hw6-1 - 2 Basic Options 87 P (S, t) and Sf (t) are known...

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Basic Options 87 where P ( S, t ), ∂P ( S, t ) ∂S and S f ( t ) are known functions obtained from the solution of the American put option problem. Let P σ = ∂P ∂σ and P D 0 = ∂P ∂D 0 . For them, the formulations are ∂P σ ∂t + 1 2 σ 2 S 2 2 P σ ∂S 2 + ( r - D 0 ) S ∂P σ ∂S - rP σ + σS 2 2 P ∂S 2 = 0 , S f ( t ) S, 0 t T, P σ ( S, T ; r, D 0 , σ ) = 0 , S f ( T ) S, P σ ( S f ( t ) , t ; r, D 0 , σ ) = 0 , 0 t T and ∂P D 0 ∂t + 1 2 σ 2 S 2 2 P D 0 ∂S 2 + ( r - D 0 ) S ∂P D 0 ∂S - rP D 0 - S ∂P ∂S = 0 , S f ( t ) S, 0 t T, P D 0 ( S, T ; r, D 0 , σ ) = 0 , S f ( T ) S, P D 0 ( S f ( t ) , t ; r, D 0 , σ ) = 0 , 0 t T, respectively. Here 2 P ∂S 2 , ∂P ∂S and S f ( t ) are known functions obtained from the solution of the American put option problem. 57. De±ne α ± = 1 σ 2 - µ r - D 0 - 1 2 σ 2 ± s µ r - D 0 - 1 2 σ 2 2 + 2 σ 2 r , where r 0 and D 0 0. a) Show that α + 1, α - 0 and - ( r - D 0 ) α ± + r 0. b) Show that 1 / (1 - 1 + ) max(1 , r/D 0 ) and 1 / (1 - 1 - ) min(1 , r/D 0 ). Solution : a) Because α + 1 is equivalent to the following inequalities: 1 σ 2 - µ r - D 0 - 1 2 σ 2 + s µ r - D 0 - 1 2 σ 2 2 + 2 σ 2 r 1 , - µ r - D 0 - 1 2 σ 2 + s µ r - D 0 - 1 2 σ 2 2 + 2 σ 2 r σ 2 , s µ r - D 0 - 1 2 σ 2 2 + 2 σ 2 r r - D 0 + 1 2 σ 2 , ( r - D 0 ) 2 - ( r - D 0 ) σ 2 + σ 4 / 4 + 2
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hw6-1 - 2 Basic Options 87 P (S, t) and Sf (t) are known...

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