11. Stochastic Interest Rates...................................
This chapter is devoted to modelling the time evolution of random interest
rates. We adopt an approach similar to the binomial model of stock prices in
Chapter 3. Modelling the evolution of interest rates can be reduced to modelling
the evolution of the bond prices, since the latter determine the former. We begin
with some properties that a model of bond prices should satisfy, emphasising
the differences between bonds and stock.
First, let us recall that the evolution of interest rates or bond prices is
described by functions of two variables, the running time and the maturity
time, whereas stock prices are functions of just one variable, the running time.
Second, there are many ways of describing the term structure: bond prices,
implied yields, forward rates, short rates. Bond prices and yields are clearly
equivalent, being linked by a simple formula. Bond prices and forward rates are
also equivalent. The short rates are different, easier to handle, but the problem
of reconstructing the term structure emerges. This may be non-trivial, since
short rates usually carry less information.
Third, the model needs to match the initial data. For a stock this is just
the current price. In the case of bonds the whole initial term structure is given,
imposing more restrictions on the model, which has to be consistent with all
currently available market information.
Fourth, bonds become non-random at maturity. This is in sharp contrast
with stock prices. The fact that a bond gives a sure dollar at maturity has to
be included in the model.
Finally, the dependence of yields on maturity must be quite special. Bonds
with similar maturities will typically behave in a similar manner. In statistical
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