31 results
Search Results
Now showing 1 - 10 of 31
- 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
- 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.
- 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.
- 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.
- 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).
- ItemTrue upper bounds for Bermudan products via non-nested Monte Carlo(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestny, Denis; Bender, Christian; Schoenmakers, John G.M.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.
- 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.
- 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.
- 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.
- 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.