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ExCH09 - PRACTICE PROBLEMS CHAPTER 9 MODELS OF YIELD CURVE...

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PRACTICE PROBLEMS CHAPTER 9: MODELS OF YIELD CURVE AND THE TERM STRUCTURE Exercise 9.1: See slide 9- 25 of the Ch. 9 lecture notes. Exercise 9.2: See slide 9-47 of the Ch. 9 lecture notes. Also, find the par bond yield curve for this problem. Exercise 9.3: Consider the lognormal interest rate process model. Suppose the current one-period rate is 6%, and let u = 1.2, d = 0.9, and p = 0.5. (a) Find the term structure, i.e., the zero yield curve: y 1 , y 2 , and y 3 . (b) Find y 1 and y 2 if p = 0.6 (rather than 0.5). How has the term structure changed? (c) Returning to the case where p = 0.5, suppose a security pays $1 at date t = 2 in the “ uu ” state but pays $0 otherwise. Find the price of this security. Can you interpret this security in terms of options? [Hint: it might help to assume that bonds have a face value of $100.] SOLUTIONS Exercise 9.1: The risk-neutral probability is p = ( ) 1 + R d u d = 0.5555. The second bond has price P * = [ p 105 + (1 p )90]/(1.04) = 94.5513. Assuming that the actual probability is q = 0.6, we have, for bond 1, E q [ P T / P ] = qu + (1 – q ) d = 1.044, Var( R 1 ) = q (1 – q )( u d ) 2 = 0.001944, σ 1 = 0.04409, φ 1 = 1 ) 1 ( ] / [ σ R P P E T + = 0.09072 .
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