hw7-2 - 3 Exotic Options 129 = c ( S, t ; r, D ) c ( bE ) 2...

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Unformatted text preview: 3 Exotic Options 129 = c ( S, t ; r, D ) c ( bE ) 2 , t ; r, D + = e- r ( T- t ) f ( bE ) 2 e- ( D + )( T- t ) Ee- r ( T- t ) , 2 ( T- t ) = e- r ( T- t ) f B 2 l ( t ) S e- D ( T- t ) Ee- r ( T- t ) , 2 ( T- t ) ! = c B 2 l ( t ) S , t ; r, D . Consequently, we finally have c ( S, t ) = c ( S, t ; r, D )- B l ( t ) S 2( r- D- - 2 / 2) / 2 c B 2 l ( t ) S , t ; r, D . 5. Let P o ( S, t ) denote the price of an American up-and-out put option. Show that under the following transformation = E 2 S , C o ( , t ) = EP o ( S, t ) S , the new function C o ( , t ) represents the price of an American down-and- out call option. Based on this result, derive the symmetry relations be- tween American down-and-out call and up-and-out put options. Solution : As we know, the price of an American up-and-out put option is the solution of the following LC problem: min - P o t- L S P o , P o ( S, t )- max( E- S, 0) = 0 , S B u , t T, P o ( S, T ) = max( E- S, 0) , S B u , P o ( B u , t ) = 0 , t T, where L S = 1 2 2 S 2 2 S 2 + ( r- D ) S S- r. Because E S max( E- S, 0) = max( - E, 0) , for C o ( , t ) the payoff and constraint are max( - E, 0). Noticing P o t = S E C o t , P o S = 1 E C o + S C o - E 2 S 2 = 1 E C o- C o , 2 P o S 2 = 3 E 3 2 C o 2 , 130 3 Exotic Options we have P o t + 1 2 2 S 2 2 P o S 2 + ( r- D ) S P o S- rP o = S E C o t + 1 2 2 2 2 C o 2 + ( D- r ) C o - D C o . When S (0 , B u ), we have ( E 2 /B u , ) Therefore the function C o ( , t ) is the solution of the following American down-and-out call option problem: min - C o t- L C o , C o ( , t )- max( - E, 0) = 0 , E 2 /B u , t T, C o ( , T ) = max( - E, 0) , E 2 /B u , C o ( E 2 /B u , t ) = 0 , t T, where L = 1 2 2 2 2 2 + ( D- r ) - D . Consequently, an American up-and-out put problem can be converted into an American down-and-out call problem. However in the two probloms, the state variable and the parameters are different. From the definitions of L S and L , we know that the volatilities of the up-and-out put and down-and-out call problems are the same, but the interest rate and the dividend yield of the call problem are equal to the dividend yield and the interest rate of the put problem, respectively. Also, we can see that if the location of the upper barrier of the put problem is B u , then the location of the lower barrier of the call problem is E 2 /B u . In order to explain these facts, we express the dependences of the options on interest rate, dividend yield, the location of the barrier explicitely. Let P o ( S, t...
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hw7-2 - 3 Exotic Options 129 = c ( S, t ; r, D ) c ( bE ) 2...

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