This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 7 Introduction to Fixedincome Market The fixedincome market is an important part of the global financial market in which various interest rate securities, such as bonds and their derivatives, are traded. The total volume of fixed income securities traded in the market is much greater than that of equities such as common stocks. Mathematical models for fixedincome derivatives are also more subtle and complex. We will introduce some basic concepts, models and derivatives in the next few lectures. 7.1 Zerocoupon bonds, yields and forward rates Assume the basic setting in Section 2.1 with a sample space Ω, an underlying probability measure P , a filtration FF , a finite horizon T , and with the following extension: the bank account B is assumed to be random and predictable with interest rate r ( t ) > 0 for all t = 1 ,...,T . This means that the interest rate r ( t ) for borrowing or lending over the period ( t 1 ,t ] is known at time t 1. The process B is usually taken as a num´ eraire , i.e. the unit of an account in which other assets are denominated. r ( t ) is called the spot rate or short rate. • Various bonds are considered as risky securities. One of them is a collection of zerocoupon or discount bonds, denoted by { B ( t,τ ) : 0 ≤ t ≤ τ, τ = 1 ,...,T } , where B ( t,τ ) represents the time t price of a zerocoupon bond with maturity τ . Sometimes we simply call a zero coupon bond with maturity τ a τbond . Assume that for each τ , the process B ( · ,τ ) is positive and adapted to FF , in particular, B ( τ,τ ) = 1 at par (the nominal value is $1 at maturity). On the other hand, for each t , the collection { B ( t,τ ) , t < τ ≤ T } is called the time t term structure of zerocoupon bond prices. Hence we are dealing with a process B ( · , · ) with double indices, which makes the analysis considerably more challenging. Now we consider a couple of other term structures equivalent to B ( t, · ), and we assume t ≤ τ . • Let Y ( t,τ ) be the constant interest rate (or return) at which B ( t,τ ), when compounded during ( t,τ ], would reach $1 at time τ , called the yield to maturity , i.e. B ( t,τ ) [1 + Y ( t,τ )] τ t = 1 , (7.1) or equivalently, Y ( t,τ ) = [ B ( t,τ )] 1 τ t 1 . (7.2) In particular, Y ( t 1 ,t ) = r ( t ), the spot rate at one period before maturity. For each t , the collection { Y ( t,τ ) , t < τ ≤ T } is called the time t term structure of interest rates or yield curve . The two term structures B ( t, · ) and Y ( t, · ) are equivalent. 1 • Let f ( t,τ ) be the “short rate” such that (i) it is locked into at time t ; (ii) it is applied to the period ( τ,τ + 1]; (iii) it is associated with a ( τ + 1)bond....
View
Full
Document
This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff

Click to edit the document details