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Unformatted text preview: Lecture 8 Some Spot Rate and Term Structure Models 8.1 Spot rate models Different expressions of the drift and volatility in the SDE (7.14) r ( t + 1) = ( t,r ( t )) + ( t,r ( t )) t lead to various spot rate models. In this lecture, we first consider timehomogeneous SDEs in which ( t,r ( t )) = ( r ( t )) and ( t,r ( t )) = ( r ( t )), i.e. the drift and volatility do not involve time t explicitly. We then study an inhomogeneous SDE in HullWhite model, along with Markov chains represented by binomial or trinomial trees. 8.1.1 Vasicek and CIR models Consider the following special cases in which a,b, are all positive constants, and is a constant with 0 1. Example 8.1 (Vasicek model) ( t,r ( t )) = a [ b r ( t )], ( t,r ( t )) = . Example 8.2 [CoxIngersollRoss (CIR) model] ( t,r ( t )) = a [ b r ( t )], ( t,r ( t )) = p r ( t ). Example 8.3 (a more general class) ( t,r ( t )) = a [ b r ( t )], ( t,r ( t )) = [ r ( t )] . A common feature of these models is the mean reversion property: the spot rate r tends to decrease if r > b and increase if r < b . An advantage of CIR over Vasicek is its capability of enforcing positive values on r . To formulate the question more precisely, assume r (1) = x > 0 and let x = inf { 2 t T : r ( t ) } . (8.1) The question is: under what conditions on the parameters a,b and in the SDEs for Vasicek and CIR respectively, Q ( x T ) = 0 will be satisfied? Note that Q ( x T ) = 0 and P ( x T ) = 0 are equivalent since Q is an EMM. Both imply that the values of r remain positive in the whole process. It turns out that those conditions needed in CIR are quite mild and reasonable, but the conditions needed in Vasicek are too strict to be realistic. Further discussion on various modeling issues will appear in continuoustime finance. 1 8.1.2 HullWhite model In a very general form, HullWhite model can be expressed via the SDE r ( t + 1) = a ( t ) [ b ( t ) r ( t )] + ( t ) [ r ( t )] t , (8.2) where 0 is still a constant, but a ( t ), b ( t ) and ( t ) are positivevalued deterministic functions. These functions greatly enhance modeling flexibility, e.g. the initial term structure can be incor porated into a ( t ) and b ( t ). The condition on { t } is often relaxed in this model, e.g. t s are independent with Q ( t = 1) = q ( t ) < 1 . Under what conditions on functions a , b and , an underlying MC X can be represented by a recombining tree such that r is defined by r ( t + 1) = g t ( n t ) (Q1 in Lecture 7)? A recombining tree requires that starting from any node, a updown combination ( t = 1 and t +1 = 1) and a downup combination ( t = 1 and t +1 = 1) should merge at the same node. In other words, starting from r ( t ), these two combinations must arrive at the same value of r ( t + 2). Standard calculation turns this into the condition...
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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.
 Spring '08
 Staff

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