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【正文】 TTTr T tT T T TKSV K e K VKSV S K e S V???? ? ? ? ??? ? ? ? ? ? ?Namely, when t=T ? ?11P r ob ( ) ( ) 1TTVV? ? ? ? Proof of Theorem (left side) ?If the American put option P is early exercised at time , then ?By Theorem , ()tT????()12( ) , ( ) ( )rtV C K eV K S S???? ? ???? ? ?? ? ? ?? ?11P r ob ( ) ( ) 1VV??? ? ? ? Proof of Theorem (left side) ?According to the arbitragefree principle and Theorem , there must be ?That is, ?The Theorem has been proved. 12( ) ( ) , ttVV? ? ? .t t tC K P S? ? ? Theorem ?Let be the price of a European call option with the strike price K. ?For with the same expiration date, ()tcK 12( ) , ( )ttc K c K122 1 1 2 0 ( ) ( ) .ttKKc K c K K K??? ? ? ? Financial Meaning of Theorem ?For 2 European call options with the same expiration date, ?the option with strike price ,leaves its holder profit room and is therefore priced , ?the difference between the two options shall not exceed the difference between the strike prices. Proof of Theorem (left side) ?Leave the right side part to reader ?Construct two portfolios at t: ?when t=T: 1 1 1 2 2 2( ) , ( ) c K K c K K? ? ? ? ? ?( ) ( )1 1 1 1 1()2 2 2( ) ( ) ( ) ,( ) ( )r T t r T tT T Tr T tTTV c K K e S K K eV S K K e? ? ???? ? ? ? ? ?? ? ? ? Proof of Theorem (left side) ?Case 1 1TSK?()11()22( ) ( 1 )( 1 ) ( ) ,r T tTTr T tTTV S K eS K e V??? ? ? ?? ? ? ? ? Proof of Theorem (left side) ?Case 2 So, 21TK S K??()11()22( ) , ( ) ( 1 )r T tTr T tTTV K eV S K e????? ? ? ?()1 2 1 2 2()12( ) ( ) ( ) ( ) ,( ) ( 1 ) 0.r T tT T Tr T tV V K K e K SK K e??? ? ? ? ? ? ?? ? ? ? Proof of Theorem (left side) ?Case 3 ?Thus, when t=T ?By Theorem Arbitrage Free Principle, for 0tT, 2TSK? ( ) ( )1 1 2 2( ) ( )r T t r T tTTV K e K e V??? ? ? ? ?? ?12P r ob ( ) ( ) 1TTVV? ? ? ?12( ) ( ) .ttVV? ? ? Theorem ?For two European put options with the same expiration date, if then 12 , KK? 1 2 1 20 ( ) ( ) .ttp K p K K K? ? ? ? Theorem ?European call (put) option price is a convex function of K, ., ( ) ( ( ) ) ttc K p K1 2 1 21212 ( 1 ) , ( 0 1 ) , ( ) ( ) ( 1 ) ( ) ,( ) ( ) ( 1 ) ( ) .t t tt t tK K K K Kc K c K c Kp K p K p K???? ? ?????? ? ? ? ? ??? ? ?? ? ? Proof of Theorem ?Only prove the first one, the second one left to the reader ?Construct two portfolios at t=0 ?On the expiration date t=T, ?Discuss in 4 cases 1 1 2 2( ) ( 1 ) ( ) , ( ) . c K c K c K ?? ? ?? ? ? ? ? ?1 1 22( ) ( ) ( 1 ) ( ) ,( ) ( ) .T T TTTV S K S KV S K ??? ???? ? ? ? ? ?? ? ? Proof of Theorem ?Case 1 ?Case 2 Therefore 1TSK? 12( ) ( ) 。TTVV? ? ? Proof of Theorem ?Case 3 ?Case 4 2 TK S K ???122( ) ( 1 ) ( ) 0( ) 。 Also consider buying 1 European option to purchase shares of the same stock at strike price K on the expiration
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