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.     

Unspanned Stochastic Volatility: Evidence from Hedging Interest Rate Derivatives

Most existing dynamic term structure models assume that interest rate derivatives are redundant securities and can be perfectly hedged using solely bonds. We find that the quadratic term structure models have serious difficulties in hedging caps and cap straddles, even though they capture bond yields well. Furthermore, at‐the‐money straddle hedging errors are highly correlated with cap‐implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. Our results strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets.     

Junp-Diffusion Interest Rate Process: An Empirical Examination

We investigate a jump-diffusion process, which is a mixture of an O-U process used by Vasicek (1977) and a compound Poisson jump process, for the term structure of interest rates. We develop a methodology for estimating the jump-diffusion model and complete an empirical study in comparing the model with the Vasicek model, for the US money market interest rates. The results show that when the short-term interest rate is low, both models predict an upward sloping term structure, with the jump-diffusion model fitting the actual term structure quite well and the Vasicek model overestimating significantly. When the short-term interest rate is high, both models predict a downward sloping term structure, with the jump-diffusion model underestimating the actual term structure more significantly than the Vasicek model.     

Option Pricing under Stochastic Interest Rates: An Empirical Investigation

Using daily data of the Nikkei 225 index, call option prices and call money rates of the Japanese financial market,a comparison is made of the pricing performance of stock option pricing modelsunder several stochastic interest rate processes proposedby the existing term structure literature.The results show that (1) one option pricing modelunder a specific stochastic interest ratedoes not significantly outperformanother option pricing model under an alternative stochasticinterest rate, and (2) incorporating stochastic interest ratesinto stock option pricing does not contribute to the performanceimprovement of the original Black–Scholes pricing formula.     

Pricing and hedging interest rate options: Evidence from cap–floor markets

We examine the pricing and hedging performance of interest rate option pricing models using daily data on US dollar cap and floor prices across both strike rates and maturities. Our results show that fitting the skew of the underlying interest rate probability distribution provides accurate pricing results within a one-factor framework. However, for hedging performance, introducing a second stochastic factor is more important than fitting the skew of the underlying distribution. This constitutes evidence against claims in the literature that correctly specified and calibrated one-factor models could replace multi-factor models for consistent pricing and hedging of interest rate contingent claims.     

The Term Structure of Real Rates and Expected Inflation

Changes in nominal interest rates must be due to either movements in real interest rates, expected inflation, or the inflation risk premium. We develop a term structure model with regime switches, time‐varying prices of risk, and inflation to identify these components of the nominal yield curve. We find that the unconditional real rate curve in the United States is fairly flat around 1.3%. In one real rate regime, the real term structure is steeply downward sloping. An inflation risk premium that increases with maturity fully accounts for the generally upward sloping nominal term structure.     

Pricing the term structure of inflation risk premia: Theory and evidence from TIPS

In this paper, we study inflation risk and the term structure of inflation risk premia in the United States' nominal interest rates through the Treasury Inflation Protection Securities (TIPS) with a multi-factor, modified quadratic term structure model with correlated real and inflation rates. We derive closed form solutions to the real and nominal term structures of interest rates that drastically facilitate the estimation of model parameters and improve the accuracy of the valuation of nominal rates and TIPS prices. In addition, we contribute to the literature by estimating the term structure of inflation risk premia implied from the TIPS market. The empirical evidence using data from the period of January 1998 through October 2007 indicates that the expected inflation rate, contrary to data derived from the consumer price indices, is very stable and the inflation risk premia exhibit a positive term structure.     

A comparison of diffusion models of the term structure

A number of different continuous time approaches that have been developed to model the term structure of interest rates are examined. These techniques span the interest rate literature over the last 20 years or so, and are the most commonly used among both academics and practitioners. We view this paper as a reference for the different term structure models, aiming to bring together the three most commonly used approaches, emphasizing their differences, analysing their respective advantages and disadvantages, and with explicit representations where they exist for prices of discount bonds.     

New No-arbitrage Conditions and the Term Structure of Interest Rate Futures

Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (J Finance 52(1); 409–430, 1997), the starting point of this paper is a model of futures prices. Using, as the input to the model, the prices of futures on interest related assets new no-arbitrage restrictions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices.     

Unspanned stochastic volatility and fixed income derivatives pricing

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.     

Interest Rate Futures Options and Interest Rate Options

This paper derives pricing models of interest rate options and interest rate futures options. The models utilize the arbitrage-free interest rate movements model of Ho and Lee. In their model, they take the initial term structure as given, and for the subsequent periods, they only require that the bond prices move relative to each other in an arbitrage-free manner. Viewing the interest rate options as contingent claims to the underlying bonds, we derive the closed-form solutions to the options. Since these models are sufficiently simple, they can be used to investigate empirically the pricing of bond options. We also empirically examine the pricing of Eurodollar futures options. The results show that the model has significant explanatory power and, on average, has smaller estimation errors than Black's model. The results suggest that the model can be used to price options relative to each other, even though they may have different expiration dates and strike prices.