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

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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....
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## This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.

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

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