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Unformatted text preview: 2 Basic Options 79 boundary to appear while in the region ( D β/r, ∞ ) it is possible for a free boundary to appear. If D β/r < β , then in the entire region ( β, ∞ ), it is possible for a free boundary to appear. Therefore if η < max( β, D β/r ) = β max(1 , D /r ) , then it is impossible for a free boundary to appear and if η > max( β, D β/r ) = β max(1 , D /r ) , then it is possible for a free boundary to appear. 53. Suppose r, D , and σ are constant. a) Derive the put–call symmetry relations. b) Explain the financial meaning of the symmetry relation. c) Explain how to use these relations when we write codes if a code for put options is quite a different from a code for call options. Solution : a) As we know, the price of an American put option is the solution of the following LC problem: min µ ∂P ∂t L S P, P ( S, t ) max( E S, 0) ¶ = 0 , ≤ S, t ≤ T, P ( S, T ) = max( E S, 0) , ≤ S, where L S = 1 2 σ 2 S 2 ∂ 2 ∂S 2 + ( r D ) S ∂ ∂S r. Let ζ = E 2 S , C ( ζ, t ) = EP ( S, t ) S . Because E S max( E S, 0) = max( ζ E, 0) , for C ( ζ, t ) the payoff and constraint are max( ζ E, 0). Noticing ∂P ∂t = S E ∂C ∂t , ∂P ∂S = 1 E • C + S ∂C ∂ζ µ E 2 S 2 ¶‚ = 1 E µ C ζ ∂C ∂ζ ¶ , ∂ 2 P ∂S 2 = ζ 3 E 3 ∂ 2 C ∂ζ 2 , we have 80 2 Basic Options ∂P ∂t + 1 2 σ 2 S 2 ∂ 2 P ∂S 2 + ( r D ) S ∂P ∂S rP = S E ‰ ∂C ∂t + 1 2 σ 2 ζ 2 ∂ 2 C ∂ζ 2 + ( D r ) ζ ∂C ∂ζ D C ¾ . Therefore the function C ( ζ, t ) is the solution of the following American call option problem: min µ ∂C ∂t L ζ C, C ( ζ, t ) max( ζ E, 0) ¶ = 0 , ≤ ζ, t ≤ T, C ( ζ, T ) = max( ζ E, 0) , ≤ ζ, where L ζ = 1 2 σ 2 ζ 2 ∂ 2 ∂ζ 2 + ( D r ) ζ ∂ ∂ζ D . Consequently, an American put problem can be converted into an American call problem. However in the two probloms, the state vari able and the parameters are different. From the definitions of L S and L ζ , we know that the volatilities of the put and 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 prob lem, respectively. In order to explain these facts, we express the depen dences of the options on interest rate and dividend yield explicitely....
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 Spring '10
 Zhu
 Math, Interest Rates, Option style, free boundary

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