Question

Suppose there are call options and forward contracts available on coal, but no put options. Show how a financial engineer could synthesize a put option using the available contracts. What does your answer tell you about the general relationship among puts, calls and forwards?

Answer 1

A synthetic put option is created when an investor take short position in a stock as well as a long option in call. Let us supplose the put option of a stock with a strike price = $ 100, with one month maturity and a premium of $ 10. Now if at the expiry the stock price is $ 120, the profit of holder of put option will be premiuim paid as the value of put option will be 0. So his profit = - $10. Similarly if stock price is 80, then profit = Striek price - Price of stock - Premium Paid = 100 -80 - 10 = 10, whereas his profit at $ 60 price will be = 100-60-10 = $ 30.

Now assume there is no put option available, however this put option position can also be created using short future contract, long call option. Lets say the current future price (short)= 100, and call option(long) of $ 100 is available at $ 10. Now the profit at different price will be as follows

price	Profit or loss on short position in future	Profit or loss on call option	Total Pofit
120	-20	10	-10
80	20	-10	10
60	40	-10	30

Thus the exact postion of having a put, i.e. same profit or loss has been created, when there is no put option available.

The above discussion tells us about the put call parity, i.e

Premium of call option + present value of the excercise price discounted at risk free rate = Premium of put + Existing price of stock