LIBOR Market Model Research Articles

An Exact and Efficient Method for Computing Cross-Gammas of Bermudan Swaptions and Cancellable Swaps Under the Libor Market Model

We introduce a new simulation algorithm for computing the Hessians of Bermudan swaptions and cancelable swaps. The resulting pathwise estimates are unbiased and accurate. Given the exercise strategy, the pathwise angularities are removed by a sequence of measure changes. The change of measure at each exercise time is chosen to be optimal in terms of minimizing the variance of the likelihood ratio terms. Numerical results for the Hessian of cancelable swaps are presented to demonstrate the speed and efficacy of the method.

The Markov-switching jump diffusion LIBOR market model

In this paper, we introduce an extension to the LIBOR Market Model (LMM) that is suitable to incorporate both sudden market shocks as well as changes in the overall economic climate into the interest rate dynamics. This is achieved by substituting the simple diffusion process of the original LMM by a regime-switching jump diffusion. We demonstrate that the new Markov-switching jump diffusion (MSJD) LMM can be embedded into a generalized regime-switching Heath–Jarrow–Morton model and prove that the considered market is arbitrage-free. We derive pricing formulas for caps, floors and swaptions using Fourier pricing techniques and show how the model can be calibrated to real market data.

The Markov-Switching Jump Diffusion Libor Market Model

In this paper, we introduce an extension to the LIBOR Market model that is suitable to incorporate both sudden market shocks as well as changes in the overall economic climate into the interest rate dynamics. This is achieved by substituting the simple diffusion process of the original LIBOR Market model by a regime-switching jump diffusion. We demonstrate that the new Markov-switching jump diffusion (MSJD) LIBOR Market model can be embedded into a generalized regime-switching Heath-Jarrow-Morton (HJM) model and prove that the considered market is arbitrage-free. We derive pricing formulas for caps, floors, and interest rate swaps using Fourier pricing techniques and show how the model can be calibrated to real data.

Chasing market dislocation: The LIBOR shifted model

We introduce a flexible multivariate extension to the LIBOR market model that is analytically tractable. The flexibility of the model is a solution to the interest rate market dislocation that happened on the credit crunch aftermath in August 2007. The extension is based on a parameter uncertainty modeled through a random variable whose value is drawn at an infinitesimal time after zero. The proposed shift captures the skew commonly seen in the cap market, whereas the uncertain volatility component of the model allows us to obtain more general implied volatility structures. We show how this model can be calibrated to cap prices. We also propose an analytical approximated formula to price swaptions from the cap calibrated model. Finally, we build a bridge between the caps and swaptions markets by calibrating the correlation structure to swaption prices, and analysing some implications of the calibrated model parameters.

A drift‐free simulation method for pricing commodity derivatives

Having in view the pricing of commodity derivatives in Libor Market Model (LMM) setting, we first analyze the set of basic rates we need to formulate the model by using the spanning tree concept taken from graph theory. Next, we present an efficient procedure for Monte Carlo simulation of the dynamics of the rates associated to LMM, avoiding the presence of the rates dependent drifts (drift‐free simulation) and the presence of negative deflated bond prices and negative forward rates. The method is based upon a new parameterization of the martingales introduced by Glasserman and Zhao and it is extended to a Cross‐Market Model for commodities. Finally, a particular example of commodity derivative (spread option) pricing problem is considered. Copyright © 2014 John Wiley & Sons, Ltd.

SABR/LIBOR market models: Pricing and calibration for some interest rate derivatives

In order to overcome the drawbacks of assuming deterministic volatility coefficients in the standard LIBOR market models to capture volatility smiles and skews in real markets, several extensions of LIBOR models to incorporate stochastic volatilities have been proposed. The efficient calibration to market data of these more complex models becomes a relevant target in practice. The main objective of the present work is to efficiently calibrate some recent SABR/LIBOR market models to real market prices of caplets and swaptions. For the calibration we propose a parallelized version of the simulated annealing algorithm for multi-GPUs. The numerical results clearly illustrate the advantages of using the proposed multi-GPUs tools when applied to real market data and popular SABR/LIBOR models.

Conditional Quadratic Semidefinite Programming: Examples and Methods

The conditional quadratic semidefinite programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace, and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

Barrier Caps and Floors under the LIBOR Market Model with Double Exponential Jumps

The LIBOR market model (LMM) has become the standard model for many kinds of interest rate derivatives, such as cap contracts. It assumes that the distribution of the one-period interest rate at each future repricing date is lognormal, so that every caplet can be easily priced using the Black model. Volatilities at different dates are tied together by assumptions that restrict the number of parameters to a manageable set. But interest rates are strongly affected by things like monetary policy decisions, which can lead to sharp, nondiffusive jumps when policy changes. The effect may be seen in the leptokurtosis of the empirical distributions of interest rates and in volatility smiles for caplets. The low probability of extreme rate moves in the standard LMM is especially problematical for barrier options. In this article, the authors introduce a new jump-diffusion process for the LLM that can capture the effects of jumps on interest rate barrier contracts. The authors allow exponential jumps, both up and down, so the model is called the LMM with a double-jump process, or LMMDJ. Explicit valuation equations are provided for the four classes of barrier contracts and are shown to be quite accurate in Monte Carlo simulations. <b>TOPICS:</b>Derivatives, quantitative methods

Levy-Driven Libor Models: Modeling and Numerical Analysis

In this survey we discuss different approaches of Libor Market Models with Levy forcing term. We begin with introduction to the Levy processes, and survey main approaches for modeling and numerical analysis of such models.

Valuation of Caps and Swaptions under a Stochastic String Model

We develop a Gaussian stochastic string model that provides closed-form expressions for the prices of caps and swaptions that, under certain conditions, reduce to Black (1976) formulas. We also propose a stochastic string LIBOR market model that generalizes the models of Brace et al. (1997) and Longstaff et al. (2001a) and allows us to obtain cap prices using the Black (1976) formula. This model can be approximated by one that is compatible with the previous Gaussian models. Our main results are as follows: we show that one of the assumptions of Longstaff et al. (2001a) is incompatible with our model and, then, we obtain a possible explanation for some problems related to the relative valuation of caps and swaptions. We attain the observational equivalence of Kerkhof and Pelsser (2002) and we prove that, under the stochastic string approach, the models proposed by Brace et al. (1997) and Longstaff et al. (2001a) are more parsimonious than previously stated in the original papers. Finally, we derive a a close-form expression for the price of a portfolio option that is a multi-factor extension of the result obtained by Jamshidian (1989) and an analytical formula for swaption prices.

A Random Field LIBOR Market Model

A random field LIBOR market model (RFLMM) is proposed by extending the LIBOR market model, with interest rate uncertainties modeled via a random field. First, closed‐form formulas for pricing caplet and swaption are derived. Then the random field LIBOR market model is integrated with the lognormal‐mixture model to capture the implied volatility skew/smile. Finally, the model is calibrated to cap volatility surface and swaption volatilities. Numerical results show that the random field LIBOR market model can potentially outperform the LIBOR market model in capturing caplet volatility smile and the pricing of swaptions, in addition to possessing other advantages documented in the previous literature (no need of frequent recalibration or to specify the number of factors in advance). © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 34:580–606, 2014

Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model

Robust and accurate Monte Carlo simulation of (cross-) Gammas for Bermudan swaptions in the LIBOR market model

A new parameterization for the drift-free simulation in the Libor Market Model

We present an efficient procedure to simulate the dynamics of Libor Market Model that avoids the use of drift dependent paths in Monte Carlo simulation. We follow a drift-free simulation methodology by first simulating certain martingales and then obtaining the involved forward Libor rates in terms of them. More precisely, we propose a parameterization of those martingales so that the desired properties of the continuous models can be maintained after the discretization procedure when using either any intermediate forward measure or the spot one. Thus, the need of using the terminal measure to maintain these properties can be overcome. Finally, some numerical results concerning caplets pricing illustrate that the proposed method outperforms other ones existing in the literature.

Kooderive: Multi-Core Graphics Cards, the Libor Market Model, Least-Squares Monte Carlo and the Pricing of Cancellable Swaps

We discuss the pricing of cancellable swaps using the displaced diusion LIBOR market model using a multi-core graphics card. We demonstrate that over one hundred times speed up can be achieved in a realistic case.

THE EFFICIENT COMPUTATION OF PRICES AND GREEKS FOR CALLABLE RANGE ACCRUALS USING THE DISPLACED-DIFFUSION LMM

We study the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model (DDLMM). We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach for simulating the range accrual coupons. We consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic.

Pricing of range accrual swap in the quantum finance Libor Market Model

We study the range accrual swap in the quantum finance formulation of the Libor Market Model (LMM). It is shown that the formulation can exactly price the path dependent instrument. An approximate price is obtained as an expansion in the volatility of Libor. The Monte Carlo simulation method is used to study the nonlinear domain of the model and determine the range of validity of the approximate formula. The price of accrual swap is analyzed by generating daily sample values by simulating a two dimension Gaussian quantum field.

Smile Pricing and Hedging Caps and Swaptions in Libor Market Model and Its Simple Extension

This paper presents an extensive test of the Libor Market on the Euro Cap and Swaption market. The deterministic LIBOR market model prices OTC cap exactly and prices OTC swaptions with errors well below two basis points. Tests for the hedging performance show that the deterministic LIBOR market model is more effective in hedging in- and at-the money OTC caps and swaptions. The simple extended LIBOR market model under the displaced diffusion dynamics is sufficient to model the Euro caplet smile surface, with pricing errors for caplets across strikes below one basis point. The implied swaption smile for the Euro market takes on a similarly symmetric shape as the cap counterpart.

Pricing Interest Rate Derivatives and Computing Pathwise Greeks in the Extended LIBOR Market Model

Classical interest rate models were formulated under the no-arbitrage assumption of a unique yield curve. However, in the outbreak of the 2007 global financial crisis, market interest rates witnessed unprecedented segmentation, and this instigated the modelling of a basis spread, whose non-negligible widening could no longer be neglected. One extension of the LIBOR market model (LMM) to incorporate basis spread was proposed by Mercurio (2010a), and in this paper, we examine how his theoretical description is translated into practical implementation to price interest rate derivatives and compute pathwise Greeks. Specifically, our contribution is to develop and implement the theory of the extended LMM in the framework of a displaced diffusion process in the spot LIBOR measure. The adjoint mode from algorithmic differentiation is premiered in this new implementation to compute pathwise deltas and vegas before the accumulated knowledge is applied in the pricing of interest rate caps and one-way floaters.

Stochastic Skew in the Interest Rate Cap Market

AbstractThe term structure of interest rates cannot be used to fully explain and hedge the prices of interest rate derivatives. Unspanned stochastic volatility improves the accuracy of interest rate derivatives valuation but is still inadequate to capture the variation of skews in the implied volatility surface. In this study, we document the stochastic variation of implied volatility skews in the interest rate cap market. To develop a term structure model that is consistent with the empirical phenomena, we incorporate the Wishart process into the standard LIBOR market model, namely, the LIBOR N‐dimensional Wishart (“LNW” for short) market model and derive a closed‐form, accurate, efficient caplet pricing formula. The capacity of the LNW model to capture stochastic skews is examined and compared with that of LIBOR multi‐Heston (“LMH” for short) market model with two and three stochastic volatilities. We find that the LNW model outperforms its LMH counterparts in terms of both in‐sample and out‐of‐sample pricing errors. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:1146–1169, 2014

A Parametric n-Dimensional Markov-Functional Model in the Terminal Measure

This article develops and tests an n-dimensional Markov-functional interest rate model in the terminal measure based on parametric functional forms of exponential type. The parametric functional forms enable analytical expressions for forward discount bonds and forward LIBORs at all times and allows for calibration of the model to caplet prices given by a displaced diffusion Black model. The analytical expressions of the model provide a theoretical tool for understanding the structure of standard Markov-functional models (MFMs) as well as comparisons with the LIBOR market model (LMM). In particular, it is shown that for ‘typical’ market data the model is close enough to the LMM to be able to calibrate using the LMM calibration set-up and machinery. This provides further information about the similarities (as well as some of the differences) between MFM and LMM. The parametric n-dimensional MFM may be used for products that require high-dimensional models for appropriate pricing and risk management. When compared with an n-factor LMM, it has the virtue of being (much) faster for certain types of products.