Abstract
In this chapter we consider the term structure of interest rates and the interest rate derivatives. The interest rates are closely related to the bond market; we therefore introduce the interest rates in connection with the simplest assets on the bond market, namely the so-called T-bonds which are contracts that guarantee a unitary amount at a given maturity T and their prices express the expectations of the market on the future value of money.
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Notes
- 1.
We shall deal with the modeling of interest rate markets starting from the next Section 4.2, where we shall analyze various approaches to assign a stochastic dynamics to interest rates and to corresponding assets.
- 2.
By convention we put D(n,n) = 1.
- 3.
In general, in an affine model the price of the T-bonds is the exponential of a linear (affine) function of the rate r.
- 4.
In the binomial model of Section 1.4.1 the martingale measure Q is uniquely identified as a product measure thanks to the independence property under Q of the sequence of random variables μ n .
- 5.
The coefficients of the system depend on the choice of the functions φ n ,N , namely on the parameters of the model.
- 6.
The choice of H n = N − n + 1 appears as a natural one since at time n one has N − n + 1 assets that are exactly the N − n zero coupon bonds relative to the maturities N = n + 1, n + 2, . . . , N and the money market account B.
- 7.
We denote by 1A the indicator function of the set A.
- 8.
Recall Notation 4.1.
- 9.
α represents the number of units invested in the underlying bond that has maturity in \( \overline{N} \) = 2 and β is the amount invested in the non-risky asset.
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© 2012 Springer-Verlag Italia
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Pascucci, A., Runggaldier, W.J. (2012). Interest rates. In: Financial Mathematics. Unitext(). Springer, Milano. https://doi.org/10.1007/978-88-470-2538-7_4
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DOI: https://doi.org/10.1007/978-88-470-2538-7_4
Publisher Name: Springer, Milano
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Online ISBN: 978-88-470-2538-7
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