Suppose that c_1, c_2, and c_3 are the prices of European call options with strike prices K_1, K_2 and K_3, respectively, where K_3 > K_2 > K_1 and K_3 - K_2 = K_2 - K_1. All options have the same maturity. Show that c_2 <= 0.5(c_1 + c_3). Hint: Consider a portfolio that is long one option with strike price K_1, long one option with strike price K_3, and short two options with strike price K_2.

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This question seems to be simple, but I'm nevertheless lost. Please help me on it:

Give a numerical example (choosing S, K, r, T-t, sigma) in which it is obvious (without any formulas) that American put price on a nondividend paying stock is larger than the corresponding European put price.

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I also have another question. Can you explain or give me a link to any website which explains how to calculate the delta of a call option using the Black-Sholes model and the mid-market price? I know that the mid-market price is the average between bid and ask prices. Is it the same as the strike price?