Question

Assume Alpha Ltd is currently trading on the NYSE with a stock price of $65. The American one-year call option on the stock is trading at $20 with strike price of $65. If the one-year rate of interest is 10% p.a. (continuously compounding), is the call price free from arbitrage or is it too cheap/expensive, assuming that the stock pays no dividends? What if the stock pays a dividend of $5 in one year?

Answer 1

We have the following data,

Current Stock Price = $65

One-Year Call Option = $20

Strike Price = $65

Rate of interest = 10% p.a.

The best way to determine whether there is any scenario of arbitrage or not is to find if there is any benefit involved in the trade or not. We can determine the same with the help of the following formula

where C is the price of call option  

X is the strike price

r is the rate of interest and t is the time period

Through this formula, we determine the present value of the stock if we purchase the call option. Let's say

So if where is the current stock price, the trade would be free from any arbitrage, however, if the , then arbitrage can take place and the trade can be called as profitable or unprofitable.

In this problem,

Since, 78.814 > 65, the call is not free from arbitrage. In such a condition, the call option is expensive as the present value of the strike price is greater than the current stock price. Had the price been below the current stock price, the call option would have been cheap.

In case of dividends, the formula changes a bit to, where D represents the dividend received during the year.

Therefore, the solution would turn out to be:

Therefore, had the current stock price been $83.33, the trade would have been free from arbitrage but since the current stock price is $65, purchasing the call option would be expensive.