# ex_2 - T . The payo² o± the Gamma swap can thus be...

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ORIE 569 Problem set 2 Feb 6, 2008 Homework must be turned in by Monday, Feb 18 at 12 noon. 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. Let S t be a stock price process. For a plain vanilla option f ( S T ) defned in terms o± a twice continuously di²erentiable ±unction f , prove the Breeden-Litzenberger formula , f ( S T ) = f ( S 0 ) + f 0 ( S 0 )( S T S 0 ) + Z S 0 0 ( K S T ) + f 00 ( K ) dK + Z S 0 ( S T K ) + f 00 ( K ) dK, which yields a static hedge o± f ( S T ) in terms o± cash, an at-the-money ±orward contract, and port±olios o± put and call options. Exercise 2: A Gamma swap or Entropy swap on S has the payo² N X i =1 S t i (log S t i log S t i 1 ) 2 , where 0 = t 0 < t 1 < . . . t N = T are the daily closing prices o± S until the maturity
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Unformatted text preview: T . The payo² o± the Gamma swap can thus be aproximated by V ( T ) = Z T S t d [log S, log S ] t . Use similar arguments as in class to derive the ±ollowing representation ±or V ( T ), which can be interpreted as a hedge ±or the Gamma swap in terms o± an at-the-money ±orward contract, a dynamic trading strategy, and port±olios o± put and call options: V ( T ) = 2(log S +1)( S T − S ) − Z T 2(log S t +1) dS t + Z S ( K − S T ) + 2 K dK + Z ∞ S ( S T − K ) + 2 K dK. Exercise 3: Exercise 7.1 on page 332 in Shreve’s book. Exercise 4: Exercise 8.3 on page 370 in Shreve’s book. 1...
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## This note was uploaded on 08/05/2008 for the course ORIE 569 taught by Professor Alexanderschied during the Spring '08 term at Cornell.

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