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 sell the put option and sell short one share of the stock.

Doing which of the following can you make arbitrage profit?:

Buy the call option and deposit D¯(t)+K at the risk-free rate.-correct

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Please help me with this variation:

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 whether arbitrage profit can be made in this scenario, we need to understand the inequality presented:

P(t) - D¯(t) - K > C(t) - S(t)

where P(t) represents the price of the put option at time t, D¯(t) represents the discounted value of future dividends at time t, K represents the strike price of both the call and put options, C(t) represents the price of the call option at time t, and S(t) represents the current price of the stock at time t.

The inequality suggests that the cost of buying the put option (P(t)) and subtracting the present value of future dividends (D¯(t)) and the strike price (K) should be greater than the difference between the call option price (C(t)) and the stock price (S(t)).

To exploit this situation for arbitrage profit, we can sell the put option and short sell one share of the stock. In doing so, we will receive the premium from selling the put option, and by short selling the stock, we would receive the current stock price (S(t)).

If we know that the put option price (P(t)) satisfies the given inequality, we can take that premium received and use it to buy the call option. Additionally, we can deposit the discounted value of future dividends (D¯(t)) plus the strike price (K) at the risk-free rate.

By doing this, we will have established a risk-free investment with the deposit plus the call option. This strategy eliminates any possibility of loss and ensures an arbitrage profit.

Therefore, the correct action to make arbitrage profit in this scenario is to buy the call option and deposit the discounted value of future dividends plus the strike price at the risk-free rate.