10. Variable Interest Rates.....................................
This chapter begins with a model in which the interest rates implied by bonds
do not depend on maturity. If the rates are deterministic, then they must
be constant and the model turns out to be too simple to describe any real-life
situation. In an extension allowing random changes of interest rates the problem
of risk management will be dealt with by introducing a mathematical tool
called thedurationof bond investments. Finally, we shall show that maturity-
dependent rates cannot be deterministic either, preparing the motivation and
notation for the next chapter, in which a model of stochastic rates will be
explored.
As in Chapter 2,B(t, T) will denote the price at timet(the running time)
of a zero-coupon unit bond maturing at timeT(the maturity time). The de-
pendence on two time variables gives rise to some difficulties in mathematical
models of bond prices. These prices are exactly what is needed to describe the
time value of money. In Chapter 2 we saw how bond prices imply the interest
rate, under the assumption that the rate is constant. Here, we want to relax
this restriction, allowing variable interest rates.
In this chapter and the next one time will be discrete, though some parts of
the theory can easily be extended to continuous time. We shall fix a time step
τ, writingt=τnfor the running time andT=τNfor the maturity time. In
the majority of examples we shall take eitherτ= 121 orτ=1.The notation
B(n, N) will be employed instead ofB(t, T) for the price of a zero-coupon unit
bond. We shall use continuous compounding, bearing in mind that it simplifies
notation and makes it possible to handle time steps of any length consistently.
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