 # HullOFOD11eSolutionsCh32.pdf - CHAPTER 32 No-Arbitrage...

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CHAPTER 32No-Arbitrage Models of the Short RatePractice Questions32.1Equilibrium models usually start with assumptions about economic variables and derive thebehavior of interest rates. The initial term structure is an output from the model. In a no-arbitrage model, the initial term structure is an input. The behavior of interest rates in a no-arbitrage model is designed to be consistent with the initial term structure.32.2No. The approach in Section 32.2 relies on the argument that, at any given time, all bond pricesare moving in the same direction. This is not true when there is more than one factor.32.3Using the notation in the text,3s,1T,100L,87K, and2 0 1 12 0 10 0151(1)0 0258860 120 1Pee      From equation (31.6),(0 1)0 94988P,(0 3)0 85092P, and1 14277h so that equation(31.20) gives the call price as call price is1000 85092(1 14277)870 94988(1 11688)2 59NN  or \$2.59.32.4As mentioned in the text, equation (32.10) for a call option is essentially the same as Black’smodel.By analogy with Black’s formulas corresponding expression for a put option is(0)()(0)()PKPT NhLPs Nh In this case, the put price is870 94988( 1 11688)1000 85092( 1 14277)0 14NN    Since the underlying bond pays no coupon, putcall parity states that the put price plus thebond price should equal the call price plus the present value of the strike price. The bond priceis 85.09 and the present value of the strike price is870 9498882 64 . Putcall parity istherefore satisfied:82 642 5985 090 1432.5As explained in Section 32.2, the first stage is to calculate the value ofrat time 2.1 yearswhich is such that the value of the bond at that time is 99. Denoting this value ofrbyr, wemust solve(2 1 2 5)(2 1 3 0)2 5 (2 1 2 5)102 5 (2 1 3 0)99BrBrAeAe        where theAandBfunctions are given by equations (31.7) and (31.8). In this case,A(2.1, 2.5) = 0.999685,A(2.1,3.0) = 0.998432,B(2.1,2.5) = 0.396027, andB(2.1, 3.0) =0.88005, and Solver shows that065989.0*r. Since434745.2)5.2,1.2(5.2*)5.2,1.2(rBeAand

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Term
Fall
Professor
IsaacOtchere
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Strike price, 81 88 32 11
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