Futures Pricing Formula How to calculate and Things to Consider

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If you are taking a conventional course on Futures trading, the pricing formula would usually be discussed at the outset. We have chosen to wait until later in our course as for most people, who are trading futures based on technical analysis, there is no real need to know the pricing formula. This does not mean advanced knowledge of it is unhelpful. However, if one wants trade using quantitative strategies such as Calendar Spreads or Index Arbitrage then understanding how futures prices are determined is essential. We will delve into these strategies more deeply later in our dedicated module on Trading Strategies; this early discussion serves to lay a foundation for those studies.

The value of a futures instrument is based on the price of its underlying. The movement of the futures instrument follows that of the underlying, with falls in the latter resulting in a corresponding decrease in the former. However, it is important to note that their prices are not identical; at present, Nifty Spot is trading at 8,845.5 while the current month contract is priced at 8,854.7 (see snapshot below). This difference between the spot and futures prices is referred to as “the basis or spread” and, in this instance, it amounts to 9.2 points (8854.7 – 8845.5).

The ‘Spot – Future Parity’ is the source of the price difference. Variables like interest rates, dividends, and time till expiry contribute to this phenomenon. It is essentially a mathematical formulation to align the underlying cost with its respective futures cost. This arrangement is otherwise referred to as the futures pricing formula.

The futures pricing formula simply states –

Futures Price = Spot price *(1+ rf )– d

Where,

rf = Risk-free rate

d – Dividend

Consider that the risk-free rate referred to here is for a year; if the duration of expiration is different, adjust the formula in accordance with the time period. Thus, an overall formulation would be –

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

Where,

x = number of days to expiry.

The RBI’s 91-day Treasury bill can be used to serve as an indication of the short-term risk-free rate.

The image above shows that the current rate is 8.3528%, and with this in mind, it’s time to analyse a pricing example: let’s say Infosys spot is trading at 2,280.5 with just seven days remaining until its expiry. What should the current month futures contract be priced at?

Futures Price = 2280.5 * [1+8.3528 % (7/365)] – 0

Infosys is not estimated to dispense any dividend for the upcoming week, thus I assumed it as 0. From this equation, the future cost amounts to 2283 – known as the ‘Fair value’ of futures. In comparison, the current trading price of the futures contract is labelled as the ‘Market Price’, which can be seen from the image below at 2284.

 

The divergence between fair value and market price is mainly attributed to costs incurred in the market, including taxes, transaction fees and margins. In most cases, this fair value reflects where futures contracts should trade at a given risk free rate and number of days left until expiration. Now let us take this thought even further and comprehend how we can determine mid- and last-month contract prices.

Mid-month calculation

The contract will terminate on 26th March 2015, with only 34 days until then.

Futures Price = 2280.5 * [1+8.3528 % (34/365)] – 0

= 2299

Far month calculation

Number of days to expiry = 80 (as the contract expires on 30th April 2015)

Futures Price = 2280.5 * [1+8.3528 % (80/365)] – 0

= 2322

 

It is evident that the fair value and market price are not the same, which can most likely be attributed to various costs. In addition, one should also consider financial year end dividends that may be factored into the market price. Ultimately, with more time until expiry, it is expected that the divergence between these two values continues to grow.

In fact, this leads us to another important commonly used market terminology – the discount and the premium.

If the futures is trading at a higher rate than the spot, mathematically speaking, this is normal. It is described as ‘premium’ in the equity derivatives market, and ‘contango’ in the commodity derivatives market. Despite the different terms for it, both are referring to a situation of futures trading higher than spot.

This plot displays the Nifty spot and correlated futures for January 2015. Throughout the series, you can observe that the futures are trading in excess of the spot.

 

I would like to direct your attention to these points. These points are important and should be noted.

  1. At the outset of the series (signified by a black arrow) there is a wide gap between the spot and futures. This is because the days left until expiry are great, which then yields a higher x/365 factor in the formulas used to calculate future prices.
  2. Throughout the series, the futures stayed at a higher rate than the spot.
  3. At the end of the series (highlighted by a blue arrow), the futures and spot will always converge, no matter what their relationship is beforehand. This phenomenon is an invariable reality.
  4. If you don’t close your futures position before the expiry date, the exchange will do it for you and the settlement value will be determined by the spot price on that day, as the gap between futures and spot closes.

Sometimes the futures does not trade more expensively than the spot. This is usually due to short-term demand and supply imbalances which can cause the futures to sell for less than its spot figure. This phenomenon is known as ‘trading at a discount’ in commodities lingo, or ‘backwardation’.

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