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When we assume a perfect market and no arbitrage opportunities, we can find some relationships between option prices that do not require any assumptions about volatility and the probabilistic behaviour of stock prices.

(a) Explain the arbitrage restrictions on option price with respect to its underlying asset price and the payoffs at the maturity. How these restrictions can be used to derive the Black-Scholes option pricing formula?

(b) Show the following relationship is true;
i) C(S,T1,E) > C(S,T2,E) where T1 > T2, and
ii) C(S,T,E1) > C(S,T,E2) where E1 < E2,
where C is an American call option, S is the stock price, T is the time to maturity and E is the exercise price.

(c) Suppose the following three options for the same underlying asset (S) with the same time-to-maturity (T), but with different exercise prices: C(S,T,E1), C(S,T,E2) and C(S,T,E3) where E2=(E1+E3)/2 and E3<E2<E1. Demonstrate that C(S,T,E1)+C(S,T,E3) > 2C(S,T,E2) and explain the implication of this boundary condition.




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关键词：期权定价 无套利 relationship Restrictions restriction following perfect between require respect

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