Help with an arbitrage opportunity

I'm helping a friend here with one of his MBA derivatives questions so maybe you guys can shed some light on what I can't.

A European call option and put option on a stock both have a strike price of $20 and an expiration dates in 3 months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in one month. Identify the arbitrage opportunity open to a trader.

My inclination is to buy the put and sell the call but I feel likes there's something I should so with the stock?

Any help would be great!

 

go long the share and short the synthetic share (buy put and sell call

 

Did you mean long the shares and then short the call and put? Or did you mean go long, sell call and buy put?

I still don't see why you would by the shares?

 

To get the dividend.

 

Call + strike = stock + put + k
k = (strike * days * interest) - dividend

We have all the info... so lets solve for stock and then compare it to where stock is...

3 + 20 = Stock + 3 + (.10 * 90/360 * 20) - 1

23 = stock + 2.50

20.50 = stock Ths is what stock is worth doing the syntheic (selling calls and buying puts)

So if you can buy stock for 19 and sell synthetic stock for 20.50 you probably should.

 

So would you sell the call and buy the put and not buy or sell any stock?

 

Buy the stock Sell call Buy put . Neutral to short position? You get the Dividend? Is that right.

 

You not only make money on the dividend for owning the stock, but you also make money because the 20.00 Call and Put are mispriced at $3.

This is a European option which means it cannot be exercised until expiration (too late to get dividend), yet the 20.00 Call has an unusual premium built-in. The 20.00 Call is 1.00 OTM while the 20.00 Put is 1.00 ITM yet they both cost the same!

If this was an American style option then you will see Call premiums b/c they can be exercised prior to ex-div date. Although given these numbers the 20.00 Call is still a bit overpriced.

You will make $2 on this. $1 from the dividend and $1 from the synthetic short (e.g. your synthetic short is already 1.00 ITM and it costs you nothing to initiate because both Put and Call cost the same).

Also, with a risk-free interest of 10%, you should look to unwind the position after ex-div date, unless cost to unwind the synthetic short costs more than 2 months of risk-free interest.

 

very nice explanation Dotom