hw13-2 - 284 4 Interest Rate Derivative Securities and...

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Unformatted text preview: 284 4 Interest Rate Derivative Securities and using the final condition, we arrive at b ( t ) = kZ r ‡ 1- e- r ( T- t ) · , which is nonnegative for t ≤ T . Define B c ( S, t ) = B c ( S, t )- b ( t ). Thus B c ( S, t ) = B c ( S, t ) + b ( t ). Substituting this relation into the original problem yields ∂ B c ∂t + db dt + 1 2 σ 2 S 2 ∂ 2 B c ∂S 2 + ( r- D ) S ∂ B c ∂S- r B c- rb + kZ = 0 , ≤ S, ≤ t ≤ T, B c ( S, T ) + b ( T ) = max( Z, nS ) , ≤ S. Noticing that b ( t ) is the solution of the ODE problem above, we have ∂ B c ∂t + 1 2 σ 2 S 2 ∂ 2 B c ∂S 2 + ( r- D ) S ∂ B c ∂S- r B c = 0 , ≤ S, ≤ t ≤ T, B c ( S, T ) = max( Z, nS ) , ≤ S. The solution of this problem is B c ( S, t ) = e- r ( T- t ) Z ∞ max( Z, nS ) G ( S , T ; S, t ) dS ≥ e- r ( T- t ) Z ∞ nS G ( S , T ; S, t ) dS = e- r ( T- t ) nSe ( r- D )( T- t ) = nSe- D ( T- t ) ....
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hw13-2 - 284 4 Interest Rate Derivative Securities and...

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