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Unformatted text preview: ORIE 569
Problem set 7 April 9, 2008 Homework must be turned in by Thursday, April 24, at 10 a.m. Solutions turned in at a later time will not be graded. Place homework in the 569 homework box on the bridge between Upson and Rhodes.
Exercise 1. a) Show that the dierentia equations
CH = 2 2 C 2 + C 1; C (T; T ) = 0; A(T; T ) = 0; and where C H = @C (t; T )=@t and AH = @A(t; T )=@t, have the solutions
C (t; T ) = AH = C; sinh( (T t)) 1 cosh( (T t)) + 2 sinh( (T
h
1 t)) t))
i
: and A(t; T ) = 1 2 2
2 Here = p e 2 (T t) log cosh( (T t)) + 1 sinh( (T 2 2 + 2 2 . b) Use the dierential equations for A and C in Part a) to deduce that the CIR model e satises the HJM noarbitrage condition under P. That is, @C (t; T ) @C H (t; T ) @AH (t; T ) @C (t; T ) 2 ( Rt ) + Rt + = Rt C (t; T );
@T @T @T @T see lectures.
Exercise 2. Exercise 10.12 on page 459 in Steve Shreve's book. 1 ...
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 Spring '08
 ALEXANDERSCHIED

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