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# Actsc 445 spring 2011 discrete interest rate models 4

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Actsc 445 (Spring 2011) Discrete Interest Rate Models 4/ 12 4 / 12

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Bonds with Embedded Options Pricing a Callable Bond - Method 1 For t = T - 1 , T - 2 , . . . , 1 , Steps: Compute B ( t , n ) and E ( t , n ) . H ( t , n ) = 1 1 + i ( t , n ) [ q ( t , n ) · V ( t + 1 , n + 1 ) + ( 1 - q ( t , n )) · V ( t + 1 , n )] . Compute V ( t , n ) = max ( E ( t , n ) , H ( t , n )) . V ( 0 , 0 ) = H ( 0 , 0 ) gives the price of the call option at time 0. In-class example. Actsc 445 (Spring 2011) Discrete Interest Rate Models 5/ 12 5 / 12
Bonds with Embedded Options Pricing a Callable Bond - Method 2 (Direct Computation) Define: V ( t , n ) = min ( Strike Price , PV(Future Cash Flows) ) It is the value of the callable bond at ( t , n ) (immediately after the coupon is paid). We use backward recursion to compute V ( t , n ) , starting with V ( T , n ) = F . Actsc 445 (Spring 2011) Discrete Interest Rate Models 6/ 12 6 / 12

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Bonds with Embedded Options Pricing a Callable Bond - Method 2 (Direct Computation) For t = T - 1 , T - 2 , . . . , 1, Steps: PV(Future Cash Flows) = 1 1 + i ( t , n ) q ( t , n ) · [ V ( t + 1 , n + 1 ) + C ( t + 1 , n + 1 )] +( 1 - q ( t , n )) · [ V ( t + 1 , n ) + C ( t + 1 , n )] . Compute V ( t , n ) = min ( Strike Price , PV(Future Cash Flows) ) . V ( 0 , 0 ) gives the price of the callable bond at time 0 and is computed as the time 0 present value of future cash flows, that is, V ( 0 , 0 ) = 1 1 + i ( 0 , 0 ) q ( 0 , 0 ) · [ V ( 1 , 1 ) + C ( 1 , 1 )] + [ 1 - q ( 0 , 0 )] · [ V ( 1 , 0 ) + C ( 1 , 0 )] Actsc 445 (Spring 2011) Discrete Interest Rate Models 7/ 12 7 / 12
Bonds with Embedded Options Putable Bond The bond holder has the right to sell the bond back to the issuer before maturity.

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• Spring '09
• ChristianeLemieux
• callable bond

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