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by RAOUL PIETERSZ AND ANTOON PELSSER This article presents a new approach to calculating swap vega per bucket in a LIBOR model. It shows that for some forms of volatility an approach based on recalibration may make estimated swap vega very uncertain, as the instantaneous volatility structure may be distorted by recalibration. This does not happen in the case of constant swap rate volatility. Discuss this paper
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Peter Jackel Riccardo Rebonato Abstract We present and approximation for the volatility of European swaptions in a forward rate based Brace-Gatarek-Musiela/Jamshidian framework [BGM97, Jam97] which enables us to calculate prices for swaptions without the need for Monte Carlo simulations. Also, we explain the mechanism behind the remarkable accuracy of these approximate prices. For cases where the yield curve varies noticeably as a function of maturity, a second, and even more accurate formula is derived.
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by A.R. Radhakrishnan Convergence rate and accuracy of the non-recombining HJM forward rate tree are tested by constructing a tree for the forward rate process equivalent to the Hull-White short rate process. Put option values on a ten-year discount bond from the forward rate tree are compared to the accurate values obtained from a recombining short rate lattice. European option values from the HJM tree converge to the true value in twelve steps for all option maturities up to twenty-five years. American option values are within a cent or two of the accurate values for one and five-year options, but do not converge to the accurate value in twenty-four steps, the maximum attempted, for higher maturities. At-the-money options are underpriced by one percent for ten-year maturity and by more than three percent for twenty-five year maturity. Out-of-the-money options are underpriced by up to nine percent. Results are independent of the shape of the initial term structure. Using an HJM tree with equal stepsizes leads to more accurate values for European options, but a tree with linearly increasing stepsizes performs better in the case of American options. It is found optimal to have the same number of forward rates maturing per year beyond option maturity as the number of steps per year through option maturity. Discuss this paper
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by Cedreece Tamagushiku The purpose of this paper is to investigate the performance of three different models in the pricing of call options on ninety-day bank bill futures traded on the Sydney Futures Exchange between 1993 and 2000. The three models analysed are embedded into the Heath, Jarrow, and Morton framework namely; the one, two, and three factor models. Principal Components Analysis was applied in order to provide the forward rate volatility functions necessary to implement several popular multi-factor versions of the Heath, Jarrow, and Morton model. Results showed that the three-factor model consistently outperforms the one and two-factor models. Also the pricing errors are positively correlated with the time to maturity of the option and that no real relationship existed between the errors of one and two-factor models and the date and the moneyness of the options. Although three-factor models exhibited lower errors as time progressed.
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Thesis by Simona Svoboda-Greenwood The LMM is an effective framework for the pricing of interest rate derivatives, not least because it models observable market quantities. In its lognormal form, calibration to market implied volatilities is intuitive and fast. The amendments required to incorporate a monotonically decreasing implied volatility skew are fairly straightforward and do not significantly reduce the ease and speed of calibration. However, the incorporation of a full implied volatility smile is significantly more challenging,from both a mathematical and computational perspective. There exist three main techniques for incorporating a volatility smile/skew in any modelling framework: allowing a local volatility function, stochastic volatility and jump dynamics. In this thesis various ways to incorporate smile/skew are studied, loosely based on the above three approaches. Both the constant-elasticity-of-variance and displaced-diffusion processes give rise to an implied volatility skew. In fact it has been experimentally shown that, for a certain parameterisation, the two processes produce closely matching prices for European call options over a variety of strikes and maturities. Here, this similarity in prices is analytically quantified, not only via an asymptotic expansion of the call prices, but also via expansion of the conditional probability density functions and a comparison of the raw and central moments of the two distributions.
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by Robert Jarrow, Haitao Li, and Feng Zhao Using more than two years of daily interest rate cap price data, this paper provides a systematic documentation of a volatility smile in cap prices. We find that Black (1976) implied volatilities exhibit an asymmetric smile (sometimes called a sneer) with a stronger skew for in-the-money caps than out-of-the-money caps. The volatility smile is time varying and is more pronounced after September 11, 2001. We also study the ability of generalized LIBOR market models to capture this smile. We show that the best performing model has constant elasticity of variance combined with uncorrelated stochastic volatility or upward jumps. However, this model still has a bias for short- and medium-term caps. In addition, it appears that large negative jumps are needed after September 11, 2001. We conclude that the existing class of LIBOR market models can not fully capture the volatility smile
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presentaiotn by Peter Jackel An approximation for the volatility of European swaptions is presented that makes it possible to calculate prices of swaptions without the need for numerical computations such as Monte Carlo simulations or lattice-based integration methods. The approximation can be used whenever the covariance matrix of an initial set of fixed income observables is known and thus applies to most interest rate models such as the extended Vasicek (also known as multi-factor Hull-White) model or the Brace- Gatarek-Musiela/Jamshidian framework. Also, the mechanism behind the remarkable accuracy of the approximation is explained.
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by Raphael Yan In this study, the implementation of the BGM model is investigated. Theoretical background and numerical techniques are presented. Derivatives on Libor are priced in this model, and numerical results are compared to existing literature
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Riccardo Rebonato It is shown in this paper that it is not only possible, but indeed expedient and advisable, to perform a simultaneous calibration of a log-normal BGM interest-rate model to the percentage volatilities of the individual rates and to the correlation surface. One of the contributions of the paper it to show that the task can be accomplished in two separate and independent steps: the first part of the calibration (i.e. to cap volatilities) can always be accomplished exactly thanks to straightforward geometrical relationships; the fitting to the correlation surface, thanks to a simple theorem, can then be carried out in a numerically efficient way so that the calibration to the volatilities is not spoiled by the second part of the procedure. The ability to carry out the two tasks separately greatly simplifies the overall task. Actual calculations are shown for a 3- and 4-factor implementation of the approach, and the quality of the overall agreement between the target and model correlation surfaces is commented upon. Finally, the dangers of overparametrization, i.e. of forcing (near) exact fitting to certain portions of the correlation matrix, are analysed by looking at the cases of a trigger swap, a Bermudan swaption and a oneway floater (resettable cap).
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John Hull and Alan White This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. The approximation is found to be very accurate for the range of market parameters normally encountered. It enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be easily calibrated to broker quotes on caps and European swap options so that a wide range of non-standard interest rate derivatives can be valued.
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Ramaprasad Bhar, Carl Chiarella This paper considers the Heath-Jarrow-Morton (HJM) model of the term structure of interest rates for a fairly general specification of forward rate volatility, including stochastic variables. Estimation of this volatility function is at the heart of the identification of the HJM model. Reduction of the model to state space form is discussed and use of the Kalman filter as an estimation technique is proposed. Since typical data sets are small, a bootstrap procedure is used to determine the statistical significance of the estimates. A Monte-Carlo experiment is used to compare the bootstrap and “true” smallsample distributions of the estimates of the parameters of the volatility function.
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Timo Salminen This thesis examines a forward rate market model which is used for simulating the development of multiple successive LIBOR forward rates. The main goal is to find the best available method for calibrating the model to current market expectations. Also the model plausibility for pricing swaption based products is studied. In the thesis different calibration methods Discuss this paper
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Jianwei Zhu In this paper we extend standard Libor Market Model (LMM) with nested stochastic volatilities. The stochastic volatility of each Libor follows a meanreverting process as in Schoebel and Zhu (1999) or in Heston (1993) under the individual forward measure of each Libor. Other than the existing stochastic volatility models, every volatility in the extended LMM is correlated with its Libor individually, and the parameters of stochastic volatility are also different over all Libors, however, are nested by some deterministic functions. With a nesting function, the same type of parameter such as mean level in all volatility processes share a certain term structure. In this model set-up, we can still derive the stochastic processes for Libors and volatilities under an arbitrary forward measure. In line with the stochastic volatility models for equity options, we obtain a closed-form solution via Fourier transform for caplets and floorlets. Finally, we use factor representation to express Libors and swap rates by some independent factors, namely principle components. The approximated analytical pricing formula for swaption can then be derived by using the characteristic functions that are just a product of the characteristic function of each factor. The numerical implementation of the nested stochastic volatility model is efficient and identical to the existing stochastic volatility models. Keywords: Libor Market Model, Stochastic Volatility, Characteristic Function, Pricinple Component, Caps, Swaptions.
With a rich spectrum of maturities and tenors to contend with, the toughest aspect of pricing interest rate options is calibrating models of forward rates to market data. Here, Damiano Brigo and Fabio Mercurio present a scheme for simultaneously calibrating swaption volatilities to covariance parameters in the forward Libor model, reducing the need for Monte Carlo simulation Discuss this paper
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Damiano Brigoƒ Cristina Capitani Fabio Mercurio Abstract In this paper we consider several parametric assumptions for the instantaneous covariance structure of the Libor market model. We examine the impact of each different parameteriza- tion on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance pa- rameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating.
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Linus Kajsajuntti This thesis deals with pricing exotic derivatives with the LIBOR market model. In addition to a perfect recovery of the cap market an accurate approximation formula for e ective calibration to swaptions is implemented. Much e ort is put on assuring a stable and accurate evolution of the forward rate structure and it is shown how to design an evolution scheme that suits a given derivative. Pricing schemes with fast convergence is developed by the use of quasi-Monte Carlo integration based on a highdimensional Sobol low-discrepancy sequence. It is shown that a clever implementation of the quasi-Monte Carlo integration implies at least a factor 10 faster convergence and that this, in contrast with theoretical results, continues to hold in very high dimensions. Discuss this paper
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LIXIN WU DEPARTMENT OF MATHEMATICS Abstract. We claim to have developed the optimal methodology for non-parametric calibration of market model to the prices of at-the-money (ATM) caps/floors and swaptions, and to the historic correlations of the LIBOR rates. We take the approach of divide-andconquer: first fit the model to historic correlations, then to the implied Black volatilities of the input options. Regularization is adopted and the calibration is cast into minimizationmaximization problems by the method of Lagrange multiplier. By utilizing the quadratic functional form of both objective function and constraints, we solve the inner maximization problems with a single matrix eigenvalue decomposition, which renders the efficiency of our method. The outer minimization problems, meanwhile, are nicely subdued by gradient-based descending methods due to the convexity of the objective functions. The well-posedness of the Lagrange multiplier problems and the convergence of the descending methods are rigorously justified. Numerical results show that we have achieved very quality calibration. We have also developed a technique to calculate the hedging ratios of a derivative security with respect to the benchmark derivative instruments, using the auxiliary results of the calibration.
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R. Pietersz This thesis presents the theory of the LMM as well as practical issues arising with a computer implementation. Also, a novel extension is made to incorporate the market observed so-called Discuss this paper
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