Options: Put call parity – Part II
by J Victor on July 20th, 2012If we recall the strategies of two investors we mentioned in our last post, one was ‘buy a call and invest the present value of exercise price in risk free assets’ and the second was ‘Buy shares on spot and buy a put’. Both these strategies were identical in payoffs. The basic relationship was also drawn. Now, let’s put down that as formula:
 S + PE = C + PV
Where,
 S = Spot buy price of a share
 PE = Buy European put
 C = Buy European Call
 PV = Present value of cash invested in risk free securities.
Those who are good at maths would have guessed one more thing by now. Since this one is a mathematical relationship, the formula can be rewritten in anyway to find any missing variable.
HOW TO READ & CONSTRUCT THE FORMULA?
When letters pass from one side of the formula to the other, their mathematical relationships will also change. Any positive figure, when taken across the equal sign to the other side, will have the opposite meaning.
Let’s assume that we want to create a transaction that’s the equivalent of ‘PE’ (Buy a European put). The formula can be turned around to look like this:
 PE = C + PV – S
So in the above case, ‘S’ or spot buy price of the share would become, spot sell. So, by buying a call + investing the money in risk free securities and then selling the share on spot, you create a complex transaction that’s equivalent to a long European put (PE).
Why do we need such combinations?
Such combinations becomes necessary when the option for a share is not available and you need to create one for your benefit.
Now, an example.
TCS 3 months call option with a strike price of 400 is sold for Rs 36. The stock is trading at 380 right now. The risk free interest rate is 8% per annum. What would be the theoretical value of a TCS put with the same maturity and exercise price?
In this case, we have to find ‘PE’. Hence the formula will be modified as:
 PE = C + PV –S
 C = Price of a call option = 36
 PV = Present value of 400 invested at 8% for 3 months.
 S = Current market price of the stock = 380
How to find the present value?
This was discussed in one of our basic lessons time value of money. We are showing the calculation once again here:
 PV=FV / (1+r) ^{N}
 Where PV is the present value ,
 FV is the future value
 ‘r’ is the rate of interest
 N is the number of years in investment. ( in this case 3 months or 0.25 years)
Applying the formula, the present value will be as follows:
 PV = 400 / ( 1+ 0.08 ) ^{0.25}
 PV = 400 / 1.019427
 PV = 392.38
Now that we have got all the values, let’s find the value of put.
 PE = C + PV –S
 PE = 36 + 392.38 – 380
 PE = 48.38 or the theoretical equivalent of call sold for 36 would be 48.38
 In other words, by buying a call, investing the value of strike money in 8% risk free deposits and by selling the stock, you create an equivalent put.
 48.38 is the theoretical In case the value of put is below 48.38, it means that the put is undervalued. If it’s above 48.38 it means that the put is over valued. Undervalued puts should be bought and overvalued put should be sold.
With this, we complete our discussion on put call parity.
Final note: The word ‘parity’ here means ‘equality in amount’.
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