LIBOR Market Model Research Articles

Calibration of stochastic models for interest rate derivatives

For the pricing of interest rate derivatives various stochastic interest rate models are used. The shape of such a model can take very different forms, such as direct modelling of the probability distribution (e.g. a generalized beta function of second kind), a short-rate model (e.g. a Hull–White model) or a forward rate model (e.g. a LIBOR market model). This article describes the general structure of optimization in the context of interest rate derivatives. Optimization in finance finds its particular application within the context of calibration problems. In this case, calibration of the (vector-valued) state of a given stochastic model to some target state, which is determined by available relevant market data, implies a continuous optimization of the model parameters such that a global minimum of the distance between the target state and the model state is achieved. In this article, a novel numerical algorithm for the optimization of parameters of stochastic interest rate models is presented. The optimization algorithm operates within the model parameter space on an adaptive lattice with a number of lattice points per dimension which is both low and fixed. In this way, a considerable performance gain is achieved, as compared to algorithms working with non-adaptive lattices requiring increasing and/or large numbers of lattice points. As compared to standard algorithms, e.g. those of the Levenberg–Marquardt type, the presented adaptive lattice algorithm also reduces the danger of getting trapped near a wrong local minimum. As a numerical example, its application is demonstrated by optimizing volatility and mean reversion parameters of the Hull–White model, such that the latter becomes calibrated to the swaption volatility market relevant for a given OTC (over-the-counter, i.e. not exchange traded) bond option.

Regression methods in pricing American and Bermudan options using consumption processes

Numerical algorithms for the efficient pricing of multidimensional discrete-time American and Bermudan options are constructed using regression methods and a new approach for computing upper bounds of the options’ price. Using the sample space with payoffs at optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach allows the constructing of both lower and upper bounds for the price by Monte Carlo simulations. The algorithms are tested by pricing Bermudan max-calls and swaptions in the Libor market model.

Calibration of Interest Rate and Option Models Using Differential Evolution

We describe in detail the Differential Evolution algorithm and tune it to be suitable for a wide range of minimization problems using a testbed of various cost functions. We then use the algorithm in the calibration of the one-factor Hull-White model to caplets and the Libor market model to European swaption data. We also calibrate the Heston model of stochastic volatility to European option data. We find that Differential Evolution consistently results in successful model calibrations and outperforms the Downhill Simplex and Levenberg-Marquardt algorithms. We propose an efficient method for using Differential Evolution to provide fast, reliable calibrations for any pricing model.

Empirical analysis of quantum finance interest rates models

Empirical forward interest rates drive the debt markets. Libor and Euribor futures data is used to calibrate and test models of interest rates based on the formulation of quantum finance. In particular, all the model parameters, including interest rate volatilities, are obtained from market data. The random noise driving the forward interest rates is taken to be a Euclidean two dimension quantum field. We analyze two models, namely the bond forward interest rates, which is a linear theory and the Libor Market Model, which is a nonlinear theory. Both the models are analyzed using Libor and Euribor data, with various approximations to match the linear and nonlinear models. The results are quite good, with the linear model having an accuracy of about 99% and the nonlinear model being slightly less accurate. We extend our analysis by directly using the Zero Coupon Yield Curve (ZCYC) data for Libor and for bonds; but due to some technical difficulties we could not derive the models parameters directly from the ZCYC data.

Analytical Formulas for Pricing CMS Products in the LIBOR Market Model with the Stochastic Volatility

In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. The derived formulas can also be used for model calibration to the market, including European swaptions and CMS products. The first technical achievement of this work is related to the optimal calculation of the measure change. For single-rate CMS products, we have used the standard linear regression of the measure change, with optimally calculated coefficients. For the CMS spread options, where the linear procedure does not work, we propose a new effective 'non-linear' measure change technique. The fit quality of the new results is confirmed numerically using Monte Carlo simulations. The second technical advance of the article is a theoretical derivation of the generalized spread option price via two-dimensional Laplace transform presented in a closed form in terms of the complex Gamma-functions.

Valuation of Interest Rate Spread Options in a Multifactor LIBOR Market Model

<i>Within the multifactor LIBOR market model, the authors examine three types of interest rate spread options: LIBOR vs. LIBOR, LIBOR vs. swap rate, and swap rate vs. swap rate. These financial products are widely traded in the marketplace or are embedded in structured notes, such as CMS range accruals and steepeners. In the first case, the authors show that the drift has an impact on the pricing which differs from the results of previous research. The authors also present a new approach to approximating the distribution of a forward swap rate under the LIBOR market model and then employ it to price CMS spread options. The numerical examples show that the approximate pricing formulas are robustly accurate as compared with Monte Carlo simulation using recent two-year data.</i> <b>TOPICS:</b>Interest-rate and currency swaps, options, simulations

Practical Policy Iteration: Generic Methods for Obtaining Rapid and Tight Bounds for Bermudan Exotic Derivatives Using Monte Carlo Simulation

We introduce a set of improvements which allow the calculation of very tight lower bounds for Bermudan derivatives using Monte Carlo simulation. These lower bounds can be computed quickly, and with minimal hand-crafting. Our focus is on accelerating policy iteration to the point where it can be used in similar computation times to the basic least-squares approach, but in doing so introduce a number of improvements which can be applied to both the least-squares approach and the calculation of upper bounds using the Andersen-Broadie method. The enhancements to the least-squares method improve both accuracy and efficiency.Results are provided for the displaced-diffusion LIBOR market model, demonstrating that our practical policy iteration algorithm can be used to obtain tight lower bounds for cancellable CMS steepener, snowball and vanilla swaps in similar times to the basic least-squares method.

Fast Monte-Carlo Greeks for Financial Products with Discontinuous Pay-Offs

We introduce a new class of numerical schemes for discretizing processes driven by Brownian motions. These allow the rapid computation of sensitivities of discontinuous integrals using pathwise methods even when the underlying densities post-discretization are singular. The two new methods presented in this paper allow Greeks for financial products with trigger features to be computed in the LIBOR market model with similar speed to that obtained by using the adjoint method for continuous pay-offs. The methods are generic with the main constraint being that the discontinuities at each step must be determined by a one-dimensional function: the proxy constraint. They are also generic with the sole interaction between the integrand and the scheme being the specification of this constraint.

Multiple stochastic volatility extension of the Libor market model and its implementation

In this paper we propose an extension of the Libor market model with a high-dimensional specially structured system of square root volatility processes, and give a road map for its calibration. As such the model is well suited for Monte Carlo simulation of derivative interest rate instruments. As a key issue, we require that the local covariance structure of the market model is preserved in the stochastic volatility extension. In a case study we demonstrate that the extended Libor model allows for stable calibration to the cap-strike matrix. The calibration algorithm is FFT based, so fast and easy to implement.

Incorporating an Interest Rate Smile in an Equity Local Volatility Model

The focus of this paper is on finding a connection between the interest rate and equity asset classes. We propose an equity interest rate hybrid model which preserves market observable smiles: the equity from plain vanilla products via a local volatility framework and the interest rate from caps and swaptions via the Stochastic Volatility Libor Market Model. We define a multi-factor short-rate process implied from the Libor Market Model via an arbitrage-free interpolation and combine it with the local volatility equity model for stochastic interest rates. We show that the interest rate smile has a significant impact on the equity local volatility. The model developed is intuitive and straightforward, enabling consistent pricing of related hybrid products. Moreover, it preserves the non-arbitrage Heath, Jarrow, Morton conditions.

Control Variates for Callable LIBOR Exotics - A Preliminary Study

Monte Carlo simulation is currently the method of choice for the pricing of callable derivatives in LIBOR market models. Lately more and more papers are surfacing in which variance reduction methods are applied to the pricing of derivatives with early exercise features. We focus on one of the conceptually easiest variance reduction methods, control variates. The basis of our method is an upper bound of the callable contract in terms of plain vanilla contracts, which is found to be a highly effective control variate. Several examples of callable LIBOR exotics demonstrate the effectiveness and wide applicability of the method.

Calibration of Libor Market Model to Caps and Swaptions Market Volatilities

We show a particular case of joint calibration of the Libor Market Model (LMM) to market-quoted implied cap and swaption volatilities using a linear-exponential parameterization. We also create a Monte Carlo vanilla swaption-pricing engine using the model in the first part of the paper. In the second part of the paper, an attempt will be made to incorporate the dynamics of the volatility skew for caplets though implementation of stochastic volatility SABR model.

Pricing Strongly Path-Dependent Options in Libor Market Models without Simulation

Path-dependent options in Fixed Income are typically priced using Monte-Carlo implementations of libor market models (LMM). We present an alternative to Monte-Carlo that scales with the number of underlying factors driving the LMM for typical strongly path-dependent options in Fixed Income such as as target range accrual notes (TARNs), and Snowblades. Our method is a generalization of that developed by [HW04] for pricing CDOs without simulation. This works by conditioning on the driving factors so that the distributions of the underlying at different times are independent. Payoff distributions are built up using an iterative scheme. The option price is then calculated by integrating over the driving factors. We extend this method to strongly path-dependent payoffs (e.g. TARNs and Snowblades) in the context of LMMs by adapting the iterative build scheme and including successive measure changes. Unlike the CDO case, TARN and Snowblade payoffs depend on the whole of the previous history. The main limitation of the extended method with the LMM is that measure changes for the drift are given by the usual freezing-the-forwards approximation. This extended method is particularly appropriate in the Fixed Income world because the observations and payments are usually at discrete times. Discretely observed analytic option pricing schemes from the Equity world are not appropriate because of the successive measure changes required by our scheme (and in general for Fixed Income). In the specific case of TARNs and Snowblades the conditional coupon payoff distributions can be expressed analytically. The method can also be combined with Monte-Carlo for integrating over the driving factors, when there are, say, 4 significant eigenvalues to the driving matrix. The advantage over straight Monte-Carlo lies in the summarizing capacity of the conditional distributions and is dependent on the ease with which these can be calculated. We also describe an extension of the model to include volatility smiles.

Multi-Factor Cross Currency Libor Market Models: Implementation, Calibration and Examples

We review multi-factor cross-currency LIBOR market models. We present a new method for the calibration of cross-currency market models to FX markets. We study the case of Power Reverse Dual Currency derivatives. We also present a new version of Least Square monte carlo method which makes handling of complex Bermudan callable structured derivatives much simpler.

Risk Management with the CEV LIBOR Market Model

We analyze the pricing accuracy and risk properties of the constant-elasticity-volatility (CEV) market model. It is shown that the calibration results to both caps and swaptions show small pricing errors. The whole model fit confirms the existence of strike bias in the US interest rate derivative market (the bias is much weaker in the corresponding Euro market). Daily VaRs are computed using Monte Carlo simulation. The back testing results suggest an underestimation problem for the cap-calibration-based model while the problem is reduced in the swaption-calibration-based model. Stress test regarding price sensitivities to yield curve shape changes is conducted. The results indicate that delta-gamma hedged portfolio has strong resistance to forward curve changes.

FOURIER TRANSFORM METHOD WITH AN ASYMPTOTIC EXPANSION APPROACH: AN APPLICATION TO CURRENCY OPTIONS

This paper develops a Fourier transform method with an asymptotic expansion approach for option pricing. The method is applied to European currency options with a libor market model of interest rates and jump-diffusion stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas of the characteristic functions of log-prices of the underlying assets and the prices of currency options based on a third order asymptotic expansion scheme; we use a jump-diffusion model with a mean-reverting stochastic variance process such as in Heston [7]/Bates [1] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

Valuation of floating range notes in a LIBOR market model

AbstractThis study derives an approximate pricing formula of floating range notes (FRNs) within the multifactor LIBOR market model (LMM) framework. The LMM features the ease for calibration procedure, and the resulting pricing formula is more tractable. In addition, since the underlying rate of FRNs is usually the LIBOR rate, the pricing of the FRNs under the LMM is more direct and full of intuition. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:697–710, 2008

Partial proxy simulation schemes for generic and robust Monte Carlo Greeks

We consider a generic framework which allows to calculate robust Monte-Carlo sensitivities seamlessly through simple finite difference approximation. The method proposed is a generalization and improvement of the proxy simulation scheme method (Fries and Kampen, 2005). As a benchmark we apply the method to the pricing of digital caplets and target redemption notes using LIBOR and CMS indices under a LIBOR Market Model. The framework is generic in the sense that it is model and almost product independent. The only product dependent part is the specification of the proxy constraint. This allows for an elegant implementation, where new products may be included at small additional costs.

Fast swaption pricing under the market model with a square-root volatility process

In this paper we study a correlation-based LIBOR market model with a square-root volatility process. This model captures downward volatility skews through taking negative correlations between forward rates and the multiplier. An approximate pricing formula is developed for swaptions, and the formula is implemented via fast Fourier transform. Numerical results on pricing accuracy are presented, which strongly support the approximations made in deriving the formula. ‡In the literature both ‘skew’ and ‘smirk’ are used to name a slanted smile.

Swaptions in Libor Market Model with Local Volatility

The original Libor Market Model (LMM) has been extended to several dynamics (or local volatilities) for the underlying Libor rates. The main result presented here is a generic approximation that provides an explicit swaptions price for local volatilities LMM. The approximation is base on an initial freeze approximation very efficient in the Bond Market Model and a corrector or Runge-Kutta approach. The approximation is not done at the path level but at the global level for a given strike allowing a smile calibration. The approximation efficiency is analyzed in details in the displaced diffusion case; it is analyzed by comparison to precise Monte Carlo simulations.