L7 - Lecture 7 Introduction to Fixed-income Market The...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 7 Introduction to Fixed-income Market The fixed-income 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 fixed-income derivatives are also more subtle and complex. We will introduce some basic concepts, models and derivatives in the next few lectures. 7.1 Zero-coupon 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 zero-coupon 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 zero-coupon bond with maturity τ . Sometimes we simply call a zero coupon bond with maturity τ a τ-bond . 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 zero-coupon 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.

Page1 / 11

L7 - Lecture 7 Introduction to Fixed-income Market The...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online