The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 112 Friday, March 12, 2004 12:39 PM

112 The Mathematics of Financial Modeling and Investment Management

df x()

d^2 fx

d--------------

()  dx 

f′′()x = ---------------- = -----------------------

dx^2 dx

Provided that the derivatives exist, this process can be iterated, pro-

ducing derivatives of any order. A derivative of order n is written in the

following way:

df n –^1 ()x

d------------------------

()x d fx  dxn –^1 

n

fn()= -----------------()= ----------------------------------

dxn dx

Application to Bond Analysis

Two concepts used in bond portfolio management, duration and con-

vexity, provide an illustration of derivatives. A bond is a contract that

provides a predetermined stream of positive cash flows at fixed dates

assuming that the issuer does not default nor prepay the bond issue

prior to the stated maturity date. If the interest rate is the same for each

period, the present value of a risk-free bond has the following expres-

sion:

C C CM+

V = ------------------+ ------------------+ ...+ --------------------, i = 1,...,N

( 1 + i)^1 ( 1 + i)^2 ( 1 + i)N

If interest rates are different for each period, the previous formula

becomes

C C CM+

V = ---------------------+ ---------------------+ ...+ ------------------------, i = 1,...,N

( 1 + i 1 )^1 ( 1 + i 2 )^2 ( 1 + iN)N

In Chapter 8, we introduce the concept of continuous compound-

ing. With continuous compounding, if the short-term interest rate is

constant, the bond valuation formula becomes^6

(^6) If the short-term rate is variable:

• ∫^1 is()sd –∫^2 is() sd –∫Nis()sd
  V = Ce^0 + Ce^0 + ...+ (CM+ )e^0