3. A 6-month European put option with a strike price of $20 sells for $1.44. The...

Here, for given 6-month European put option,

Strike price, X = $20, price of the put option, p = $1.44

Current stock price, S₀ = $17.50

Risk-free rate, r = 10% per annum

a. Upper bound = Max(0, X - S₀) = Max(0, 20 - 17.5) = Max(0, 2.5) = $ 2.5

Lower bound = Max[{X/(1+r)^t} - S₀, 0]

here, t = 6 months = 0.5 years

Lower bound = Max[{X/(1+0.1)^0.5} - S₀, 0] = Max[{20/1.1^0.5} - 17.5, 0] = Max[19.07 - 17.5, 0] = $1.57

So, Upper bound of the put option = $ 2.50

Lower bound of the put option = $ 1.57

b. Here, Option price, p < Lower bound

So, there is an arbitrage opportunity.

Conducting an arbitrage for 100 shares,

Buy the put option for 100 shares at $1.44 per share, cost of buying put option = $ 144

Buy 100 shares at current price $ 17.5 per share, cost of buying 100 shares = $ 1750

Total amount of money required to buy put option and 100 shares = $ 1894

So, we will have to borrow $ 1894 at 10% per annum interest.

At the expiration date, i.e., after 6 months, we must pay back the loan

Total amount to be paid = $ 1894 * (1 +0.1)^0.5 = $ 1894 * 1.1^0.5 = $ 1986.44

Now, after 6 months

Case - I, if the stock price S_t < $20, the put option will be exercised and pay-off will be = 2000 - 1986.44 = $ 13.56

Case - II, if the stock price S_t is > $20, the put option won't be exercised and we will sell those shares at market price, i.e., S_t. In that case, the pay-off will be 100*S_t - 1986.44 > $ 13.56

So, Arbitrage profit will be greater than or equal to $13.56.