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

Risk free interest rate per month = 7%/12 =0.005833

Present value of Dividends = 1/(1+0.005833*1) + 1/(1+0.005833*7) = $1.9549 (say $1.96)

Adjusted Stock price = $25- $1.96 =$23.04

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

As the Put option is available for $2 it is cheaper than its intrinsic value and may present arbitrage opportunities

Arbitrage will work as follows

1. Today borrow $25+$2 =$27 and purchase the stock as well as the put option. Of the amount borrowed , borrow $0.99 for one month (maturing to $1 after 1 month), $0.97 for 7 months (maturing to $1 after 7 months)  

and remaining amount of $25.04 for 9 months , maturity amount to be repaid after 9 months

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

.2. Get $1 after 1 month and $1 after 7 months and repay loans taken for 1 month and 7 months respectively

3. After 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, make 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, make exactly $0.64 as arbitrage profit

So, in all possible situations, arbitrage profit can be made.