The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 620 Wednesday, February 4, 2004 1:33 PM

620 The Mathematics of Financial Modeling and Investment Management

P = Me–it = Me–i(T–s)

If the interest rate is a known function of time, the above equation

becomes

dP = –i(t)Pdt

This too is an equation with separable variables whose solution is

• ∫Tiu()ud
  P = Me s

where M is the principal to be repaid. The equivalence pathwise between

capital appreciation and present value is valid only if interest rates are

known.

In the above expression, the interest rate i is the instantaneous rate

of interest, also called the short-term rate. In continuous time, the short-

term rate is the limit of the interest rate over a short time interval when

the interval goes to zero. As observations can only be performed at dis-

crete dates, the short-term rate is a function i(t) such that

t 2

∫tis()sd

e^1

represents the interest earned over the interval (t 1 ,t 2 ).

We can now examine these valuation formulas in the limiting case

where the interval between two coupon payments goes to zero. This

means that coupon payments are replaced by a continuous stream of

cash flows with rate c(s). As discussed in Chapter 15 on arbitrage pric-

ing, a continuous cash flow rate means that

t 2

C = ∫c s()sd

t 1

is the cash received in the interval (t 1 ,t 2 ). To gain a better understanding

of these valuation relationships, let’s now explicitly compute the present

value of a continuous cash-flow rate c(s). We will arrive at the formula

for the present value of a known, deterministic continuous cash flow

rate c(t) in two different ways. We can thus illustrate in a simple context

two lines of reasoning that will be widely used later.