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Note-10 - Options-2

Note-10 - Options-2 - Economics 106V Investments Lecture...

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Economics 106V Investments : Lecture Note-10 Daisuke Miyakawa UCLA Department of Economics July 11, 2008 10th lecture covers the following items: (1) The basic ideas of option pricing, (2) an option pricing theory (APT), and (3) the Sketch of Black °Scholes option pricing formula. After discussing those items, we solve some exercise problems for the lecture-9 and -10. 1 Basic Ideas for Option Pricing 1.1 Starting Point On the last lecture note, we studied how the payo/s for various options depend on the underlying stock price on the maturity (or at the expiration date). Although the discussion is useful to understand the basic characteristics of those options, it is actually di¢ cult to evaluate/price those options solely based on such a setup. This is simply because we usually do not know the stock price at the expiration date when we buy the option but we only know the current price of the stock. On this section, as a preparation for the exact option pricing, we overview the basic idea of the option value. The key tool is the "option-stock price diagram" which graphs the option price/value against its concurrent (NOT at the expiration!) underlying stock price depicted in Figure-1. We keep using the same notation as in the lecture note-9. Note that the horizontal axis represents today±s stock price (i.e., t=0) and the vertical axis stands for the value of the targeted option today. S(0) Value of Option on Today (t=0) 0 X Figure-1: Option-Stock Price Diagram 1
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1.2 The Upper and Lower bounds of Option Value We will construct the upper and lower bounds of option value. By considering that there is no obvious arbitrage opportunity in the market (i.e., the investors cannot make a pro²t for free), we can derive those bounds. 1.2.1 Call Option First, the call option value must be non-negative. Remember the investor can always give up the right to exercise this call option. Hence, if the option value is negative, which means the investor can receive some money by buying the option, they are willing to buy an in²nite amount of options and make a pro²t for free (how?). This is obviously inconsistent with the non-arbitrage argument. Thus, the option value cannot be below the horizontal axis in Figure-2 (i.e., cannot be placed at the dotted region below (1)). S(0) Value of Option on Today (t=0) 0 45 degree line X 45 degree line (2) (1) (3) Figure-2: Upper and Lower Bounds for Call Option Second, the option price cannot exceed the current stock price. Suppose the price is higher than the current stock price. Do you want to buy this expensive call option? If the current stock price is lower than the option price, I would rather buy the stock itself not the call option since the stock is the object I can obtain by exercising the call option. There is no point to pay more to get the "right" to buy the stock.
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