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LectureNotes08Print - IEOR E4731 Credit Derivatives Lecture...

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IEOR E4731: Credit Derivatives Lecture 08: CDS Indices, Forwards, and Swaptions Notes originally written by Prof. Rama Cont Instructor: Xuedong He Spring, 2012 1 / 29
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Cash Flow Structure of CDS Portfolio Indices 2 / 29
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Value of the protection leg I Let T be the maturity of the index, t be the current time, C be the coupon rate, M be the number of underlying credits alive at t . Without loss of generality, we assume the notional of the contract remained at t is 1. I The protection leg is the summation of the protection legs of each underlying credits with maturity T and notional 1 /M . I Thus, the value of the protection leg is 1 M · M X i =1 - (1 - R m ) Z T t B ( t, u ) ∂u Q m ( t, u ) du . 3 / 29
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Value of the premium leg I Let RPV01 m ( t, T ) be the risky PV01 of credit m , i.e., the value at t of the premium leg payment starting from t to T with spread 1: RPV01 m ( t, T ) = n X i =1 ( T i - T i - 1 ) B ( t, T i ) Q m ( t, T i ) - Z T i T i - 1 B ( t, u )( u - T i - 1 ) ∂Q m ∂u ( t, u ) du. I Then, the value of the premium leg of the index is 1 M M X i =1 C · RPV01 m ( t, T ) . 4 / 29
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Value of the index I The value of a short protection position is 1 M M X i =1 C · RPV01 m ( t, T ) - - (1 - R ) Z T t B ( t, u ) ∂u Q m ( t, u ) du I Let S m ( t, T ) be the CDS spread of credit m I Then, the value of a short protection position is 1 M M X i =1 ( C - S m ( t, T )) RPV01 m ( t, T ) . 5 / 29
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CDS Index Spread Curve I To simplify things, the market creates an index curve as though the index was a single name CDS with a spread S I ( t, T ) . I Market convention dictates that the index curve used to value an index has a flat term structure and the recovery rate is 40%. I The upfront value of the CDS index can be calculated as V I ( t ) = ( S I ( t, T ) - C ) · RPV01 I ( t, T ) where RPV01 I ( t, T ) is the risky PV01 of the CDS index as though the index was a single name CDS. 6 / 29
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CDS Index Spread Curve (Cont’d) 7 / 29
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CDS Index Spread Curve (Cont’d) I Equating the intrinsic upfront and the index curve upfront, we get 1 M M X i =1 ( C - S m ( t, T )) RPV01 m ( t, T ) = ( S I ( t, T ) - C ) · RPV01 I ( t, T ) I Sometimes, we can make the following approximation RPV01 I ( t, T ) 1 M M X i =1 RPV01 m ( t, T ) I Then, S I ( t, T ) M i =1 RPV01 m ( t, T ) · S m ( t, T ) M i =1 RPV01 m ( t, T ) 8 / 29
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CDS Index Spread Curve (Cont’d) 9 / 29
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