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2020 | OriginalPaper | Chapter

7. Interest Rate Models

Author : Jiří Witzany

Published in: Derivatives

Publisher: Springer International Publishing

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Abstract

The Standard Market Model developed and applied in the previous chapter assumes that interest rates or bond prices are lognormally distributed. The model does not describe the stochastic dynamics of interest rates over time, and so it cannot be applied to value American-style options, callable bonds, or other more complex interest rate derivatives. In this chapter, we are going to introduce the most important interest rate models, which can be classified into two categories: short-rate and term-structure models. The short-rate models focus on the instantaneous interest rate stochastic dynamics. The rest of the term-structure is derived from the short rate at a point in time, and from the model parameters. Term-structure models, on the other hand, specify equations for (forward) interest rates in all maturities, and these equations are tied by certain consistency (non-arbitrage) conditions. In both cases, the models are developed and applied under a risk-neutral measure, but can be calibrated from the real-world data.

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previous chapter Stochastic Interest Rates and the Standard Market Model
next chapter Exotic Options, Volatility Smile, and Alternative Stochastic Models
Footnotes
1
The growth of the money market account from a time t to t + Δt can be bounded from below by an expression of the form exp(exp(y)Δt), where y = N(m, s2) is obtained by averaging the generalized Wiener process lnr(s). The expected value of the growth lower bound is calculated as an integral, where the values are weighted by the normal density of y which is exponential in a negative quadratic function of y. Since exp(y) overgrows any polynomial of y, the expected value of the money market account must be infinite.
 
2
Changing the measure from the real-world one to the risk-neutral one, the coefficient of dt is reduced by −λσ. Provided the price of risk λ is constant the parameter b is changed to b′ = b − λσ/a.
 
3
If dP(t, tm) = (…)dt + vmP(t, tm)dz then according to Ito’s lemma d ln P(t, tm) = (…)dt + vmdz. Hence d ln (P(t, tm)/P(t, tk+1)) = (…)dt + (vm − vk+1)dz, and so the volatility of P(t, tm)/P(t, tk+1) is vm − vk+1.
 
4
Note that x = yW′ and var[x] = E[xx′] = E[yW′Wy′] = E[yy′] =  var [y], since WW′ = I.
 
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Metadata
Title
Interest Rate Models
Author
Jiří Witzany
Copyright Year
2020
Publisher
Springer International Publishing
Book
Derivatives

Print ISBN: 978-3-030-51750-2

Electronic ISBN: 978-3-030-51751-9

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-51751-9_7