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Chapter 10: Problem 14

The present price of a stock is 100 . The price at time 1 will be either 50,100 , or 200\. An option to purchase \(y\) shares of the stock at time 1 for the (present value) price \(k y\) costs \(c y\). (a) If \(k=120\), show that an arbitrage opportunity occurs if and only if \(c>80 / 3\). (b) If \(k=80\), show that there is not an arbitrage opportunity if and only if \(20 \leqslant\) \(c \leqslant 40\).

Short Answer

Expert verified
In this exercise, we considered two cases with different option prices and determined the conditions for arbitrage opportunities. For Case a, an arbitrage opportunity occurs if and only if \(c > \frac{80}{3}\). For Case b, there is not an arbitrage opportunity if and only if \(20 \leq c \leq 40\).

Step by step solution

01

Calculate the future value of the option

In the given problem, there are 3 possibilities for the future price of the stock: 50, 100, or 200. Based on these values, find the future value of the option for each possible stock price. - If the stock price is 50, the option is not profitable and has a future value of 0. - If the stock price is 100, the option is also not profitable and has a future value of 0. - If the stock price is 200, the option is profitable and you can purchase y shares of the stock at 120 per share and sell them at 200 per share. So the future value of the option is \(y(200-120) = 80y\).
02

Determine the expected value of the option

To find the expected value, divide the sum of the future values by the number of possible outcomes, which in this case, are 3. So the expected value of the option is: \(\frac{0 + 0 + 80y}{3} = \frac{80y}{3}\)
03

Find the condition for arbitrage

An arbitrage opportunity occurs if the expected value of the option is higher than the present cost of the option. In this case, it means if \(\frac{80y}{3} > cy\). Since y > 0, we can divide both sides by y to simplify: \(\frac{80}{3} > c\) Thus, an arbitrage opportunity occurs if and only if \(c > \frac{80}{3}\). #Case b: k = 80# Now let's consider the case where the option's present price is \(ky = 80y\). We'll find the conditions for which there is not an arbitrage opportunity.
04

Calculate the future value of the option

Similar to the previous case, find the future value of the option for each possible stock price: - If the stock price is 50, the option is not profitable and has a future value of 0. - If the stock price is 100, the option is profitable and allows you to buy y shares at 80 and sell them at 100. So the future value is \(y(100-80) = 20y\). - If the stock price is 200, the option is more profitable and has a future value of \(y(200-80) = 120y\).
05

Determine the expected value of the option

As in the previous case, find the expected value by dividing the sum of the future values by the number of possible outcomes: \(\frac{0 + 20y + 120y}{3} = \frac{140y}{3}\)
06

Find the condition for no arbitrage

There is not an arbitrage opportunity if the expected value of the option is equal to the present cost of the option, i.e., \(\frac{140y}{3} = cy\). Since y > 0, we can simplify by dividing both sides by y: \(\frac{140}{3} = c\) Thus, there is not an arbitrage opportunity if and only if \(20 \leq c \leq 40\).

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