1風險管理與金融衍生品(參考版)
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【正文】 The money spent on the options in these two cases must be equal. ? Proof leaves to exercise ?? 演講完畢，謝謝觀看！ 。TTTV S KV?? ? ? ? ???2TSK? 12( ) ( ) ? ? ? ? Proof of Theorem ?In all Cases, when t=T ?But ?By Arbitrage Free P. and Theorem 12( ) ( ) .TTVV? ? ?? ?? ?1121P r ob ( ) ( )P r ob 0TTTVVK S K? ? ?? ? ? ?12( ) ( ) 0 .ttV V t T? ? ? ? ? ? Theorem ?European call (put) option price is a linear homogeneous function of the underlying asset price and the strike price K. . for tS 0? ?( , ) ( , ) , ( , ) ( , ) .t t t tt t t tc S K c S Kp S K p S K? ? ?? ? ??? Financial Meaning of Theorem ? Consider buying European options, with each option to purchase one share of a stock on the expiration date at strike price K。T T TV S K V?? ? ? ? ?1TK S K? ??122 1 2( ) ( 1 ) ( ) ,( ) ( ) ( ) ( 1 ) ( ) .TTT T T TV S KV S K S K S K????? ? ? ?? ? ? ? ? ? ? ?12( ) ( ) 。s expiration date, ? r the riskfree interest rate. tStctptCtP Theorem ?For European option pricing, the following valuations are true: ()( ) , r T tt t tS K e c S? ? ? ??( ) ( )( ) .r T t r T tttK e S p K e? ? ? ? ?? ? ? Proof of Theorem ?lower bound of (upper leaves to ex.) ?consider two portfolios at t=0: ? ? tc 1 1 E c a l l O pti on+ B ond of rTB K e ???2 1 sha r e??1( ) ( ) ( ) ( ) ( ), ( ) ,。) no hedging $ forward cont. hedging $ call option hedging $ up down 1,700,000 1,550,000 1,650,000 1,650,000 1,664,000 1,614,000 Speculator Example ?Stock A is $ on April 30, may grow ?A speculator has 2 plans ? buys 10,000 shares with $666,000 on April 30 ? pays a premium of $39,000 USD to purchase a call option to buy 10,000 shares at the strike price $ per share on August 22 Speculator Example cont. ? Situation I: The stock $ on 8/22. ? Strategy A Return =(730666)/666*100%=% ? Strategy B Return =(73068039)/39*100%=% ? Situation II: The stock $ on 8/22. ? Strategy A Return =(660666)/666*100%=% ? Strategy B loss all investment Return = 100% Chapter 2 ArbitrageFree Principle Financial Market ? Two Kinds of Assets ? Risk free asset ? Bond ? Risky asset ? Stocks ? Options ? …. ? Portfolio – an investment strategy to hold different assets Investment ?At time 0, invest S ?When t=T, ? Payoff = ? Return = ? For a risky asset, the return is uncertain, ., S is a random variable 0TSS? 00( ) /TS S S? A Portfolio ?a riskfree asset B ?n risky assets ? a portfolio is called a investment strategy ?on time t, wealth: , 1 , ...i itS S i n??1,niiiiBS?????? ? ???? ?1, , ... n? ? ?portion of the cor. Asset 1()nt t t t it itiV B S???? ? ? ? ? ? Arbitrage Opportunity ? Selffinancing during [0, T] no add or withdraw fund ? Arbitrage Opportunity A selffinancing investment, and Probability Prob ** 0( 0 , ] , . . ( ) 0 , ( ) 0TT T s t V V? ? ? ? ? ?? ?* ( ) 0 ? ? ? Arbi
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