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$110 130

As illustrated, these two strategies—buying a share of stock or buying a call and investing in the risk-free asset—have exactly the same payoffs in the future.

Because these two strategies have the same future payoffs, they must have the same value today or else there would be an arbitrage opportunity. The stock sells for $100 today, so the value of the call option today, C0, is:

Where did we get the $87.50? This is just the present value of the exercise price on the option, calculated at the risk-free rate:

Given this, our example shows that the value of a call option in this simple case is given by:

In words, the value of the call option is equal to the stock price minus the present value of the exercise price.

A More Complicated Case Obviously, our assumption that the stock price in one year will be either $110 or $130 is a vast oversimplification. We can now develop a more realistic model by assuming that the stock price in one year can be anything greater than or equal to the exercise price. Once again, we don't know how likely the different possibilities are, but we are certain that the option will finish somewhere in the money.

We again let S1 stand for the stock price in one year. Now consider our strategy of investing $87.50 in a riskless asset and buying one call option. The riskless asset will again be worth $105 in one year, and the option will be worth S1 - $105, the value of which will depend on what the stock price is.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

V. Risk and Return

14. Options and Corporate Finance

© The McGraw-Hill Companies, 2002

CHAPTER 14 Options and Corporate Finance

When we investigate the combined value of the option and the riskless asset, we observe something very interesting:

Combined value

Option value

Just as we had before, buying a share of stock has exactly the same payoff as buying a call option and investing the present value of the exercise price in the riskless asset.

Once again, to prevent arbitrage, these two strategies must have the same cost, so the value of the call option is equal to the stock price less the present value of the exercise price:2

Our conclusion from this discussion is that determining the value of a call option is not difficult as long as we are certain that the option will finish somewhere in the money.

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