LIBOR Market Model

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.

Bermudan Swaptions in the LIBOR Market Model

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

A number of models have been successfully developed wherein the theoretical prices for caps, floors, and swaptions produced by the model are of the Black-76 form. In these models, discrete market rates are modelled like LIBOR rates in the LIBOR market models or forward swap rates in the swap market models; and under a suitable choice of numeraires, market rates can be modelled log normally. LIBOR caps and the market practice for pricing and quoting these instruments are discussed. It is shown that given a swap market model, the LIBOR rates will not be lognormal; thus, LIBOR market models and swap models are generally incompatible. Practice exercises are included.

This paper introduces a short rate model in continuous time that adds one or more memory (delay) components to the Merton model [R.C. Merton, A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm, Working Paper 497, Sloan School of Management, MIT, Cambridge, 1970; R.C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci. 4 (1973), pp. 141–183.] or the Vasiček model [Vasiček, An equilibrium characterization of the term structure, J. Financ. Econ. 5 (1977), pp. 177–188.] for the short rate. The distribution of the short rate in this model is normal, with the mean depending on past values of the short rate, and a limiting distribution exists for certain values of the parameters. The zero coupon bond price is an affine function of the short rate, whose coefficients satisfy a system of delay differential equations. This system can be solved analytically, obtaining a closed formula. An analytical expression for the instantaneous forward rate is given: it satisfies the risk neutral dynamics of the Heath-Jarrow-Morton model. Formulae for both forward looking and backward looking caplets on overnight risk free rates are presented. Finally, the proposed model is calibrated against forward looking caplets on SONIA rates and the United States yield curve.

ABSTRACTThis paper evaluates the forecasting and finite sample performance of short‐term interest rate models in a number of countries. Specifically, we run a series of in‐sample and out‐of‐sample tests for both the conditional mean and volatility of one‐factor short rate models, and compare the results to the random walk model. Overall, we find that the out‐of‐sample forecasting performance of one‐factor short rate models is poor, stemming from the inability of the models to accommodate jumps and discontinuities in the time series data. In addition, we perform a series of Monte Carlo analyses similar to Chapman and Pearson to document the finite sample performance of the short rate models when β3 is not restricted to be equal to one. Our results indicate the potential dangers of over‐parameterization and highlight the limitations of short‐term interest rate models.

In this paper an efficient calibration method for the multi-factor LIBOR Market Model (LMM) is proposed and is applied for the Japanese interest rate market. At first the joint calibration method in the cap and swaption market is demonstrated using a new parameterization for the correlation matrix in the LMM. Then we implement the proposed methodology for calibrating the Japanese cap and swaption markets, where the computational procedure is shown to be tractable and provides a practical estimation for the implied correlation matrix in the LMM. The empirical analysis also illustrates that Black's swaption volatilities through our calibration fit the market data almost exactly and that the estimated implied correlation matrix is smooth and stable.

Implementation of Short-Rate Models - A Case Study of the Black-Derman-Toy Model of Interest Rate

LIBOR Market Model with Stochastic Volatility

In the preceding chapters we considered different term structure models, for instance, the Gaussian short rate models or models for the instantaneous forward rate.

The Libor market model has established itself as the benchmark model for interest rate derivatives. If the observed correlation and volatility surfaces cannot be reproduced by a model, we cannot hope to get meaningful prices; therefore the crucial task, before it comes to pricing and hedging, is to calibrate the model to given market data. An overview of the Libor market model is given and it is shown how to obtain a robust calibration. We show how to extend the model to include the market smile by making use of stochastic volatility. For these stochastic volatility Libor market models a new time homogeneous skew parametrization is introduced. Furthermore a new approximative terminal correlation formula based on the same parameter averaging technique used for pricing in a stochastic volatility Libor market model is presented. We look at a very robust initial calibration procedure; here we introduce some new approximations making it possible to use global optimizers. The optimal choice of local and global optimizers for each step of the calibration will be dealt with and an analysis of the robustness of the calibrated parameters is performed. Two extensions to the cross currency Libor market model introduced by Schlögl are developed. These make use of displacements and stochastic volatility to fit the observed skew and smile respectively. The calculation of the Greeks for exotic interest rate derivatives is crucial for hedging and therefore for trading these products. For a Libor market model this task can only be fulfilled by means of nontrivial Monte Carlo based methods. A new method for the calculation of the weights for the proxy simulation scheme method is developed making use of the conditional independence of the underlying from the stochastic volatility process, which can be used for any model for which this independence holds. The developed estimators are fast enough to be used in everyday practice.

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.

PurposeRecently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016).Design/methodology/approachBond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise.FindingsThe authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases.Originality/valueBond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

On the Information in the Interest Rate Term Structure and Option Prices

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.KeywordsSABR volatility modelsSABR/LIBOR market modelsParallel simulated annealingGPUs