Models for the Interest Rate and Interest Rate Derivatives
Pricing interest rate derivatives fundamentally depends on the underlying term structure. The often made assumptions of constant risk free interest rate and its independence of equity prices will not be reasonable when considering interest rate derivative
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Models for the Interest Rate and Interest Rate Derivatives
Human fortunes are as unpredictable as the weather.
Pricing interest rate derivatives fundamentally depends on the underlying term structure. The often made assumptions of constant risk free interest rate and its independence of equity prices will not be reasonable when considering interest rate derivatives. Just as the dynamics of a stock price are modeled via a stochastic process, the term structure of interest rates is modeled stochastically. As interest rate derivatives have become increasingly popular, especially among institutional investors, the standard models for the term structure have become a core part of financial engineering. It is therefore important to practice the basic tools of pricing interest rate derivatives. For interest rate dynamics, there are one-factor and twofactor short rate models, the Heath Jarrow Morton framework and the LIBOR Market Model. Exercise 10.1 (Forward Rate Agreements and Receiver Interest Rate Swap). Consider the setup in Table 10.1 with the face value of the considered bonds as 1 EUR. (a) Calculate the value of the forward rate agreements. (b) Calculate the value of a receiver interest rate swap. (c) Determine the swap rate. (a) A forward rate agreement FRARK ;S fr.t/; t; T g is an agreement at time t that a certain interest rate RK will apply to a principal amount for a certain period of time .T; S/, in exchange for an interest rate payment at the future interest rates R.T; S /, with t < T < S .
S. Borak et al., Statistics of Financial Markets, Universitext, DOI 10.1007/978-3-642-33929-5 10, © Springer-Verlag Berlin Heidelberg 2013
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10 Models for the Interest Rate and Interest Rate Derivatives
Table 10.1 Dataset
Maturity(years)
0:5
1
1:5
2
Bond value Strike rate (%)
0:97 7:50
0:94 7:50
0:91 7:50
0:87 7:50
D V .t; S / .T; S /RK C V .t; S / V .t; T /
(10.1)
The value of a forward rate agreement is determined by: FRARK ;S fr.t/; t; T g D
.T; S/fRK R.T; S /g 1 C R.t; S /.t; S /
where t is the current time, the time when FRAs come into place is T , and the maturity of the FRAs is S . Here RK stands for the strike interest rate. The term structure of interest rates is therefore not needed. .T; S/ D 0:5 for all FRAs. Plug in (10.1), we now calculate: FRA0:075;0:5 fr.t/; 0; 0:0g D 0:97 0:5 0:075 C 0:97 1:00 D 0:0064 FRA0:075;1:0 fr.t/; 0; 0:5g D 0:94 0:5 0:075 C 0:94 0:97 D 0:0053 FRA0:075;1:5 fr.t/; 0; 0:5g D 0:91 0:5 0:075 C 0:91 0:94 D 0:0041 FRA0:075;2:0 fr.t/; 0; 0:5g D 0:87 0:5 0:075 C 0:87 0:91 D 0:0074 These results are given in Table 10.2. (b) An Interest Rate Swap IRSRK ;T fr.t/; tg is an agreement to exchange payments of a fixed rate RK against a variable rate R.t; ti / over a period .t; T / at certain time points ti , with t ti T . When we consider a receiver interest rate swap, we receive the fixed interest rate in exchange for paying the floating rate. For the valuation of the receiver interest rate swap, we can apply two different methods
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