Assume that the future dividends on a given stock S are known, and denote their discounted value at the present time t by D¯(t). For American call and put options values C(t) ,P(t), suppose we have that
P(t)−D¯(t)−K>C(t)−S(t)
Suppose you buy the call option and deposit D¯(t)+K at the risk-free rate.
Doing which of the following can you make arbitrage profit?:
a. Sell the put option and sell short one share of the stock.
b. Sell the put option and buy one share of the stock.
c. Buy the put option and sell short one share of the stock.
d. Buy the put option and buy one share of the stock.
To determine which action can generate arbitrage profit, we need to analyze the given inequality and the options presented.
The inequality P(t) - D¯(t) - K > C(t) - S(t) suggests that the put option's value minus the discounted dividends and strike price is greater than the call option's value minus the stock price.
Now let's consider the options presented:
a. Sell the put option and sell short one share of the stock.
Selling the put option provides a profit of P(t) but selling short one share of the stock results in a loss of S(t). Therefore, the net profit would be P(t) - S(t).
b. Sell the put option and buy one share of the stock.
Selling the put option provides a profit of P(t), and buying one share of the stock results in a cost of S(t). Therefore, the net profit would be P(t) - S(t).
c. Buy the put option and sell short one share of the stock.
Buying the put option results in a cost of P(t), and selling short one share of the stock provides a profit of S(t). Therefore, the net profit would be -P(t) + S(t).
d. Buy the put option and buy one share of the stock.
Buying the put option results in a cost of P(t), and buying one share of the stock results in a cost of S(t). Therefore, the net profit would be -P(t) - S(t).
To make an arbitrage profit, we need to find an option that has a positive net profit.
Based on the analysis, option a (selling the put option and selling short one share of the stock) would generate a positive net profit of P(t) - S(t), which satisfies the inequality condition. Therefore, by choosing option a, you can make an arbitrage profit.
It's important to note that this analysis assumes no transaction costs, borrowing/lending rates for selling short/buying stock, and that the underlying assumptions of the model hold. Real-world factors may impact the profitability of these strategies.