The Pricing of Derivatives and Arbitrage
Amit K Ghosh
March 3, 2019
Suppose, we want to buy shares of Godrej Industries. 1500 shares of Godrej Industries will cost 1500*494.55 = 741825 INR.
Now 1 lot of Godrej Mar Future will allow holding 1500 shares as well (Lot size of Godrej Futures Contract is 1500.) But in this case, it will cost 132361 which is the margin. The only condition here is  You need to settle the MTM.
Settling MTM means you need to fill in the loss if happens and similarly, you can take out the profit if happens. The margin has to be maintained.
So, instead of paying an upfront of 741825, We are paying 132361. It is saving us 741825132361 = 609464 INR.
As there is nothing called free money. Let’s assume, we are given a loan of 609464 INR here whose interest rate is same as risk free rate of return.
The RiskFree rate of return is the rate of return where there is no default risk or no risk of loss. The Indian government 10 year Bond rate could be taken as a benchmark for the same.
To check, that, let’s go to Reserve Bank of India website.
Basic Mathematics  What is the interest of 609464 INR for one month if You’re charged 7.5558% annual?
Instead of complicating 
 One year interest of 609464 INR@7.5558% = 46049.8809 INR
 One month interest = 46049.8809/12 = 3837.49007 INR
Case 1: Godrej Industry ends above the buying price
Suppose, Godrej Industry ends at 500.Suppose, Godrej Industry ends at 500 at the end of March expiry.
The guy who has 1500 shares of Godrej Industries will make a profit of (500494.55)*1500 = 8175 INR.
The guy who has 1 lot futures of Godrej Industries of March expiry will make a profit of (500496.5)*1500 = 5250 INR.
The derivative buyer guy is making less money of (81755250) = 2925 INR.
Let’s refine the maths even more 
 Today is 3rd March. Expiry is on 28th March. It takes 26 days to expire.
 One year interest of 609464 INR@7.5558% = 46049.8809 INR
 One day interest = 46049.8809/365 = 126.164057 INR
 26 days interest = 126.164057*26 = 3280.26548 INR
Let’s forget about March futures.
Let’s say Godrej Industries is ending at 500 at the end of April expiry.
The guy who has 1500 shares of Godrej Industries will make a profit of (500494.55)*1500 = 8175 INR.
The guy who has 1 lot futures of Godrej Industries of April expiry will make a profit of (500498.8)*1500 = 1800 INR
The derivative buyer guy is making less money of (81751800) = 6375 INR.
Expiry of April is at 25th.
DTE = Days to Expiry = 54
54 days interest = 126.164057*54 = 6812.85908 INR
The derivatives prices are ripping off interest in a passive way as you can see here. This is called the Cost Of Carry (COC).
The sellers are the beneficiary. The buyers are getting leeched. Why? It is assumed, stocks are supposed to rise due to our concept of inflation. Sellers are assumed to take more risk. Hence, the system is designing itself to benefit the weak.
Futures price always trade at a premium. Future Price = Stock Price + COC.
At the end of the expiry, this COC decays and Future Price became Stock Price. No one is charging the interest rate. That is just a figure of speech to normalize the topic.
The concept is similar like theta but not same!
Case 2: Godrej Industry ends at 490 at the end of March expiry
The guy who has 1500 shares of Godrej Industries will make a loss of (490494.55)*1500 = 6825 INR.
The guy who has 1 lot futures of Godrej Industries of March expiry will make a loss of (490496.5)*1500 = 9750 INR
The derivative buyer guy is making more loss of (97506825) = 2925 INR.
So as you can see, in case of profit the derivative guy will make less money of 2925 INR and in case of loss, the derivative guy will make more loss of the same amount. But what else we are missing here?
In this case, the amount of money needed to take the trade is more as the margin is fixed and we need to settle the loss from the pocket. Similarly, in the previous case, it is assumed, we are booking profit at each day’s end.
Let’s move to slightly complicated maths now by studying the effect of a stock’s falling in the premium of future pricing.
For that, let’s say Godrej fell to 400 before March expiry.
What happens now?
The guy who has 1 lot futures of Godrej Industries of March expiry will make a loss of (400496.5)*1500 = 144750 INR.
The guy who has 1 lot futures of Godrej Industries will also have to maintain a margin of 132361. (We are considering the margin is fixed for simplicity’s sake.)
Now he has to engage 144750+132361 = 277111 INR.
Now if someone buys 1500 shares of Godrej, he has to pay 1500*400 = 600000 INR.
If we are thinking of someone who is buying 1 lot futures fresh, he has to pay 132361 INR to get stocks worth of 600000 INR.
 He is saving 600000132361 = 467639.
 One year interest of 467639 INR@7.5558% = 35333.8676 INR
 One month interest = 35333.8676/12 = 2944.48897 INR
 In the previous case, one month interest was 3837.49007 INR
So, when the stock fell from 494.55 to 400. The interest rate is also declining. Hence, if the lot size if stays same it is expected the premium to drop!
Point proved. KO.
*Lot size doesn’t change till the contract expires unless there is a corporate action (Like demerger, stock split etc).
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Topic: What other things affect Future’s price?
Prices are made by people. So, let’s discuss people. Suppose a stock is falling since infinity and beyond. Will you like to long it taking an interest in the middle of the fall?
Obviously no.
What happens then? It is a saying that  If the mountain will not come to Muhammad, then Muhammad must go to the mountain
The rate of interest falls. (Read, the premium of future prices drop!)
As we have seen in the bond market, even Government bonds fluctuate. But it is not supposed to fluctuate right? As it is supposed to give 7.5% at the end of 10 years anyway!
Now, let’s have a look at our heatmap 
Let’s analyze a poor stock which got Thor’s hammer (read, fell) on the last trading day. Airtel took that spot.
Now if we see Airtel’s future price. The premium is 0.1 which is too low!
Maths:
Lot size  1700
Guy buying Share will have to put  1700*307.65 = 523005 INR
Guy buying Future will have to put  93028 INR
Guy buying Future will save  (52300593028) = 429977 INR
Interest of 429977 INR @7.5558% for 26 days = 2314.2281 INR
But,
Future’s COC is = .1*1700 = 170 INR
So, (Interest with respect to Date to Expiry)/Future’s COC is an effective indicator used by FNO analyst to gauge the sentiment of a stock.
In fact, some times, the future’s COC goes into negative too. In that case, stock’s price stays above the future’s price!
Here are some hopeless cases of March Expiry 
Note, in case of a dividend, future prices trade at discount like this. That’s doesn’t imply anything on the sentiment.
Now there are two more topics 

How mean reversion trader is benefitted from this kind of pricing? What are the best opportunities in the next week on this case?

Effect of Physical Settlement and Arbitrage Trades of the various option strikes.