Question

3. A 6-month European put option with a strike price of $20 sells for $1.44. The stock is priced at $17.50 and the risk-free rate is 10% per annum. (a) (5 points) What are the upper and lower bounds for this option? (b) (10 points) Is there an arbitrage opportunity in part (a)? If so, conduct an arbitrage with 100 shares of stock (clearly illustrate the steps of an arbitrage). What is the arbitrage profit?

Answer 1

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.