《金融期货与期权》（英文版） Chapter 27 Credit derivatives

27.1 Chapter 27 Credit derivatives Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.1 Chapter 27 Credit Derivatives

27.2 Credit Default Swap Company a buys default protection from b to protect against default on a reference bond issued by the reference entity, C a makes periodic payments to B In the event of a default by c A has the right to sell the reference bond to b for its face value, or B pays a the difference between the market value and the face value Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.2 Credit Default Swap • Company A buys default protection from B to protect against default on a reference bond issued by the reference entity, C. • A makes periodic payments to B • In the event of a default by C – A has the right to sell the reference bond to B for its face value, or – B pays A the difference between the market value and the face value

27,3 CDS Structure go bps per year Default Default Protection Protection Buyer, A Seller. B Payment if default by reference entity, C Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.3 CDS Structure Default Protection Buyer, A Default Protection Seller, B 90 bps per year Payment if default by reference entity,C

27,4 CDS Final Payments Notation Face value of bond. notional value of cds A(t): Accrued interest on bond per s of principal at time R: Recovery rate, market price as a percent of face value plus accrued interest CDS payment rate per year. Annual payment SL t Time since last CDs payment A pays tsl and b pays L-RL[1+ A(tI Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.4 CDS Final Payments Notation: L: Face value of bond, notional value of CDS A(t): Accrued interest on bond per $ of principal at time t R: Recovery rate, market price as a percent of face value plus accrued interest s: CDS payment rate per year. Annual payment = sL : Time since last CDS payment A pays sL and B pays L - RL[1 + A(t)]

27.5 Sample Quotes ( jan 2001) Company Rating 3yr 5 avr 10yr oyota Aa/AAA16/2420/3026/3732/53 Merrill Lynch Aa3/AA 21/4140/5541/83 56/96 Ford A+/A 59/8085/10095/136118/159 Enron Baa1/BBB+105/125115/135117/158182/233 ssan Ba1/BB+115/145125/155200/230244274 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.5 Sample Quotes (Jan 2001) Company Rating 3yr 5yr 7yr 10yr Toyota Aa1/AAA 16/24 20/30 26/37 32/53 Merrill Lynch Aa3/AA- 21/41 40/55 41/83 56/96 Ford A+/A 59/80 85/100 95/136 118/159 Enron Baa1/BBB+ 105/125 115/135 117/158 182/233 Nissan Ba1/BB+ 115/145 125/155 200/230 244/274

CDS Valuation 27.6 T: Life of credit default swap Pi: Risk-neutral default probability density at time t u(t Present value of $1 per year on payment dates between time zero and time t e(t): Present value of an accrual payment at time t Present value of $l received at time t w: Total payments per year made by CDs buyer s: Value of w for which cds value is zero T: Risk-neutral probability of no credit event during the life of the swap A(t: Accrued interest on the reference obligation at time t as a percent of face value Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.6 CDS Valuation T : Life of credit default swap pi: Risk-neutral default probability density at time ti u(t): Present value of $1 per year on payment dates between time zero and time t e(t) : Present value of an accrual payment at time t v(t): Present value of $1 received at time t w : Total payments per year made by CDS buyer s : Value of w for which CDS value is zero  : Risk-neutral probability of no credit event during the life of the swap A(t): Accrued interest on the reference obligation at time t as a percent of face value

PV of cDs Payments per S1 of27. Notional If default event occurs at t< Pv of payments is wu(t+e(t)] If no default event, PV of payments is Expected Pv is ∑p[(t)+(+)+7(T Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull PV of CDS Payments per $1 of 27.7 Notional • If default event occurs at t < T, PV of payments is • If no default event, PV of payments is • Expected PV is = + +  n i w pi u t i e t i w u T 1 [ ( ) ( )] ( ) w[u(t) + e(t)] wu(T)

PV of CDs Costs per $l of 2.8 Notional Principal If default event occurs at t<T cost is 1-[1+A(t)R]=1-R-A(t)R Expected cost is ∑[1-R-4(4)Rp() Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull PV of CDS Costs per $1 of 27.8 Notional Principal • If default event occurs at t < T cost is • Expected cost is A t R R A t R ˆ ( ) ˆ ] 1 ˆ 1−[1+ ( ) = − − ] ( ) ˆ ( ) ˆ [1 1 i i n i i  R A t R p v t = − −

27.9 Value of cds to buyer Value is expected PV of payments less expected Pv of costs ∑[1-R-4(t1)RPv(t) i=1 -w∑p[()+e()+7(T Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.9 Value of CDS to Buyer • Value is expected PV of payments less expected PV of costs [ ( ) ( )] ( ) ] ( ) ˆ ( ) ˆ [1 1 1 w p u t e t u T R A t R p v t n i i i i n i i i i − + +  − −   = =

27.10 CDS Rate CDS rate sets value to zero ∑[1-R-A(1)Rpv(t i=1 S ∑P[(t)+e()+(m Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 27.10 CDS Rate CDS rate sets value to zero   = = + +  − − = n i i i i n i i i i p u t e t u T R A t R p v t s 1 1 [ ( ) ( )] ( ) ] ( ) ˆ ( ) ˆ [1