The Valuation of American-style Swaptions in a Two-factor Spot Futures Model (original) (raw)
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The Valuation of American-style Swaptions in a Two-factor Spot-Futures Model1
1999
We build a no-arbitrage model of the term structure of interest rates using two stochastic factors, the short-term interest rate and the premium of the futures rate over the short-term interest rate. The model provides and extension of the lognormal interest rate model of Black and Karasinski (1991) to two factors, both of which can exhibit mean-reversion. The method is
The Term Structure of Interest-Rate Futures Prices
SSRN Electronic Journal, 2000
We derive general properties of two-factor models of the term structure of interest rates and, in particular, the process for futures prices and rates. Then, as a special case, we derive a no-arbitrage model of the term structure in which any two futures rates act as factors. In this model, the term structure shifts and tilts as the factor rates vary. The cross-sectional properties of the model derive from the solution of a two-dimensional, autoregressive process for the short-term rate, which exhibits both mean-reversion and a lagged persistence parameter. We show that the correlation of the futures rates is restricted by the no-arbitrage conditions of the model. In addition, we investigate the determinants of the volatilities and the correlations of the futures rates of various maturities. These are shown to be related to the volatility of the short rate, the volatility of the second factor, the degree of meanreversion and the persistence of the second factor shock. We also discuss the extension of our model to three or more factors. We obtain specific results for futures rates in the case where the logarithm of the short-term rate [e.g., the London Inter-Bank Offer Rate (LIBOR)] follows a two-dimensional process. We calibrate the model using data from Eurocurrency interest rate futures contracts.
The analysis and valuation of interest rate options
Journal of Banking & Finance, 1993
This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the BlackkScholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general. and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.
Interest rate futures: estimation of volatility parameters in an arbitrage-free framework
1997
Hedging interest rate exposures using interest rate futures contracts requires some knowledge of the volatility function of the interest rates. Use of historical data as well as interest rate options like caps and swaptions to estimate this volatility function have been proposed in the literature. In this paper the interest rate futures price is modelled within an arbitrage-free framework for
One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities
The Journal of Financial and Quantitative Analysis, 1993
This paper compares different approaches to developing arbitrage-free models of the term structure. It presents a numerical procedure that can be used to construct a wide range of one-factor models of the short rate that are both Markov and consistent with the initial term structure of interest rates.
On the Information in the Interest Rate Term Structure and Option Prices
Review of Derivatives Research, 2000
We examine whether the information in cap and swaption prices is consistent with realized movements of the interest rate term structure. To extract an option-implied interest rate covariance matrix from cap and swaption prices, we use Libor market models as a modelling framework. We propose a flexible parameterization of the interest rate covariance matrix, which cannot be generated by standard low-factor term structure models. The empirical analysis, based on US data from 1995 to 1999, shows that option prices imply an interest rate covariance matrix that is significantly different from the covariance matrix estimated from interest rate data. If one uses the latter covariance matrix to price caps and swaptions, one significantly underprices these options. We discuss and analyze several explanations for our findings.
Interest Rate Swaptions: A Review and Derivation of Swaption Pricing Formulae
In this paper we outline the European interest rate swaption pricing formula from first principles using the Martingale Representation Theorem and the annuity measure. This leads to an expression that allows us to apply the generalized Black-Scholes result. We show that a swaption pricing formula is nothing more than the Black-76 formula scaled by the underlying swap annuity factor. Firstly, we review the Martingale Representation Theorem for pricing options, which allows us to price options under a numeraire of our choice. We also highlight and consider European call and put option pricing payoffs. Next, we discuss how to evaluate and price an interest swap, which is the swaption underlying instrument. We proceed to examine how to price interest rate swaptions using the martingale representation theorem with the annuity measure to simplify the calculation. Finally, applying the Radon-Nikodym derivative to change measure from the annuity measure to the savings account measure we arrive at the swaption pricing formula expressed in terms of the Black-76 formula. We also provide a full derivation of the generalized Black-Scholes formula for completeness.
Forward Rate Volatilities, Swap Rate Volatilities, and Implementation of the LIBOR Market Model
The Journal of Fixed Income, 2000
This paper presents a number of new ideas concerned with the implementation of the LIBOR market model and its extensions. It develops and tests an analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used to price interest rate caps. The approximation is very accurate for the range of market parameters normally encountered and enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be calibrated to broker quotes on caps and European swap options so that other interest rate derivatives can be valued.
Journal of Futures Markets, 2003
We propose a generalization of the Shirakawa (1991) model to capture the jump component in fixed income markets. The model is formulated under the Heath, Jarrow and Morton (1992) framework, and allows the presence of a Wiener noise and a finite number of Poisson noises, each associated with a time deterministic volatility function. We derive the evolution of the futures price and use this evolution to estimate the model parameters via the likelihood transformation technique of Duan (1994). We apply the method to the short term futures contracts traded on CME, SFE, LIFFE and TIFFE, and find that each market is characterized by very different behaviour.