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# Chapter 10 Futures Pricing Formula How is the price of a stock determined in the futures market?

A futures contract is nothing more than a standardized forwards contract. The price of a futures contract is determined by the spot price of the underlying asset, adjusted for time and dividend accrued till the expiry of the contract.

When the futures contract is initially agreed to, the net present value must be equal for both the buyer and the seller else there would be no consensus between the two.

This difference in price between the futures price and the spot price is called the “basis or spread”.

The futures pricing formula is used to determine the price of the futures contract and it is the main reason for the difference in price between the spot and the futures market. The spread between the two is the maximum at the start of the series and tends to converge as the settlement date approaches. The price of the futures contract and its underlying asset must necessarily converge on the expiry date.

The spot future parity i.e. difference between the spot and futures price arises due to variables such as interest rates, dividends, time to expiry, etc. It is a mathematical expression to equate the underlying price and its corresponding futures price.

According to the futures pricing formula:
Futures price = (Spot Price*(1+rf))- Div)
Where,
Spot Price is the price of the stock in the cash market.
rf = Risk free rate (T Bill/ Government securities)
d – Dividend paid by the company

A key point to take note of is ‘r’ is the risk free interest that we can earn for the entire year but since the future contracts expires in 1, 2 or 3 months, we require to adjust the formula proportionately.

Futures price = Spot price * [1+ rf*(x/365) – d]

x = number of days to expiry

One can take the RBI’s 91 or 182 days Treasury bill as a proxy for the short term risk free rate. The ongoing rate can be referred from RBI’s website. The prevailing rate in the market for 91 and 182 day t bill is ~6.68% and ~6.92% respectively.

### Fair Value vs. Futures Price

Sometimes we observe that there is a difference in price between the value calculated through the futures pricing formula (fair value) and value trade in the market (futures price). The futures price may be different from the fair value due to the short term influences of supply and demand for the futures contract. A large deviation between the two could result in an arbitrage opportunity assuming that the futures price will eventually revert back to the fair value.

Let us consider a practical example to understand the concept better.

We will calculate the futures price of ITC through the pricing formula and compare it with the current futures price in the market.

The spot price of ITC is 302.55 as of 9/8/2018

The stock has not declared any dividend in the current August series.

August series expiry is on 30th i.e last Thursday of the month.

There are 22 days for expiry 30-9 (taking both the days into consideration)

What should ITC’s current month futures contract be priced at?

Futures price = 302.55* [1+6.68 %( 22/365)] – 0

Futures price =303.7

Solving the above equation, we obtain the futures price of 303.7, which is called its fair value. However, the actual price in the market, of ITC Aug Futures is 303.5, which is called its market price. The price difference between the fair value and prevailing market price occurs due to supply /demand, liquidity as well as factors such as transaction charges, taxes and margins. But on most occasions, the theoretical future price would match the market price.

Similarly, we could determine the future price for the mid month and far month contracts.

Mid month calculation (September series)

Spot Price =302.65

Div =0

Days to expiry =22+28 (22 days of Aug series +28 days of Sep series)

Futures Price = 302.65 * [1+ 6.68%( 50/365)] – 0

= 305.4 Far month calculation (October series)

Spot Price =302.65

Div =0

Days to expiry =22+28 +28 (22 days of Aug series +28 days of Sep series+28 days of October Series)

Futures price = 302.65 * [1+ 6.68%( 78/365)] – 0>

= 307.04 There is a difference of Rs. 2.66 between the fair value and the market price. This is due to the low liquidity that is present. There are hardly any contracts of ITC October Futures which are traded in the market, hence the premium.

If the price of the security is trading higher in the futures market vs. spot price (which is usually the case) then the futures price is said to be at a premium. While it is said to be trading at a discount if the futures price is less vs. the spot price. We will now see how a trader can benefit if there is a major difference in the price of security in the cash and futures market.

Arbitrage

An arbitrage is known as simultaneous buying and selling of securities, currency, or commodities in different markets or in derivative forms in order to take advantage of differing prices for the same asset.

Here we will see how arbitrage strategies can be made use of when there is a difference in price between the spot market and the futures market.

It is considered a riskless strategy as the gains are locked right at the start and thereafter, it does not matter in which direction the asset moves.

Consider a scenario where Infosys -
Spot = 1380
Rf – 6.68%
Days to expiry (x) = 22
div = 0
Having this information we calculate the future price using the mathematical formula as earlier
Futures price = 1380*(1+6.68 %( 22/365)) – 0
= 1385.55
This is nearly the rate at which it is trading in the futures market at present But what if Infosys Aug Futures is trading at 1410 drastically deviating from its theoretical price.

According to the formula there should ideally be only a Rs5 difference between the spot and future price. But if a huge gap is witnessed due to supply demand imbalances, an arbitrage opportunity arises.

How can a trader benefit in such a scenario?

Clearly Infosys is trading above its fair value, and according the arbitrage strategy, we sell the expensive asset and purchase the cheaper one. In this case, we will sell Infosys futures and purchase the same quantity of shares in the cash market, knowing that by the end of the series the price of the futures and spot will converge.

Sell Infosys Futures at 1410

Since the lot size of Infosys is 600 we purchase 600 shares of Infosys in the cash market at 1380.

Once we have executed the trade at the expected price you have locked in the spread. So irrespective of where the market goes by expiry, profit is guaranteed. The strategy requires us to square off our positions before expiry i.e. we would have to buy Infosys in the futures market and sell the 600 shares in the cash market.

This arbitrage strategy between the spot and futures market is known as cash and carry arbitrage.

Pay-off of cash and carry arbitrage

Expiry Value Spot Trade P&L (Long) Futures Trade P&L (Short) Net P&L
1390 1390 – 1380 = +10 1410 – 1390 = +20 +10 + 20 = +30

Similarly, if the stock is trading at discount in the futures market compared to its spot price, we can implement the reverse cash and carry arbitrage strategy. In this strategy it is required that the trader already has shares in his DP equal to or more than the lot size or its multiple.

Here the trader purchases the future contract and sells the stock in the cash market. At the time of expiry, a reverse trade has to be executed, as we had seen in the cash and carry arbitrage.

Example: Price of Infosys in cash market 1380

Price of Infosys August Futures 1352.

Clearly, the stock is trading at a discount to its fair value, which has been calculated through the mathematical formula and which had come up to 1385.

Hence, in order to take benefit of the situation, the trader sells 600 shares of Infosys (or multiples of 600 lot size) in the cash market and purchase the proportionate quantity of Infosys in futures market. In this strategy too, the spread is locked right at the start, and hence, it does not matter at what price the stock ends on expiry.

Pay-off of reverse cash and carry arbitrage

Expiry Value Spot Trade P&L (Short) Futures Trade P&L (Long) Net P&L
1390 1380 – 1390 = -10 1390 – 1352 = +38 -10 + 38 = +28

Through these two examples we see how a trader can benefit if there is a price difference in the spot and futures market.