Abstract
The theory of interest-rate modeling was originally based on the assumption of specific one-dimensional dynamics for the instantaneous spot rate process r. Modeling directly such dynamics is very convenient since all fundamental quantities (rates and bonds) are readily defined, by no-arbitrage arguments, as the expectation of a functional of the process r.
“It will be short, the interim is mine. And a man’s life is no more than to say ‘one’” Hamlet, V.2
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References
See for instance Björk (1997).
The Radon-Nikodym derivative and the Girsanov change of measure are briefly reviewed in Appendix A. For a formal treatment see Musiela and Rutkowski (1998).
This is the reason why they have been also referred to as “endogenous term structure models”.
Indeed, the market price of risk under the Vasicek model is usually chosen to be constant, i.e., λ(t) = λ. However, this is just another possible formulation.
We should stress that mean reversion under Q does not necessarily imply mean reversion under Q0. However, we can assume that the change of measure does not affect the asymptotic behavior of the process r.
We thank our colleague Francesco Rapisarda for kindly providing us with such a figure.
We thank our colleague Gianvittorio Mauri (aka “ The Master”) for kindly providing us with such a graphic “masterpiece”!
Maghsoodi (1996) derived different formulas for the prices of bonds and bond options, but still relying on numerical integration.
The Black and Karasinski (1991) model is actually a generalization of the continuous-time formulation of the Black, Derman and Toy (1990) model.
Notice, however, that a lognormal instantaneous short-rate process does not lead to lognormal simple forward rates or lognormal swap rates.
What we are saying here is a little redundant with the material appearing in other chapters. This is done to maintain the single chapters self-contained to a certain degree.
The specification of the function a is motivated by its interpretation in the above Gaussian case.
We again omit to express the dependence on the index i to lighten notation.
If no variance reduction technique is employed, p typically ranges from p = 100, 000 to p = 1, 000, 000.
These parameters values derive from the model calibration to the actual Euro ATM caps volatility curve on January 17, 2000.
Notice that, when studying the implied cap volatility curve, varying θ is equivalent to varying θ̄ as soon as k is fixed and both θ and k are positive.
We just concentrate on these two models for practical purposes. Indeed, among the short rate models developed in this chapter, the BK and EEV models are likely to imply the best fitting to the swaption volatility surfaces in many concrete market situations.
These initial parameters values derive from the models calibration to the actual Euro ATM swaption volatility surface on January 17, 2000.
We use analytical formulas for the HW model, whereas, for the other models, we build a trinomial tree with a variable (small) time step in the first year, and an average of 50 time steps per year afterwards.
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© 2001 Springer-Verlag Berlin Heidelberg
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Brigo, D., Mercurio, F. (2001). One-factor short-rate models. In: Interest Rate Models Theory and Practice. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04553-4_3
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DOI: https://doi.org/10.1007/978-3-662-04553-4_3
Publisher Name: Springer, Berlin, Heidelberg
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