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Midterm Solutions_2007

# Midterm Solutions_2007 - 1 = 0.25 0 = Cu Cd 0 10 = = 0.25...

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δ 1 = C u - C d su - sd = 0 - 10 120 - 80 = - 0 . 25 0 . 25 δ 0 = 1 1 + r ( C u - δ 1 su ) = ... = 28 . 5714 C (0) = δ 0 · 1 + δ 1 · s 0 = ... = 3 . 5714 LHS : ( S ( T ) - K ) + + K RHS : ( K - S ( T )) + + S ( T ) LHS : S ( T ) - K + K = S ( T ) RHS : S ( T ) LHS : K RHS : K - S ( T ) + S ( T ) = K S ( T ) = K r = 0 c (0) = p (0) S (0) = K 0 = S (0) - K C (0) - P (0) S (0) - Ke - rT = 0 P (0) = 10 C (0) = 10 f ( t, x ) = x α dS = rSdt + σ SdW * df = f t dt + f x dX + 1 2 f xx dX · dX df ( S ) = α S α - 1 dS + 1 2 α ( α - 1) S α - 2 σ 2 S 2 dt dS α = S α ( α r + 1 2 α ( α - 1) σ 2 ) dt + ασ dW *

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f ( t, x ) = x 4 - A t 0 x 2 ( u ) du dW = 0 · dt + 1 · dW df = f t dt + f x dX + 1 2 f xx dX · dX df ( W ) = - AW 2 dt + 4 W 3 dW + 1 2 12 W 2 dt df ( W ) = (6 - A ) W 2 dt + 4 W 2 dW f ( t, W ) A = 6 F ( t ) = s 0 e r ( T - t ) F (0) = S (0) < S (0) e rT S ( T ) - F (0) : - S ( T ) : S (0) e rT : - - - - -- - - - - - - - - - - - - - - - - - = S (0) e rT - F (0) : Δ F = dC dS = 1 δ = ( δ 0 , δ 1 ) = ( - Ke - rT , 1) . C ( t, S ( t )) = δ 0 · e rt + δ 1 · S ( t ) = S ( t ) - Ke - r ( T - t ) E * [ C ] = E * [ S ( T ) - Ke - rT ] = E * [ S ( T )] - Ke - rT = S (0) - Ke - rT = C (0 , S (0)) S C = S - Ke - r ( T - t ) C t = - rKe - r ( T - t ) C ss = 0
p * = (1 + r ) - d u - d = 0 . 75 C (0) = E * [ C ] = 100( p * ) 4 1 . 05 4 = 28 . 699 δ = C u - C d S u - S d δ i C ( i ) i C ( i ) = 1 1 . 05 ( p * C ( i + 1)) δ i = C ( i + 1) - 0 su i +1 - sdu i = 1 s · C ( i + 1)
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