Interest structure model

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An interest structure model is a financial mathematical model that describes the entire interest structure , i.e. the interest rates for different terms.

Properties and purpose

With the help of interest structure models one would like to explain the empirically observable relationships between interest rates of different maturities by as few interest structure factors as possible and describe the possible development of interest rates over time. These models are used to value bonds and interest rate derivatives , i.e. financial transactions whose value depends on interest rates. Since the various interest rates depend on or influence one another, modeling individual interest rates independently of one another does not make sense. For this reason, an attempt is made to represent the entire term structure in one model.

Interest rate structure models are among the most demanding of the financial market models. This is particularly the case when the “explanatory” interest rate structure factors are described using stochastic processes .

The aim of interest structure models is to describe the possible future development of the interest structure. It is less about predicting interest rates than about their likely future distribution. This is analogous to the procedure of the Black-Scholes model , which, however, cannot be used to value interest rate derivatives for a number of reasons:

  • Instead of an underlying asset (the share), the interest structure consists of a multitude of interest rates, all of which are to be modeled simultaneously.
  • Unlike stocks, interest rates are not traded directly, only derivatives on interest rates.
  • Shares have a (theoretically) unlimited term, whereas bonds usually have a limited term.

For the reasons mentioned, modeling the interest rate structure is considerably more difficult than the valuation of equity derivatives and the corresponding models are much more complex. Depending on the underlying explanatory variable ("factor"), a distinction is often made:

Current interest models

Models in which the current interest rate is the explanatory variable and thus the only source of uncertainty are called current interest rate models . The current rate is of a theoretical nature and cannot be observed on the market. It denotes the interest rate of a safe investment for the current, infinitesimal (infinitely) short period of time. Usually one- or three-month interest rates (see EURIBOR ) are used to calculate the current interest rate. A distinction is made between different approaches depending on the modeling of the instantaneous rate process. One of the first comes from Vasicek (1977), who used a Gaussian Ornstein-Uhlenbeck process to develop the instantaneous interest rate. The current interest rate is normally distributed, i.e. H. Interest rates can have negative values ​​with a positive probability. Other significant approaches are e.g. B. the models by Cox / Ingersoll / Ross (1985) and Hull / White (1990).

One advantage of the instantaneous interest rate models is the mostly simple implementation and the great freedom in the choice of parameters. Usually they provide closed valuation equations for bonds and simple interest rate derivatives. The calibration process, i.e. the adaptation of the model parameters to the real market data, is often criticized. This adaptation is more complicated, the more realistic the model is. Furthermore, empirical studies have shown that instantaneous interest rate models with (only) one factor have poor explanatory power. The use of the instantaneous interest rate as the only explanatory variable means that all interest rates of the interest rate structure are perfectly correlated and real interest rate structures cannot be adequately reproduced. The addition of other factors such as the inflation rate or the long-term interest rate improves the adaptability, but makes the models more complicated to use.

Forward rate models

To overcome the shortcomings of instantaneous interest rate models, designed Heath / Jarrow / Morton (1992) a general model framework that instead of a single point of the yield, the development of the entire appointment -Momentanzinsen (also called "current forward rates", English often imprecise shortening forward rates called). The current interest rate is the current rate for the future point in time specified as the date. This models the yield curve as a whole. In their work, the authors show that in addition to the initial interest rate structure, the only additional input variable required is the volatility function of the forward instantaneous interest rates ( drift restriction ). This is analogous to the Black-Scholes model, where the value of a stock option also only depends on the current value of the underlying asset and its volatility. This enables the preference-free valuation of interest rate derivatives. The choice of volatility function is important; it essentially determines the respective model properties.

The Gaussian interest rate models represent an important subclass . Their volatility function is deterministic, which leads to normally distributed forward instantaneous interest rates (and thus lognormally distributed bond prices). Closed valuation equations exist for the special case of constant or exponentially falling volatilities.

Forward rate models allow the use (theoretically) of any number of input parameters in order to enable a more realistic representation of the interest rate structure. At the same time, however, the instantaneous forward rate is a theoretical construct that cannot be directly observed on the market. The market models overcome this disadvantage.

Market models

Market models represent a further development of the forward interest rate models. Instead of the fictitious forward instantaneous interest rates, they use interest rates actually observable on the market. This methodology was first used in the work of Brace / Gatarek / Musiela (1997) and Miltersen / Sandmann / Sondermann (1997) for LIBOR rates ( LIBOR market model ) and by Jamshidian (1997) for swap rates ( swap market -Model ) applied. These models lead to valuation equations similar to the Black-Scholes formula and have therefore become established in practice today.

literature

Original work

  • Alan Brace, Dariusz Gatarek, Marek Musiela: The Market Model of Interest Rate Dynamics. In: Mathematical Finance. Vol. 7, No. 2, 1997, ISSN  0960-1627 , pp. 127-147, doi : 10.1111 / 1467-9965.00028 .
  • John C. Cox, Jonathan E. Ingersoll, Jr., Stephen A. Ross: A Theory of the Term Structure of Interest Rates. In: Econometrica. Vol. 53, No. 2, 1985, ISSN  0012-9682 , pp. 385-407.
  • David Heath, Robert Jarrow, Andrew Morton: Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. In: Econometrica. Vol. 60, No. 1, 1992, pp. 77-105.
  • John Hull, Alan White: Pricing Interest-Rate-Derivative Securities. In: The Review of Financial Studies. Vol. 3, No. 4, 1990, ISSN  0893-9454 , pp. 573-592.
  • Farshid Jamshidian: LIBOR and Swap Market Models and Measures. In: Finance and Stochastics. Vol. 1, No. 4, 1997, ISSN  1432-1122 , pp. 293-330.
  • Kristian R. Miltersen, Klaus Sandmann, Dieter Sondermann: Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates. In: Journal of Finance. Vol. 52, No. 1, 1997, ISSN  0022-1082 , pp. 409-430.
  • Oldrich Vasicek: An Equilibrium Characterization of the Term Structure. In: Journal of Financial Economics. Vol. 5, No. 2, 1977, ISSN  0304-405X , pp. 177-188, doi : 10.1016 / 0304-405X (77) 90016-2 .

Books