CH. Bayer, Smoothing the payoff for computation of basket options, 13th International Conference in Monte Carlo & Quasi-Monte Carlo Methods in Scientific Computing, July 1 - 6, 2018, University of Rennes, Faculty of Economics, France, July 3, 2018.

CH. Bayer, Smoothing the payoff for computation of basket options, Stochastic Methods in Finance and Physics, July 23 - 27, 2018, National Technical University of Athens, Department of Mathematics, Heraklion, Greece.

CH. Bayer, Smoothing the payoff for efficient computation of basket option, Financial Math Seminar, Princeton University, Operations Research & Financial Engineering, USA, October 11, 2017.

J.G.M. Schoenmakers, Multilevel dual evaluation and multilevel policy iteration for optimal stopping/American options, Advances in Financial Mathematics, January 7 - 10, 2014, l'Institut Louis Bachelier, Paris, France, January 9, 2014.

M. Ladkau, Multilevel policy iteration for pricing American options, 26th European Conference on Operational Research, June 30 - July 4, 2013, Università La Sapienza, Rome, Italy, July 2, 2013.

M. Ladkau, Multilevel policy iteration for pricing american options, Workshop on Stochastic Models and Control, March 18 - 22, 2013, Humboldt-Universität zu Berlin, March 21, 2013.

M. Ladkau, Multilevel policy iteration for pricing American options, PreMoLab: Moscow-Berlin Stochastic and Predictive Modeling, May 31 - June 1, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, June 1, 2012.

J.G.M. Schoenmakers, Multilevel dual approach for pricing American options, Mini-workshop CWI-EUR Backward Stochastic Differential Equations (BSDE's), January 17 - 18, 2012, Eindhoven University of Technology, European Institute for Statistics, Probability, Stochastic Operations Research and its Applications (EURANDOM), January 18, 2012.

J.G.M. Schoenmakers, Optimal dual martingales, their analysis and application to new algorithms for Bermudan products, 10th German Probalility and Statistic Days 2012, March 6 - 8, 2012, Johannes Gutenberg Universität Mainz, March 7, 2012.

J.G.M. Schoenmakers , Multilevel dual approach for pricing American options, PreMoLab: Moscow-Berlin Stochastic and Predictive Modeling, May 31 - June 1, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, June 1, 2012.

J.G.M. Schoenmakers , Multilevel primal and dual approaches for pricing American options, 21st International Symposium on Mathematical Programming (ISMP), August 20 - 24, 2012, Technische Universität Berlin, August 24, 2012.

J.G.M. Schoenmakers, New dual methods for single and multiple exercise option, Universität Ulm, Institut für Numerische Mathematik, May 27, 2011.

J.G.M. Schoenmakers, New dual methods for single and multiple exercise options, Workshop ``Quantitative Methods in Financial and Insurance Mathematics'', April 18 - 21, 2011, Lorentz Center, Leiden, Netherlands, April 21, 2011.

J.G.M. Schoenmakers , Multi-level dual approach for pricing American options, Workshop on Rough Paths and Numerical Integration Methods, September 21 - 23, 2011, Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, September 23, 2011.

J.G.M. Schoenmakers , New dual methods for single and multiple exercise options, International Workshop on Numerical Algorithms in Computational Finance, July 20 - 22, 2011, Goethe Universität Frankfurt, Goethe Center for Scientific Computing, July 22, 2011.

V. Krätschmer, Representations for optimal stopping under dynamic monetary utility functionals, Leipziger Stochastik Tage, March 1 - 5, 2010, Universität Leipzig, Fakultät für Mathematik und Informatik, March 3, 2010.

V. Panov, Non-Gaussian component classification with applications to American Option Pricing, Haindorf Seminar 2010 (Klausurtagung des SFB 649), February 11 - 14, 2010, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, February 13, 2010.

R.L. Loeffen, Absolute ruin in the insurance risk model of Ornstein--Uhlenbeck type, 24th European Conference on Operational Research (EURO XXIV LISBON), July 11 - 14, 2010, Universidade de Lisboa, Faculdade de Ciéncias, Portugal, July 14, 2010.

J.G.M. Schoenmakers, Advanced Libor modeling, Postbank Bonn, February 25, 2010.

J.G.M. Schoenmakers, On three innovations in financial modeling, Colloquium, University of Twente, Faculty of Electrical Engineering, Mathematics and Computer Science, Netherlands, August 24, 2010.

J.G.M. Schoenmakers, The real multiple dual, Leipziger Stochastik Tage, March 1 - 5, 2010, Universität Leipzig, Fakultät für Mathematik und Informatik, March 2, 2010.

V. Panov, Pricing Bermudan options via dimension reduction, Klausurtagung des SFB 649, June 4 - 6, 2009, Humboldt-Universität zu Berlin, Motzen, June 5, 2009.

D. Peschka, Dewetting of thin liquid films on viscoelastic substrates, European Coating Symposium, September 7 - 9, 2009, Karlsruhe, September 7, 2009.

D. Belomestny, Regression methods for stochastic control problems and their convergence analysis, Fourth General Conference on Advanced Mathematical Methods in Finance, May 4 - 10, 2009, University of Oslo, Norway, May 9, 2009.

J.G.M. Schoenmakers, Holomorphic transforms with application to affine processes, Workshop ``Computational Finance'', August 10 - 12, 2009, Kyoto University, Faculty of Sciences, Japan, August 10, 2009.

J.G.M. Schoenmakers, Monte Carlo methods for pricing of complex structured callable derivatives, Rhein-Main Arbeitskreis em Mathematics of Computation, Johann Wolfgang Goethe-Universität Frankfurt, Villa mboxGiersch, January 16, 2009.

J.G.M. Schoenmakers, Regression methods for stochastic control problems and their convergence analysis, Workshop ``Computational Finance'', August 10 - 12, 2009, Kyoto University, Faculty of Sciences, Japan, August 10, 2009.

J.G.M. Schoenmakers, Statistical and numerical methods for evaluation for financial derivates and risk, Center Days 2009 (DFG Research Center scshape Matheon), March 30 - April 1, 2009, Technische Universität Berlin, March 31, 2009.

J. Kampen, Higher order WKB expansions of the fundamental solution and pricing of options, Credit Suisse, Zurich, Switzerland, March 28, 2008.

J. Kampen, Monte carlo greeks for financial prods via approximate transition densities, 5th World Congress, Bachelier Finance Society, July 16 - 19, 2008, Royal Geographical Institute, London, UK, July 19, 2008.

D. Belomestny, New series representations for the characteristic functions of affine Feller processes with applications to option pricing, Financial and Actuarial Mathematics, September 22 - 26, 2008, Technical University of Vienna, Austria, September 23, 2008.

A. Kolodko, Monte carlo greeks for financial products via approximative transition densities, 3rd General AMAMEF Conference, May 8 - 10, 2008, Romanian Academy, Institute of Matematics ``Simion Stoilow'', Piteşti, May 9, 2008.

J.G.M. Schoenmakers, Holomorphic transforms and affine processes, Technische Universität Braunschweig, May 20, 2008.

J.G.M. Schoenmakers, Holomorphic transforms with application to affine processes, 2nd Meeting in the winter semester 2008/2009 of the Research Seminar ``Stochastic Analysis and Stochastics of Financial Markets'', Technische Universität Berlin, November 6, 2008.

J.G.M. Schoenmakers, Monte Carlo methods for pricing of complex structured callable derivatives, 2nd International Conference on Numerical Methods for Finance, June 4 - 5, 2008, Institute for Numerical Computation and Analysis, Dublin, Ireland, June 4, 2008.

J.G.M. Schoenmakers, New Monte Carlo methods for pricing high-dimensional callable derivatives, Conference on Numerical Methods in Finance, June 26 - 27, 2008, Università di Udine, Italy, June 27, 2008.

J.G.M. Schoenmakers, Pricing and hedging exotic interest rate derivatives with Monte Carlo simulation, finance master class$^rm TM$ workshop, March 19 - 20, 2008, Concentric Italy, Milan.

J.G.M. Schoenmakers, Regression methods for high-dimensional Bermudan derivatives and stochastic control problems, Conference on Numerical Methods for American and Bermudan Options, October 17 - 18, 2008, Wolfgang Pauli Institute (WPI), Fakultät für Mathematik, Vienna, Austria, October 17, 2008.

J. Kampen, Closed form analytic expansion formulas for characteristic functions of affine jump diffusion processes, Workshop on Numerics in Finance, November 5 - 6, 2007, Commerzbank AG, Frankfurt/Main, November 6, 2007.

A. Kolodko, Iterative procedure for pricing Bermudan options, Workshop on Methods for Pricing Financial Options, March 8, 2007, The University of Melbourne, Department of Mathematics and Statistics, Australia, March 8, 2007.

J.G.M. Schoenmakers, Enhanced policy iteration via scenario selection, International Multidisciplinary Workshop on Stochastic Modeling, June 25 - 29, 2007, Sevilla, Spain, June 27, 2007.

J.G.M. Schoenmakers, Iterative procedures for the Bermudan stopping problem, International Multidisciplinary Workshop on Stochastic Modeling, June 25 - 29, 2007, Sevilla, Spain, June 26, 2007.

J.G.M. Schoenmakers, Policy iteration for American/Bermudan style derivatives, 6th Winter School on Mathematical Finance, January 22 - 24, 2007, CongresHotel De Werelt, Lunteren, Netherlands, January 23, 2007.

J.G.M. Schoenmakers, Robust Libor modelling and calibration, International Multidisciplinary Workshop on Stochastic Modeling, June 25 - 29, 2007, Sevilla, Spain, June 29, 2007.

J.G.M. Schoenmakers, True upper bounds for Bermudan style derivatives, International Multidisciplinary Workshop on Stochastic Modeling, June 25 - 29, 2007, Sevilla, Spain, June 28, 2007.

A. Kolodko, Iterative procedure for pricing callable options, 4th Actuarial and Financial Mathematics Day, February 10, 2006, Brussels, Belgium, February 10, 2006.

J.G.M. Schoenmakers, Interest rate modelling: Practical calibration and implementation techniques, June 15 - 16, 2006, Risk, London, UK.

J.G.M. Schoenmakers, Iterative Methoden zur Bewertung komplex strukturierter Finanzderivate mit vorzeitigen Ausübungsrechten, WIAS-Day, WIAS, Berlin, February 24, 2006.

J.G.M. Schoenmakers, Iterative methods for complex structured callable products, 7th GOR Workshop on Financial Optimization and Optimal Pricing Strategies, May 22 - 23, 2006, BASF AG, Ludwigshafen, May 23, 2006.

J.G.M. Schoenmakers, Iterative procedures for the Bermudan stopping problem, 42nd Dutch Mathematical Congress, March 27 - 28, 2006, Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Netherlands, March 27, 2006.

J.G.M. Schoenmakers, Iterative construction of the optimal Bermudan stopping time, Frankfurt MathFinance Workshop ``Derivatives and risk management in theory and practice'', April 14 - 15, 2005, HfB --- Business School of Finance and Management, Frankfurt am Main, April 14, 2005.

J.G.M. Schoenmakers, Robust Libor modelling and pricing of derivative products, Delft University of Technology, Netherlands, June 9, 2005.

J.G.M. Schoenmakers, Interactive construction of the optimal Bermudan stopping time, Eidgenössische Technische Hochschule Zürich, Institut für Mathematik, Switzerland, May 6, 2004.

J.G.M. Schoenmakers, Interest rate modelling --- Practical calibration and implementation techniques, Incisive Media Events, Hong Kong, China, December 8, 2004.

J.G.M. Schoenmakers, Iterative construction of the optimal Bermudan stopping time, 2nd IASTED International Conference on Financial Engineering and Applications (FEA 2004), November 8 - 10, 2004, Cambridge, USA, November 8, 2004.

J.G.M. Schoenmakers, New Monte Carlo Methods for American and Bermudan style derivatives, Delft University of Technology, Faculty of Information Technology and Systems Numerical Analysis Group, Netherlands, January 30, 2004.

J.G.M. Schoenmakers, Monte Carlo methods for pricing and hedging American options, IV IMACS Seminar on Monte Carlo Methods; (Workshop Financial Models and Simulation), September 15 - 19, 2003, Berlin, September 19, 2003.

J.G.M. Schoenmakers, Monte Carlo simulation of Bermudan derivatives by dual upper bounds, Miniworkshop ``Risikomaße und ihre Anwendungen'', Humboldt-Universität zu Berlin, December 1, 2003.

J.G.M. Schoenmakers, Monte Carlo simulation of Bermudan derivatives by dual upper bounds, Graduiertenkolleg Angewandte Algorithmische Mathematik, Workshop on the Interface of Numerical Analysis, Optimisation and Applications, November 13 - 14, 2003, Technische Universität München, November 14, 2003.

J.G.M. Schoenmakers, Robust calibration of LIBOR market models, Petit Dejeuner de la Finance, November 4 - 5, 2003, Paris, November 5, 2003.

J.G.M. Schoenmakers, Transition density estimation for stochastic differential equations via forward reverse representations, IV IMACS Seminar on Monte Carlo Methods (MCM 2003), September 15 - 19, 2003, Berlin, September 16, 2003.

CH. Bayer, R. Tempone, S. Wolfers, Pricing American options by exercise rate optimization, Preprint no. arXiv:1809.07300, Cornell University Library, arXiv.org, 2018.
Abstract
We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal emphexercise regions, which consist of points in time and space at which the option is exercised. In contrast, our method determines emphexercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. Starting in a neutral strategy with constant exercise rate then allows us to globally optimize this function in a gradual manner. Numerical experiments on vanilla put options in the multivariate Black--Scholes model and preliminary theoretical analysis underline the efficiency of our method both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of exercise rates. Finally, the flexibility of our method is demonstrated by numerical experiments on max call options in the Black--Scholes model and vanilla put options in Heston model and the non-Markovian rough Bergomi model.

J.G.M. Schoenmakers, J. Huang, Optimal dual martingales, their analysis and application to new algorithms for Bermudan products, Preprint no. 1825944, Social Science Research Network (SSRN) Working Paper Series, 2011.