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Unformatted text preview: 280 4 Interest Rate Derivative Securities ˜ λ r = ( u λw ) . Therefore the equation for any convertible bond is ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 + ρσSw ∂ 2 V ∂S∂r + 1 2 w 2 ∂ 2 V ∂r 2 + ( rS D ( S, t )) ∂V ∂S + ( u λw ) ∂V ∂r rV + kZ = 0 . In this equation, only the market price of risk for the interest rate needs to be given. 26. Formulate the twofactor convertible couponpaying bond problem as a linear complementarity problem. Solution : Because for any derivative securities dependent on S, r, T and paying coupons with a rate k , the partial differential equation is ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 + ρσSw ∂ 2 V ∂S∂r + 1 2 w 2 ∂ 2 V ∂r 2 +( r D ) S ∂V ∂S + ( u λw ) ∂V ∂r rV + kZ = 0 and the constraint and the payoff for convertible bonds are V ( S, r, t ) ≥ nS, and V ( S, r, T ) = max( nS, Z ) . Therefore the linear complementarity problem for the twofactor convert ible bond is µ ∂V ∂t + L S r V + kZ ¶ ( V nS ) = 0 , ≤ S, r l ≤ r ≤ r u , t ≤ T, ∂V ∂t + L S r V + kZ ≤ , ≤ S, r l ≤ r ≤ r u , t ≤ T, V nS ≥ , ≤ S, r l ≤ r ≤ r u , t ≤ T, V ( S, r, T ) = max( nS, Z ) ≥ nS, ≤ S, r l ≤ r ≤ r u , where L S r = 1 2 σ 2...
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 Spring '10
 Zhu
 Derivative, Convertible bond, Complementarity theory, free boundary

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