Valuation Of Interest Rate Derivatives Research Articles

Multi-Curve Modeling Using Trees

Since 2008 the valuation of derivatives has evolved so that OIS discounting rather than LIBOR discounting is used. Payoffs from interest rate derivatives usually depend on LIBOR. This means that the valuation of interest rate derivatives depends on the evolution of two different term structures. The spread between OIS and LIBOR rates is often assumed to be constant or deterministic. This paper explores how this assumption can be relaxed. It shows how well-established methods used to represent one-factor interest rate models in the form of a binomial or trinomial tree can be extended so that the OIS rate and a LIBOR rate are jointly modelled in a three-dimensional tree. The procedures are illustrated with the valuation of spread options and Bermudan swap options. The tree is constructed so that LIBOR swap rates are matched.

A Lévy HJM multiple-curve model with application to CVA computation

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Lévy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.

A L evy HJM Multiple-Curve Model with Application to CVA Computation

We consider the problem of valuation of interest rate derivatives in the post-crisis setup. We develop a multiple-curve model, set in the HJM framework and driven by a Levy process. We proceed with joint calibration to caps and swaptions of different tenors, the calibration to caps guaranteeing that the model correctly captures volatility smile e ffects (in strike) and the calibration to at-the-money swaptions ensuring an appropriate term structure of the volatility in the model. To account for counter-party risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counter-party risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration we study the case of a basis swap, for which we compute the counter-party risk and funding adjustments.

A flexible matrix Libor model with smiles

We present a flexible approach for the valuation of interest rate derivatives based on affine processes. We extend the methodology proposed in Keller-Ressel et al. (in press) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in this multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of Wishart Libor model in describing the movements of the implied volatility surface.

NO‐ARBITRAGE ONE‐FACTOR MODELS OF THE SOUTH AFRICAN TERM STRUCTURE OF INTEREST RATES

AbstractShort‐term interest rate processes determine the term structure of interest rates in an arbitrage‐free market and are central to the valuation of interest rate derivatives. We obtain parameter estimates and compare the empirical fit of alternative one‐factor continuous‐time processes for the South African short‐term interest rate (and hence of arbitrage‐free term structure models) using Gaussian estimation methods. We find support only for diffusions where the interest rate volatility is moderately sensitive to the level of the interest rate. Other common models with restrictions that either preclude this effect, or restrict it to be too high, do not fit the data. Differences in the specification of the drift function have no evident effect on model performance.

A Flexible Matrix Libor Model with Smiles

We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in a multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility ofWishart Libor model in describing the movements of the implied volatility surface.

Estimating value-at-risk under a Heath–Jarrow–Morton framework with jump

This article proposes a new methodology for measuring Value-at-Risk (hereafter VaR) using a model that incorporates both volatility and jumps. Heath–Jarrow–Morton (HJM) model has been used for the valuation of interest rate derivatives. This study extends the use of HJM model to the estimation VaR. This article specifically uses a two-factor HJM jump-diffusion model for the computation. The study models the Eurodollar futures prices using its derivatives. In addition, this article uses a new volatility specification of Ze-To (2002) to construct the HJM dynamics. The result indicates that the VaR model using HJM jump-diffusion framework performs well in capturing the nonnormality and in providing accurate VaR forecasts in the in-sample and out-sample tests.

Delta-Hedging of Interest Rate Risk in Longterm Contracts - An Application of the Cairns-Model

Long-term portfolios consisting of assets and liabilities often exhibit a significant sensitivity to changes in interest rates. For the management of the interest rate risk arising from such portfolios asset managers usually use duration based approaches like the PV01-method. In the meanwhile the model family proposed by Cairns (2004) proclaims to deliver arbitrage-free valuation of interest rate derivatives and an ideal framework for the management of long-term interest rate risks. In the following paper, we use the two-factor version of the Cairns-model in order to hedge interest rates of long-term portfolios and in order to test the hedge-effiectiveness of the commonly used PV01-approach. For this purpose we first derive interest rate sensitivity measures on the basis of the Cairns-model. Based on the extended Kalman-fillter approach, we calibrate the model and demonstrate the behavior of the sensitivity measures derived. We then use these measures in order to hedge interest rate risks of a pension-fund-like portfolio. The hedging-strategy we introduce for this purpose is based on plain vanilla interest rate swaps, it is rule-based and model independent. We then put the Cairns-model and PV01 approach to the test using one historical path (European interest rates, June 2000 - June 2010) as well as on the basis of a Monte Carlo simulation. Both analysis show that the strategy delivered is quite robust in eliminating interest rate risks. Interestingly, our analysis actually underlines the practicability and compatibility of the PV01-approach.

Generalized Swap Market Model and the Valuation of Interest Rate Derivatives

In this paper we will establish a generalized Swap Market Model (GSMM) by unifying the stochastic process of swap rates with constant tenors under a single swap measure. GSMM is a natural extension of Libor Market Model (LMM) for swap rates, and LMM can be considered as a special case of GSMM since Libors is a special swap rate with the constant tenor of one period. GSMM can be applied for pricing and hedging any interest rate derivatives, and is suited especially for CMS and swap rate products. There are a number advantages of GSMM: (1) GSMM models swap rates directly, and therefore achieves the best match between products and model. (2) GSMM can be calibrated to the term structure of swaption volatilities easily and quickly. (3) There is no translation of risk sensitivities with respect to swap rates within GSMM. In contrast, risk sensitives such as Vega for swap rates can not be derived directly, and must be translated in an inefficient, inaccurate and nontransparent manner in the most existing interest rate models. (4) All smile modelings for LMM can be taken over for GSMM since GSMM and LMM share an almost identical mathematical structure. (5) GSMM avoids the inconsistency of the market conventions in cap and swaptions markets. Accompanied by these favorite features, GSMM should be a promising interest rate model for pricing and hedging most traded swap rate structures in financial market.

Dynamic Term Structure Modeling: The Preface

Dynamic Term Structure Modeling, the second book in the trilogy on the Fixed Income Valuation Course, shows how to value interest rate derivatives and credit derivatives using a variety of affine, quadratic, HJM, and LIBOR market models. Using a new taxonomy, this book classifies all term structure models as either fundamental models, or preference-free single-plus, double-plus, and triple-plus models. Filled with in-depth insights and expert advice, this book shows how to price basic interest rate and credit derivative products, such as Treasury and Eurodollar futures, bond options, interest rate options (e.g., caps, floors, and swaptions), forward rate agreements, interest rate swaps, credit default swaps, credit spread options, and others. With intuitive explanations and fully developed examples, this book provides new transforms for building efficient trees under state-dependent volatility models, stochastic volatility models, and jump-diffusion models, for pricing American options, and describes fast computational methods such as the Fourier inversion method (including the FFT) and the cumulant expansion method, for valuing interest rate derivatives and credit derivatives, under a variety of affine, quadratic, and LIBOR market models. This book is also accompanied by an informative CD-ROM, which contains various Excel®/VBA® spreadsheets. This software allows for valuation of interest rate derivatives by building interest rate trees for low-dimensional affine models, as well as computing solutions using quasi-analytical formulas for higher-dimensional affine, quadratic, and LIBOR market models. Though most of the programs require coding in advanced scientific languages, such as C and C++, the final output is presented in user-friendly Excel/VBA spreadsheets.

A New Approach to the Valuation of Interest Rate Derivatives: Arrow-Debreu Prices Implicit in the Term Structure of Interest Rates

In a complete, arbitrage-free securities market, the value of a discount bond is modeled in terms of the pricing kernel and the transition density function of the spot interest rate process. The prices of discount bonds are taken from the current term structure of interest rates, and the transition density function is estimated from historical term structure data, using both parametric and nonparametric techniques. The pricing kernel and associated prices of Arrow-Debreu securities are then determined. The resulting Arrow-Debreu prices have two merits: they are consistent with the current term structure of interest rates, and therefore arbitrage-free, and, in addition, they embody the observed historical behavior of the term structure.

Monte Carlo Valuation of Interest Rate Derivatives Under Stochastic Volatility

Monte Carlo Valuation of Interest Rate Derivatives Under Stochastic Volatility