Lecture_17 - 17 Spot Rate Models The purpose of this...

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1 17 Spot Rate Models The purpose of this chapter is to provide a brief explanation of an alternative approach to Heath, Jarrow, and Morton. This alternative approach is based on an exogenous specification of the spot rate process, instead of the entire forward rate curve. The problem with the spot rate model approach is that to match an initial zero-coupon bond price curve, the entire spot rate process must be inverted. Unfortunately, this inversion is computationally intensive and not always possible.
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2 A Bond Pricing 1 One-Factor Economy Given exogenously is the evolution of the spot interest rate process. In the notation of Chapter 4 this is written as > + + > + + = + + . 0 ) t s ( t q 1 y probabilit with ) d t s ; 2 t , 1 t ( d 0 ) t s ( t q y probabilit with ) u t s ; 2 t , 1 t ( u ) 1 t s ; 1 t ( r (17.1)
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3 To perform this analysis, we reparameterize this process as = + = + = + + . d t s 1 t s if ) t s ; t ( b ) t s ; t ( r u t s 1 t s if ) t s ; t ( a ) t s ; t ( r ) 1 t s ; 1 t ( r (17.2)
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4 In addition, this approach also assumes that we are given the unique pseudo probabilities π (t;s t ) for all t , s t such that τ = = T t 0 and t s all for 1 T t j ) t s ; j ( r 1 t E ~ ) t s ; T , t ( P (17.3) where ) ( t E ~ is expectation under the given pseudo probabilities. For convenience, the pseudo probabilities can be set equal to 1 / 2: . 1 T t 0 and t s all for ) 2 / 1 ( ) t s ; t (
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5 Because the pseudo probabilities are independent of any particular zero-coupon bond, from Chapter 9 we know that there are no arbitrage opportunities with respect to the largest trading strategy set Φ τ . The evolution of the zero-coupon bond price curve is given (in abstract form) by expression (17.3). Given this evolution, the same analysis as in Chapter 9 shows that the market is complete with respect to the trading strategy set involving all the zero-coupon bonds.
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6 EXAMPLE: BOND PRICE VALUATION.
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7 Figure 17.1: An Example of a One-Factor Spot Rate Process. Actual Probabilities Along Each Branch of the Tree 1.02 3/4 1/4 1.017606 1.022406 3/4 1/4 3/4 1/4 1.016031 1.020393 1.019193 1.024436 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 1.014918 1.018972 1.017857 1.022727 1.017155 1.021830 1.020543 1.026165 r(0) =
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8 Let the pseudo probabilities be π (t;s t ) = 1 / 2 for all s t and 0 t T τ - 1. The risk-neutral valuation analysis in Chapter 9
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Lecture_17 - 17 Spot Rate Models The purpose of this...

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