A European put option on Dow Chemical (DOW) witha strike price of 42 and 1 year

Question

A European put option on Dow Chemical (DOW) witha  strike price of 42 and 1 year to expiration is trading at a price of $5.50.  A European call option on DOW with a strike price of 42 and 1 year to expiration is trading at a price of $3.34.  The risk free interest rate is 2.0202% per year, compounded continously (the present value of a risk free dollar due in one year is $0.98).  Assuming that the market price of DOW's stock is $40 per share, determine whether there is an opportunity to make a risk free profit.  (Hint: Determine whether the prices above are consistent with put-call parity.  Then figure out a way to sell the overpriced security and hedge by purchasing the low-price substitute.)

Explanation / Answer

Put call parity formula is:

Call price = put price + spot price-strike price*present value of risk free dollar

Checking, we get call price = 5.50+40-42*0.98 = 4.34

As we can see, the actual call option is cheaper at a price of 3.34. So we have an opportunity to make a risk free profit.

As the call option is cheaper and the put option is costlier, we have to buy the call option and sell the put option, and along with that, we have to sell the stock short, and invest the proceeds at the risk free rate.

By buying call option, we pay $ 3.34

By selling put option, we get $ 5.50

By selling the stock short, we get $ 40

So we get a total of -3.34+5.50+40 = 42.16 which we invest at the risk free rate.

This is the way to make a riskfree profit.

Let us see scenarios in one year.

Let the stock price ($X) be less than strike price of $42

In this case, the call option is worthless, while we pay $42 to the put option holder and get one stock. We deliver this stock to cover our short position. The investment of 42.16 is now worth 42.16/0.98 = 43.02. So net profit on the position = 43.02-42 = $ 1.02 which is risk free.

Let the stock price ($X) be more than strike price of $42

In this case, the put option is worthless while we will exercise the call option to get one stock for $42 which we will deliver to cover our short position. The investment of 42.16 is now worth 42.16/0.98 = 43.02. So net profit on the position = 43.02-42 = $ 1.02 which is risk free.

Hope this helped ! Let me know in case of any queries.

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