Browsing by Author "Schoenmakers, John G.M."
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- ItemAffine LIBOR models with multiple curves: Theory, examples and calibration(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Grbac, Zorana; Papapantoleon, Antonis; Schoenmakers, John G.M.; Skovmand, DavidWe introduce a multiple curve LIBOR framework that combines tractable dynamics and semi-analytic pricing formulas with positive interest rates and basis spreads. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices.
- ItemDual representations for general multiple stopping problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Bender, Christian; Schoenmakers, John G.M.; Zhang, JianingIn 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.
- 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.
- ItemEfficient and accurate log-Levy approximations to Levy driven LIBOR models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Papapantoleon, Antonis; Schoenmakers, John G.M.; Skovmand, DavidThe LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a L´evy-driven LIBOR model and aim at developing accurate and efficient log-L´evy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-L´evy approximation of annuities, which offers good approximations for high volatility regimes.
- ItemForward and reverse representations for Markov chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Milstein, Grigori N.; Schoenmakers, John G.M.; Spokoiny, VladimirIn 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.
- ItemForward-reverse EM algorithm for Markov chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Mai, Hilmar; Schoenmakers, John G.M.We develop an EM algorithm for estimating parameters that determine the dynamics of a discrete time Markov chain evolving through a certain measurable state space. As a key tool for the construction of the EM method we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. These representations may be considered as an extension of the earlier work [1] on conditional diffusions. We present several experiments and consider the convergence of the new EM algorithm.
- ItemFrom rough path estimates to multilevel Monte Carlo(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bayer, Christian; Friz, Peter K.; Riedel, Sebastian; Schoenmakers, John G.M.Discrete approximations to solutions of stochastic differential equations are well-known to converge with strong rate 1=2. Such rates have played a key-role in Giles multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H > 1=4 as special case).
- 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.
- ItemA jump-diffusion Libor model and tits robust calibration(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestrny, Denis; Schoenmakers, John G.M.In this paper we propose a jump-diffusion Libor model with jumps in a high-dimensional space and test a stable non-parametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the ill-posedness of the underlying inverse problem.
- ItemLibor model with expiry-wise stochastic volatility and displacement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Ladkau, Marcel; Schoenmakers, John G.M.; Zhang, JianingWe develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed
- 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.
- ItemMonte Carlo Greeks for financial products via approximative Greenian Kernels(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Kampen, Joerg; Kolodko, Anastasia; Schoenmakers, John G.M.In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.
- 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.
- ItemOptimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Schoenmakers, John G.M.; Huang, JunboLiteraturverz. 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).
- 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.
- ItemPath-wise approximation of the Cox-Ingersoll-Ross process(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Milstein, Grigori N.; Schoenmakers, John G.M.The Doss-Sussmann (DS) approach is used for simulating the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows for expressing trajectories of the CIR process by solutions of some ordinary differential equation (ODE) that depend on realizations of the Wiener process involved. Via simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving an ODE, we approximately construct the trajectories of the CIR process. From a conceptual point of view the proposed method may be considered as an exact simulation approach.
- ItemPrimal-dual linear Monte Carlo algorithm for multiple stopping : an application to flexible caps(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Balder, Sven; Mahayni, Antje; Schoenmakers, John G.M.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.
- 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.
- ItemThe real multiple dual(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Schoenmakers, John G.M.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.