Browsing by Author "Belomestny, Denis"
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- ItemAbelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Belomestny, Denis; Panov, VladimirIn this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models $(X, V)$, where both the state process $X$ and the volatility process $V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy processes driving the jumps in $X$ and $V$ . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process $X$. We derive the convergence rates for the corresponding estimator and prove that these rates can not be improved in general.
- ItemCentral limit theorems for law-invariant coherent risk measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Belomestny, Denis; Krätschmer, VolkerIn this paper we study the asymptotic properties of the canonical plug-in estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent identically distributed data and then extend it to the case of weakly dependent ones. Finally, a number of illustrating examples is presented.
- ItemFrom optimal martingales to randomized dual optimal stopping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Belomestny, Denis; Schoenmakers, John G. M.In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that does`nt require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.
- ItemGeneralized Post-Widder inversion formula with application to statistics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Belomestny, Denis; Mai, Hilmar; Schoenmakers, JohnIn this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models.We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.
- ItemHolomorphic transforms with application to affine processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Belomestny, Denis; Kampen, Joerg; Schoenmakers, John G.M.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 multidimensional 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.
- 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.
- ItemMultilevel dual approach for pricing American style derivatives(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Belomestny, Denis; Schoenmakers, John G.M.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.
- 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
- ItemOptimal stopping via deeply boosted backward regression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Belomestny, Denis; Schoenmakers, John G.M.; Spokoiny, Vladimir; Tavyrikov, YuriIn 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.
- ItemOptimal stopping via pathwise dual empirical maximisation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Belomestny, Denis; Hildebrand, Roland; Schoenmakers, John G.M.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.
- ItemPricing Bermudan options by nonparametric regression : optimal rates of convergence for lower estimates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Belomestny, DenisThe problem of pricing Bermudan options using simulations and nonparametric regression is considered. We derive optimal non-asymptotic bounds for the low biased estimate based on a suboptimal stopping rule constructed from some estimates of the optimal continuation values. These estimates may be of different nature, they may be local or global, with the only requirement being that the deviations of these estimates from the true continuation values can be uniformly bounded in probability. As an illustration, we discuss a class of local polynomial estimates which, under some regularity conditions, yield continuation values estimates possessing the required property.
- ItemPricing CMS spreads in the Libor market model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Belomestny, Denis; Kolodko, Anastasia; Schoenmakers, JohnWe present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.
- ItemProjected particle methods for solving McKean-Vlasov equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Belomestny, Denis; Schoenmakers, John G.M.We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a signficant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift.
- ItemRandomized optimal stopping algorithms and their convergence analysis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Belomestny, Denis; Hager, Paul; Pigato, Paolo; Schoenmakers, John G. M.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.
- ItemRegression methods for stochastic control problems and their convergence analysis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Belomestny, Denis; Kolodko, Anastasia; Schoenmakers, John G.M.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.
- ItemRegression methods in pricing American and Bermudan options using consumption processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestny, Denis; Milstein, Grigor N.; Spokoiny, VladimirHere 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.
- 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.
- ItemReinforced optimal control(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Belomestny, Denis; Hager, Paul; Pigato, Paolo; Schoenmakers, John G. M.; Spokoiny, VladimirLeast squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay, Commun. Math. Sci., 18(1):109?121, 2020] proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method?s efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.
- ItemRKHS regularization of singular local stochastic volatility McKean--Vlasov models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Bayer, Christian; Belomestny, Denis; Butkovsky, Oleg; Schoenmakers, John G. M.Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equation, which represents a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is well-posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.
- ItemSemi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Belomestny, Denis; Kaledin, Maxim; Schoenmakers, John G.M.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.