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Unformatted text preview: Chapter 21 The BlackScholes Equation Question 21.1. If V (S, t) = e r(T t) then the partial derivatives are V S = V SS = 0 and V t = rV . Hence V t + (r ) SV S + S 2 2 V SS / 2 = rV . Question 21.2. If V (S, t) = AS a e t then V t = V , V S = aS a 1 e t = aV/S , and V SS = a (a 1 ) S a 2 e t = a (a 1 ) V/S 2 . Therefore the left hand side of the BlackScholes equation (21.11) is V t + (r ) V S S + V SS S 2 2 / 2 rV = r + (r ) a + 2 2 a (a 1 ) V. (1) We can rewrite the coefFcient of V as + (r ) a + 2 2 a (a 1 ) = 2 2 a 2 + r 2 2 a + r. (2) rom the quadratic formula, this has roots a = r 2 2 2 r r 2 2 2 4 2 2 ( r) 2 . (3) Simplifying, a = 1 2 r 2 s r 2 1 2 2 + 2 (r ) 2 . (4) Note, for a given , these are the only values for a that will satisfy the PDE. Question 21.3. If V (S, t) = e r(T t) S a exp (( a (r ) + 1 2 a (a 1 ) 2 ) (T t) ) ,wehave V (S, T ) = S a T ,hence the boundary condition is satisFed. Note that V is of the form AS a e t , where = r a (r ) 1 2 a (a 1 ) 2 . The previous problems result shows must solve a = 1 2 r 2 s r 2 1 2 2 + 2 (r ) 2 . (5) 264 Chapter 21 The BlackScholes Equation Letting k = 1 2 r 2 , we have to check a ? = k r k 2 + 2 (r ) 2 . (6) This is equivalent to checking k 2 + 2 (r ) 2 ? = (a k) 2 . (7) Expanding, this becomes 2 (r ) 2 ? = a 2 2 a 1 2 r 2 . (8) Solving for , ? = r 2 a 2 2 + a 2 2 (r ) = r a (r ) 2 2 a (a 1 ) (9) which is conFrmed. One could also do this as a partial derivative exercise. Question 21.4. DeFning V (S, t) = Ke r(T t) + Se (T t) wehave V t = rKe r(T t) + Se (T t) , V S = e (T t) and V SS = 0. The BlackScholes equation is satisFed for V t + (r )V S S + V SS S 2 2 / 2 is rKe r(T t) + Se (T t) + (r ) e (T t) S (10) = r Ke r(T t) + Se (T t) = rV. (11) This also follows from the result that linear combinations of solutions of the PDE are also solutions. The boundary condition is V (S, T ) = K + S T , i.e. we receive one share and K dollars. Similarly, a long forward contract with value Se (T t) Ke r(T t) will solve the PDE....
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 Spring '06
 Adam

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