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【正文】 K KK S K? ? ??? ? ? ? ???? ? ? ??? Proof of Theorem cont. ? At t=T, ? and ? By Theorem ? . 12( ) ( ) , T T TV S V? ? ? ?? ? ? ?12P r ob ( ) ( ) P r ob 0 T TV V K S? ? ? ? ? ?12[ 0 , ] , ( ) ( ) , ttt T V V? ? ? ? ?() . r T tttc K e S???? Proof of Theorem cont. cont. ?Now consider a European call option c ?Since ?and ?By Theorem when tT ?. ?Together with last inequality, proved. ( ) ( ) 0 , TTV c S K ???? ? ? ?P r ob ( ) 0 P r ob 0 0 , TTV c S K? ? ? ?( ) 0 ,tVc ? ? Theorem ?For European Option pricing, there holds callput parity ()r T tt t tc K e p S??? ? ? Proof of Theorem ?2 portfolios when t=0 ?when t=T 12 ,rTc K e p S?? ? ? ? ? ?? ?? ?12( ) ( ) ( )( ) m a x , ,( ) ( ) ( )( ) m a x , .rTT T TTTT T TT T TV V c V K eS K K K SV V p V SK S S K S???? ? ?? ? ? ?? ? ?? ? ? ? Proof of Theorem cont. ?So that ?By Corollary ?. callput parity holds 12( ) ( )TTVV? ? ?12( ) ( ) , ,ttV V t T? ? ? ? ? Theorem ?For American option pricing, ?if the market is arbitragefree, ?then [ 0 , ]tT??()()ttttC S KP K S?????? Proof of Theorem ? Take American call option as example. ? Suppose not true, ., ? At time t, take cash to buy the American call option and exercise it, ., to buy the stock S with cash K, ? then sell the stock in the stock market to receive in cash. ? Thus the trader gains a riskless profit instantly. ? But this is impossible since the market is assumed to be arbitragefree. ? Therefore, must be true. ? can be proved similarly. [ 0 , )tT?? ttC S K? tCtS 0,ttS C K? ? ? ( )C S K ?? ( )P K S ??? American Option . European Option ?For an American option and a European option with the same expiration date T and the same strike price K, ?since the American option can be early exercised, its gaining opportunity must be = that of the European option. ?Therefore ,.t t t tC c P p?? Theorem ?If a stock S does not pay dividend, then ?., the ``early exercise term is of no use for American call option on a nondividendpaying stock. ttCc? Proof of Theorem ?By above inequalities, there holds ?This indicates it is unwise to early exercise this option 0,tT? ? ? () ( ) ( ) .r T tt t t tC c S K e S K? ? ? ?? ? ? Theorem ?If C,P are nondividendpaying American call and put options respectively, ?then, () r T tt t t tS K C P S K e??? ? ? ? ? Proof of Theorem (right side) ?It follows from callput party, and Theorem , ?thus the right side of the inequality in Theorem is proved. ()() , r T tt t t tr T tttP p c K e SC K e S????? ? ? ?? ? ? Proof of Theorem (left side) ?Construct two portfolios at time t ?If in [t, T], the American put option $P$ is not early exercised, then 12 ,.C K P S? ? ? ? ? ?()12( ) ( ) , ( ) ( )r T tTTT T TV S K K eV K S S???? ? ? ?? ? ? Proof of Theorem (left side) ()12()22w he n ,( ) ( ) , w he n ,( ) ( 1 ) ( ) .Tr T t
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