Interest rate derivative

Summary

In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of different interest rate indices that can be used in this definition. IRDs are popular with all financial market participants given the need for almost any area of finance to either hedge or speculate on the movement of interest rates. Modeling of interest rate derivatives is usually done on a time-dependent multi-dimensional Lattice ("tree") or using specialized simulation models. Both are calibrated to the underlying risk drivers, usually domestic or foreign short rates and foreign exchange market rates, and incorporate delivery- and day count conventions. The Heath–Jarrow–Morton framework is often used instead of short rates. Types The most basic subclassification of interest rate derivatives (IRDs) is to define linear and non-lin

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We develop a tractable and flexible stochastic volatility multifactor model of the term structure of interest rates. It features unspanned stochastic volatility factors, correlation between innovations to forward rates and their volatilities, quasi-analytical prices of zero coupon bond options, and dynamics of the forward rate curve, under both the actual and risk-neutral measures, in terms of a finite-dimensional affine state vector. The model has a very good fit to an extensive panel data set of interest rates, swaptions, and caps. In particular, the model matches the implied cap skews and the dynamics of implied volatilities.

Oxford University Press2009

This thesis studies the valuation and hedging of financial derivatives, which is fundamental for trading and risk-management operations in financial institutions. The three chapters in this thesis deal with derivatives whose payoffs are linked to interest rates, equity prices, and dividend payments. The first chapter introduces a flexible framework based on polynomial jump-diffusions (PJD) to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. Option prices are approximated efficiently using a moment matching technique based on the principle of maximum entropy. An extensive calibration exercise shows that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 index dividend futures and dividend options, and Euro Stoxx 50 index options. The second chapter revisits the problem of pricing a continuously sampled arithmetic Asian option in the classical Black-Scholes setting. An identity in law links the integrated stock price to a one-dimensional polynomial diffusion, a particular instance of the PJD encountered in the first chapter. The Asian option price is approximated by a series expansion based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however numerical experiments show that the bias can safely be ignored in practice. The last chapter presents a non-parametric method to construct a maximally smooth discount curve from observed market prices of linear interest rate products such as swaps, forward rate agreements, or coupon bonds. The discount curve is given in closed form and only requires basic linear algebra operations. The method is illustrated with several practical examples.

EPFL2019

Applying the HJM-approach when volatility is stochastic

Pierre Collin Dufresne, Wei Shi

We propose a simple approach to extend the Heath, Jarrow and Morton (1992) model to a framework where the volatility of bond prices and forward rates is stochastic and where volatility-specific risk cannot necessarily be hedged in trading bonds. The arbitrage-free modeling of Heath, Jarrow and Morton cannot readily be transposed to such an environment. HJM’s major insight is that the drift of forward rates must be a function of the volatility structure of forward rates to preclude arbitrage opportunities. This implies that to price interest-rate dependent claims, one need only specify the volatility structure of forward rates. Given the initial term structure (observed in the market), any contingent claim can be priced. While the HJM-restriction still applies when the volatility of forward rates is stochastic, it is not sufficient to price interest-rate contingent claims. Indeed one needs to specify the arbitrage-free process of volatility under the risk-neutral measure. The traditional HJM restriction does not provide any guidance in that respect. We solve that problem by transposing HJM’s idea to the volatility term structure. We show that given an initial term structure of futures prices on interest-rate yields, one need only specify the volatility structure of the futures price on interest-rate yields to derive the arbitrage-free dynamics of the volatility of forward rates. Thus, taking as input the initial term structure of futures prices, the term strucuture of forward rates and specifying a volatility structure for the futures price on yields, one can price all interest rate contingent claims in our model. Our model thus requires as input two term structures, but only one volatlity structure need be specified. We provide a detailed analysis of a gaussian specification of the model, for which volatility exhibits mean reversion. It is parsimonious as it requires only three parameters to be estimated. We use that specification to draw comparisons with a simple extended-Vasicek model in order to investigate the effects of stochastic volatility on interest-rate options. The results support the widespread assertion that even though stochastic volatility might not be essential for the modeling of the term structure, it might have an important impact on interest-rate derivatives.

Carnegie Mellon University1997