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- 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.
- 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.
- ItemA stochastic volatility Libor model and its robust calibration(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Belomestny, Denis; Mathew, Stanley; Schoenmakers, John G.M.In this paper we propose a Libor model with a high-dimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement.
- 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.
- ItemUniform approximation of the CIR process via exact simulation at random times(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Milstein, Grigori N.; Schoenmakers, John G.M.In this paper we uniformly approximate the trajectories of the Cox-Ingersoll-Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid.
- 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).
- 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.
- 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.
- 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.
- 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