Question

What should you do if you come across a 2-year option pair with a strike price of $100, a spot price of $100, a call premium of $22, a put premium of $15, and the risk-free rate is 5%?

Answer 1

There is a risk free arbitrage available and we will do the following:

a. Sell 1 unit of the stock at spot price $ 100

b. Sell 1 put option at strike price $100 for $15 premium & buy 1 call option at the strike price of $100 for $ 22

This will result in initial cash inflow of (100 + 15 - 22) $ 93. This 93 invested at 5% for 2 years will become (93 * (1.05)²?) as $ 102.53.

Now at the end of 2 years, the if the price is below $ 100: then we make profit on the short stock which will be equal to the loss on the short put and the call expires worthless. Hence there would be risk free profit of $2.53 . We can show this with an example. Lets say the stock price is $ 80. Hence out of the $ 102.53, which we have after 2 years, we use $ 80 to purchase the stock back (& close the short stock leg) and we use $ 20 to pay the put buyer. We dont have to do anything on the call option and we are left with $2.53 profit.

If the price is at $ 100: out of $ 102.53, we use $ 100 to purchase the stock in spot (& close the short sale leg). Both the call and put end up being zero and we are left with profit of $ 2.53

If the price is above $ 100: We have a loss on the short stock and profit on call option which will set off each other. The put option is zero and we will have profit of $2.53. We can show this with an example where lets say the stock price is $ 120. In this case we have cash inflow of $20 from the call option and we use $ 100 from the $ 102.53 to purchase the stock in spot market and close our short sale leg. After this we are left with $ 2.53 profit

Hence we see that in any case we will have $2.53 profit (arbitrage) with this strategy.