Application "Calibration of stock and interest rate models"

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

A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Efficient and accurate log-Lévy approximations to Lévy driven LIBOR models, J. Comput. Finance, 15 (2012) pp. 3--44.
Abstract

The 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 Lévy-driven LIBOR model and aim at developing accurate and efficient log-Lévy 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 and swaptions show that the approximations perform very well. In addition, we also consider the log-Lévy approximation of annuities, which offers good approximations for high volatility regimes.

P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm, On refined volatility smile expansion in the Heston model, Quant. Finance, 11 (2011) pp. 1151--1164.

D. Belomestny, J.G.M. Schoenmakers, A jump-diffusion Libor model and its robust calibration, Quant. Finance, 11 (2011) pp. 529--546.

D. Belomestny, A. Kolodko, J.G.M. Schoenmakers, Pricing CMS spreads in the Libor market model, Int. J. Theor. Appl. Finance, 13 (2010) pp. 45--62.
Abstract

We 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.

D. Belomestny, Spectral estimation of the fractional order of a Lévy process, Ann. Statist., 38 (2010) pp. 317--351.

P. Friz, S. Benaim, Regular variation and smile asymptotics, Math. Finance, 19 (2009) pp. 1--12.

D. Belomestny, S. Mathew, J.G.M. Schoenmakers, Multiple stochastic volatility extension of the Libor market model and its implementation, Monte Carlo Methods Appl., 15 (2009) pp. 285-310.
Abstract

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.

D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quant. 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.

R. Krämer, P. Mathé, Modulus of continuity of Nemytskiĭ operators with application to a problem of option pricing, J. Inverse Ill-Posed Probl., 16 (2008) pp. 435--461.

O. Reiss, J.G.M. Schoenmakers, M. Schweizer, From structural assumptions to a link between assets and interest rates, J. Econom. Dynam. Control, 31 (2007) pp. 593--612.
Abstract

We derive a link between assets and interest rates in a standard multi-asset diffusion economy from two structural assumptions ? one on the volatility and one on the short rate function. Our main result is economically intuitive and testable from data since it only involves empirically observable quantities. A preliminary study illustrates how this could be done.

J.G.M. Schoenmakers, B. Coffey, Systematic generation of parametric correlation structures for the LIBOR market model, Int. J. Theor. Appl. Finance, 6 (2003) pp. 507-519.
Abstract

We present a conceptual approach of deriving parsimonious correlation structures suitable for implementation in the LIBOR market model. By imposing additional constraints on a known ratio correlation structure, motivated by economically sensible assumptions concerning forward LIBOR correlations, we yield a semi-parametric framework of non-degenerate correlation structures with realistic properties. Within this framework we derive systematically low parametric structures with, in principal, any desired number of parameters. As illustrated, such structures may be used for smoothing a matrix of historically estimated LIBOR return correlations. In combination with a suitably parametrized deterministic LIBOR volatility norm we so obtain a parsimonious multi-factor market model which allows for joint calibration to caps and swaptions. See Schoenmakers [2002] for a stable full implied calibration procedure based on the correlation structures developed in this paper.

P. Friz, M. Keller-Ressel, Moment explosions in financial models, in: Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley, Chichester, 2010, pp. 1247--1253.

P. Friz, Implied volatility: Large strike asymptotics, in: Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley, Chichester, 2010, pp. 909--913.

H. Mai, Efficient maximum likelihood estimation for Lévy-driven Ornstein--Uhlenbeck processes, Preprint no. 1717, WIAS, Berlin, 2012.
Abstract, Postscript (717 kByte), PDF (326 kByte)

We consider the problem of efficient estimation of the drift parameter of an Ornstein-Uhlenbeck type process driven by a Lévy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions we prove asymptotic normality and efficiency in the Hájek-Le Cam sense for the resulting drift estimator. To obtain these results we prove an estimate for the Markov generator of a pure jump Lévy process. Finally, we investigate the finite sample behavior of the method and compare our approach to least squares estimation.

M. Ladkau, J.G.M. Schoenmakers, J. Zhang, Libor model with expiry-wise stochastic volatility and displacement, Preprint no. 1702, WIAS, Berlin, 2012.
Abstract, PDF (1247 kByte)

We 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.

S. Balder, A. Mahayni, J.G.M. Schoenmakers, Primal-dual linear Monte Carlo algorithm for multiple stopping --- An application to flexible caps, Preprint no. 1666, WIAS, Berlin, 2011.
Abstract, Postscript (3857 kByte), PDF (517 kByte)

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.

CH. Bayer, Asymptotics can beat Monte-Carlo, Global Derivatives 2013, April 16 - 18, 2013, The International Centre for Business Information (ICBI), Amsterdam, The Netherlands, April 18, 2013.

M. Ladkau, A new multi-factor stochastic volatility model with displacement, International Workshop on Numerical Algorithms in Computational Finance, July 20 - 22, 2011, Goethe Universität Frankfurt, Goethe Center for Scientific Computing (G-CSC), July 21, 2011.

P. Friz, On refined density and smile expansion in the Heston model, Workshop ``Stochastic Analysis in Finance and Insurance'', January 23 - 29, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 29, 2012.

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

J.G.M. Schoenmakers, Holomorphic transforms with application to affine processes, 5th General Conference in Advanced Mathematical Methods in Finance, May 4 - 8, 2010, University of Ljubljana, Faculty of Mathematics and Physics, Slovenia, May 6, 2010.

P. Friz, From numerical aspects of stochastic financial models to the foundations of stochastic differential equations (and back), Annual Meeting of the Deutsche Mathematiker-Vereinigung and 17th Congress of the Österreichische Mathematische Gesellschaft, Section ``Financial and Actuarial Mathematics'', September 20 - 25, 2009, Technische Universität Graz, Austria, September 25, 2009.

D. Belomestny, Estimation of the jump activity of a Lévy process from low frequency data, Haindorf Seminar 2009, February 12 - 15, 2009, Humboldt-Universität zu Berlin, CASE -- Center for Applied Statistics and Economics, Hejnice, Czech Republic, February 12, 2009.

D. Belomestny, Spectral estimation of the fractional order of a Lévy process, Workshop ``Statistical Inference for Lévy Processes with Applications to Finance'', July 15 - 17, 2009, EURANDOM, Eindhoven, The Netherlands, July 16, 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.

P. Mathé, On non-stability of some inverse problem in option pricing, Workshop on Inverse and Partial Information Problems: Methodology and Applications, October 27 - 31, 2008, Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, October 30, 2008.

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, A jump-diffusion Libor model and its robust calibration, 4th World Congress of the Bachelier Finance Society, August 17 - 20, 2006, National Center of Sciences, Hitotsubashi University, ICS, Tokyo, Japan, August 20, 2006.

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

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

J.G.M. Schoenmakers, Accuracy and stability of LIBOR model calibration via parametric correlation structures and approximative swaption pricing, Risk Conference 2002, April 23 - 24, 2002, Paris, France, April 23, 2002.

J.G.M. Schoenmakers, Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion, Johann Wolfgang Goethe-Universität, MathFinance Institute, Frankfurt am Main, November 7, 2002.

J.G.M. Schoenmakers, Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion, Quantitative Finance 2002, Risk Waters Group, London, UK, November 26, 2002.

J.G.M. Schoenmakers, Endogenous interest rates in asset markets, 2nd World Congress of the Bachelier Finance Society, June 12 - 15, 2002, Crete, Greece, June 14, 2002.

J.G.M. Schoenmakers, Kalibrierung im LIBOR Modell, Reuters AG, Düsseldorf, March 11, 2002.

J.G.M. Schoenmakers, Correlation structure in LIBOR market models, calibration to caps and swaptions, Technical University of Delft, The Netherlands, May 8, 2001.

J.G.M. Schoenmakers, Term structure dynamics endogenously induced by multi-asset markets, Conference Risk 2001 Europe, April 10 - 11, 2001, Paris, France, April 10, 2001.

J.G.M. Schoenmakers, HJM term structure dynamics from a multi asset market; finite factor models, Hamburger Stochastik-Tage 2000, March 21 - 24, 2000, Universität Hamburg, March 21, 2000.

J.G.M. Schoenmakers, HJM term structure dynamics from a multi asset market; finite factor models, WIAS-Kolloquium, Berlin, May 15, 2000.

J.G.M. Schoenmakers, Stable calibration of multi-factor LIBOR market models via a semi-parametric correlation structure, "`ICBI 2000 Conference"', December 6 - 7, 2000, Genf, Schweiz, December 7, 2000.

J.G.M. Schoenmakers, Stable implied calibration of multi-factor LIBOR models by semi-parametric correlation structure, Risk Conference Math Week 2000, November 13 - 17, 2000, New York, USA, November 15, 2000.

P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm, On refined volatility smile expansion in the Heston model, Preprint no. arXiv:1001.3003, Cornell University Library, arXiv.org, 2010.