Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

Using 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.     

The Relative Valuation of Caps and Swaptions: Theory and Empirical Evidence

Although traded as distinct products, caps and swaptions are linked by no-arbitrage relations through the correlation structure of interest rates. Using a string market model, we solve for the correlation matrix implied by swaptions and examine the relative valuation of caps and swaptions. We find that swaption prices are generated by four factors and that implied correlations are lower than historical correlations. Long-dated swaptions appear mispriced and there were major pricing distortions during the 1998 hedge-fund crisis. Cap prices periodically deviate significantly from the no-arbitrage values implied by the swaptions market.     

Hedging in the Possible Presence of Unspanned Stochastic Volatility: Evidence from Swaption Markets

This paper examines whether higher order multifactor models, with state variables linked solely to underlying LIBOR‐swap rates, are by themselves capable of explaining and hedging interest rate derivatives, or whether models explicitly exhibiting features such as unspanned stochastic volatility are necessary. Our research shows that swaptions and even swaption straddles can be well hedged with LIBOR bonds alone. We examine the potential benefits of looking outside the LIBOR market for factors that might impact swaption prices without impacting swap rates, and find them to be minor, indicating that the swaption market is well integrated with the LIBOR‐swap market.     

Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis

We empirically compare Libor and Swap Market Models for the pricing of interest rate derivatives, using panel data on prices of US caplets and swaptions. A Libor Market Model can directly be calibrated to observed prices of caplets, whereas a Swap Market Model is calibrated to a certain set of swaption prices. For both models we analyze how well they price caplets and swaptions that were not used for calibration. We show that the Libor Market Model in general leads to better prediction of derivative prices that were not used for calibration than the Swap Market Model. Also, we find that Market Models with a declining volatility function give much better pricing results than a specification with a constant volatility function. Finally, we find that models that are chosen to exactly match certain derivative prices are overfitted; more parsimonious models lead to better predictions for derivative prices that were not used for calibration.     

A displaced-diffusion stochastic volatility LIBOR market model: motivation,definition and implementation

Abstract

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced-diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that (i) the caplet market can still be efficiently and accurately fit; (ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; (iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward rate process; and (iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the co-terminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities.     

Libor Market Models versus Swap Market Models for Pricing Interest Rate Derivatives: An Empirical Analysis

We empirically compare Libor and Swap Market Models for thepricing of interest rate derivatives, using panel data on pricesof US caplets and swaptions. A Libor Market Model can directlybe calibrated to observed prices of caplets, whereas a SwapMarket Model is calibrated to a certain set of swaption prices.For both models we analyze how well they price caplets and swaptionsthat were not used for calibration. We show that the Libor MarketModel in general leads to better prediction of derivative pricesthat were not used for calibration than the Swap Market Model.Also, we find that Market Models with a declining volatilityfunction give much better pricing results than a specificationwith a constant volatility function. Finally, we find that modelsthat arechosen to exactly match certain derivative prices areoverfitted; more parsimonious models lead to better predictionsfor derivative prices that were not used for calibration. JELClassification: G12, G13, E43.     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

Approximate pricing of swaptions in affine and quadratic models

This paper proposes new bounds on the prices of European-style swaptions for affine and quadratic interest rate models. These bounds are computable whenever the joint characteristic function of the state variables is known. In particular, our lower bound involves the computation of a one-dimensional Fourier transform independently of the swap length. In addition, we control the error of our method by providing a new upper bound on swaption price that is applicable to all considered models. We test our bounds on different affine models and on a quadratic Gaussian model. We also apply our procedure to the multiple curve framework. The bounds are found to be accurate and computationally efficient.     

Affine Term Structure Models and the Forward Premium Anomaly

One of the most puzzling features of currency prices is the forward premium anomaly : the tendency for high interest rate currencies to appreciate. We characterize the anomaly in the context of affine models of the term structure of interest rates. In affine models, the anomaly requires either that state variables have asymmetric effects on state prices in different currencies or that nominal interest rates take on negative values with positive probability. We find the quantitative properties of either alternative to have important shortcomings.