(精选)金融衍生物定价理论第一章 risk management financial derivative课件.pptVIP

Assumptions The market is arbitrage-free All transactions are free of charge The risk-free interest rate r is a constant 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 options strike price, T ------ the options expiration date, r ------ the risk-free interest rate. Theorem 2.2 For European option pricing, the following valuations are true: Proof of Theorem 2.2 lower bound of (upper leaves to ex.) consider two portfolios at t=0: Proof of Theorem 2.2 cont. At t=T, and By Theorem 2.1 i.e. Proof of Theorem 2.2 cont. cont. Now consider a European call option c Since and By Theorem 2.1 when tT i.e. Together with last inequality, 2.2 proved. Theorem 2.3 For European Option pricing, there holds call-put parity Proof of Theorem 2.3 2 portfolios when t=0 when t=T Proof of Theorem 2.3 cont. So that By Corollary 2.1 i.e. call-put parity holds Theorem 2.4 For American option pricing, if the market is arbitrage-free, then Proof of Theorem 2.4 Take American call option as example. Suppose not true, i.e., s.t At time t, take cash to buy the American call option and exercise it, i.e., 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 arbitrage-free. Therefore, must be true. can be proved similarly. American Option v.s. 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 Theorem 2.5 If a stock S does