1風(fēng)險(xiǎn)管理與金融衍生品-文庫吧資料
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【正文】 trage Free Theorem ?Theorem ? the market is arbitragefree in time [0, T], ? are any 2 portfolios satisfying 1,?2 12( ) ( ) , TTVV? ? ?12P r ob { ( ) ( ) } 0TTVV? ? ? ?[ 0 , ) ,tT??( ) ( ) .tt? ? ? Proof of Theorem ?Suppose false, ., ? Denote ?B is a riskfree bond satisfying ?Construct a portfolio at *** 12[ 0 , ) , . . ( ) ( )ttt T s t V V? ? ? ? ?**21( ) ( ) 0ttE V V? ? ? ? ?c? *tt? ? ?*c 1 2= + / B tEB? ? ? ?** ()ttB V B* * * * *12( ) ( ) ( ) { / } ( ) 0ct t t t tV V V E B V B? ? ? ? ? ? ? Proof of Theorem cont. ?r – risk free interest rate, at t=T ?Then ?From the supposition *12( ) ( ) ( ) { / } ( )T c T T TtV V V E B V B? ? ? ? ? ?*( ) [ 1 ( ) ] 0 ,TcV E r T t? ? ? ? ?****( ) ( ) [ 1 ( ) ] [ 1 ( ) ]T ttV B V B r T t B r T t? ? ? ? Proof of Theorem cont. ?It follows ?Contradiction! ? ?? ?12P r ob ( ) 0P r ob ( ) ( ) 0 0TcTTVVV??? ? ? ? ? ? Corollary ?Market is arbitrage free ?if portfolios satisfying ?then for any 12?? 12( ) ( ) ,TTVV? ? ?[ 0 , ] ,tT? ( ) ( ) .tt? ? ? Proof of Corollary ?Consider ?Then ?By Theorem, for ?Namely 12c B?? ? ? ? ? ?( ) ( ) 0T c TV V B?? ? ?12( ) ( ) ( ) ( ) 0t c t t tV V V V B?? ? ? ? ? ? ?( ) ( ) ( )t t tV V V B?? ? ? ?[ 0 , ],tT? Proof of Corollary ?In the same way ?Then ?Corollary has been proved. 12( ) ( )ttVV? ? ?12( ) ( ) , [ 0 , ]ttV V t T? ? ? ? ?120 , ( ) ( ) .ttVV? ? ? ? ? ? Option Pricing ?European Option Pricing ?CallPut Parity for European Option ?American Option Pricing ?Early Exercise for American Option ?Dependence of Option Pricing on the Strike Price Assumptions 1. The market is arbitragefree 2. All transactions are free of charge 3. The riskfree interest rate r is a constant 4. The underlying asset pays no dividends Notations ? the risky asset price, ? European call option price, ? European put option price, ? American call option price, ? American put option price, ? K the option39。1000,000 with $1,600,000 90 days later. A pays a premium of $64,000 (4%) current rate ($/ 163。1000,000 ? To avoid risk, A has 2 plans ? Purchase a forward contract to buy 163。 Chapter 1 Risk Management Financial Derivative Risk ?Risk uncertainty of the oute ? bring unexpected gains ? cause unforeseen losses ?Risks in Financial Market ? asset (stocks, …), interest rate, foreign exchange, credit, modity, …… ?Two attitudes toward risks ? Risk aversion ? Risk seeking Financial Derivatives Many forms of financial derivatives instruments exist in the financial markets. Among them, the 3 most fundamental financial derivatives instruments: ? Forward contracts ? Future ? Options If the underlying assets are stocks, bonds etc., then the corresponding risk management instruments are: stock futures, bond futures, etc.. Risk Management ?risk management underlying assets ?Method – hedging using financial derivatives . holds two positions of equal amounts but opposite directions, o
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