Question

A nine-month European put option on a dividend-paying stock is currently selling for $2. The stock
price is $25, the strike price is $27, and the risk-free interest rate is 7% per annum. The stock is expected
to pay a dividend of $1 one month later and another dividend of $1 seven months later. Explain the
arbitrage opportunities available to the arbitrageur by demonstrating what would happen under
different scenarios.

Answer 1

Sol:

Current stock price = $25

European put option Strike price = $27

Risk free rate = 7% per annum, Monthly will be 7%/12 = 0.5833%

PV of dividends = 1/(1+0.5833% x 1) + 1/(1+0.5833% x 7)

PV of dividends = 1/(1.005833) + 1/(1.005833 x 7) = 1.9549 or 1.96%

Intrinsic value of put option = $27 - $23.04 = $3.96

Put option is currently trading at $2 which is cheaper than its intrinsic value and could provide arbitrage opportunities.

Borrow today $25 + $2 = $27 and then purchase stock and put option. Of the amount borrowed, borrow $0.99 for a period of one month (maturing to $1 after 1 month) and $0.97 for 7 months (Maturing to $1 after 7 months)  

1) Remaining amount of $25.04 for a period of 9 months, maturity amount to be repaid after a period of 9 months

= 25.04 x (1+0.07 x 9/12) = $26.36

2) Now get $1 after 1 month and $1 after 7 months and repay loans taken for 1 month and 7 months respectively.

3) After a period of 9 months,

If stock price > $27

Sell the stock at the market price which is more than $27 and repay the loan of $26.36, making at least $0.64 as arbitrage profit.

If stock price < $32

Sell the stock at $27 using the put option and repay the loan of $26.36, making exactly $0.64 as arbitrage profit.

Therefore from the above different scenarios, profit from arbitrage is possible.