# Chapter 6: Pricing of Futures

Have you wondered why some listed shares have different prices? Usually, this happens when a stock is traded simultaneously in cash and the futures market. Let’s look at the example of Mr Nivesh.

He was bullish on Reliance Industries Ltd and prefered to buy Reliance futures even though they were trading at a premium to the spot price in the cash markets.

One explanation for this Mr Nivesh could be that he had no other choice.

He was tempted by the lucrative returns by taking a “long position” in the futures, instead of buying in the cash markets. But is this the only explanation for the price difference? Not necessarily.

The pricing of futures contracts depends on the price of an underlying asset. But that’s not all. Different assets have different demand and supply patterns, varied characteristics, and cyclical cash flows.

Based on such differences, futures contracts may have different pricing than their underlying asset. These factors mentioned above make it even more complicated to design a single methodology for price calculation.

Market participants like traders, investors, and arbitrageurs use various models for pricing futures contracts. Let’s dive into the most popular futures pricing models.

## 5.1 Cash and Carry Model

According to the Cash and Carry Model, the pricing of a futures contract is a simple addition of the carrying charge the asset to the spot price.

Futures Price = Spot Price + the Cost of Carry

*where, *

*Spot price refers to the current market price of the underlying asset; *

*Cost of Carry refers to the cost incurred to carry the underlying asset from today to a future date of delivery.*

Costs for a financial asset may include finance costs, transaction costs, custodial charges, etc. For commodities, the cost may also include warehousing costs, insurance etc. Known as the non-arbitrage model, the cash and carry model is based on certain assumptions.

The model assumes that in an efficient market, arbitrage opportunities cannot exist. Because, as soon as there is an opportunity to make money due to the mispricing of an asset, arbitrageurs will try and take advantage to make profits.

Traders will continue to benefit from such an arbitrage opportunity until the prices are aligned across all the markets or products. The other assumption is that contracts are held till maturity.

Meet Mr Sonawala, a second-generation jewellery store owner, keeps buying gold from the bullion market.

Mr Sonawala decides to purchase gold. The spot price of gold is Rs. 50,000 per 10 grams. He buys the gold at the spot rate.

He notices that the 3-month futures contract is currently trading at Rs. 50,200 per 10 grams. He finds an arbitrage opportunity and instantly sells the future contract at that price.

Mr Sonawala figures that the cost of financing storage and insurance for carrying the gold for three months is Rs. 150 per 10 grams. The fair price of the futures contract should be Rs. 50,150 per 10 grams.

- Spot price = Rs. 50,000 per 10 grams
- Fair Price = Rs. 50,150 per 10 grams
- 3-month futures contract Price = Rs. 50,200 per 10 grams

Thus, Mr Sonwala bought gold spot price of Rs. 50,000 and after three months.

##### Mr Sonawala's Net Profit

Price of Futures – Cost Price (incl. cost of carrying) = Net Profit

50,200 – 50,150 = Rs. 50 net profit

More and more sellers will find such opportunities until the cash market prices and future contract prices are aligned. Similarly, if the futures prices are less than the fair price of the asset, it will trigger reverse cash and carry arbitrage.

This means Mr Sonawala will buy gold futures and sell gold in cash markets. Even if he doesn’t have the gold to sell, he may borrow gold and sell in the cash markets to benefit from such an arbitrage.

## 5.2 Extension to the Cash and Carry Model

The model can also work on the assets generating returns by adding the inflows during the holding period of the underlying asset.

Assets like equity or bonds may have certain inflows like dividends on equity or interest payments on bonds during the holding period.

Thus, these inflows are adjusted in the futures fair price which can be calculated as follows.

Fair Price = Spot price + Cost of Carry – Inflows

In Mathematical terms, we can calculate the pricing of futures as follows:

F= S(1 + r-q)^T

Let us apply this formula to calculate the fair price of 3-month index futures.

##### Fair Price of 3-Month Index Futures

Spot price of the index (S) = 5,000

Cost of financing = 12%

Return on Index = 4%

Time to expiry = 3 months

= 5,000(1+0.12-0.04) ^90/365

= **5,095.79**

The Cash and Carry model has certain assumptions, some of which are not known to be practical.

For example, the underlying asset being available in surplus in cash markets, having no transaction costs, no taxes, and no margin requirements. All these assumptions don’t work in the real world.

## 5.3 Convenience Yield

This concept influences the pricing of a futures contract. To understand it better let us look back at the formula for the fair price of futures contracts.

Fair Price = Spot price + Cost of Carry – Inflows

Here, inflows for assets like equity and bonds may be in the form of dividends and interest.

However, sometimes, inflows may be intangible that effectively means the values perceived by the market participants just by holding these assets.

It shows the perceived mental comfort of people holding such assets. For instance, if there is a natural disaster like earthquakes, floods, or a pandemic like Covid 19, people may start hoarding essential commodities like food & food products, vegetables and other products like oil etc.

Imagine if every person starts to behave similarly, which suddenly creates a temporary demand for the underlying asset in the cash markets. We will see a meteoric rise in prices.

In such situations, people derive convenience just by holding the asset. Thus, it is termed as convenience return or convenience yield. Convenience yield may sometimes overpower the cost of carry which leads futures to trade at a discount to the spot price of the underlying asset.

## 5.4 The Expectancy Model

According to this model, the price of a futures contract should be based on the expected demand for the underlying asset at a future date. The model argues that futures pricing is nothing but the expected spot price of an asset in the future.

Futures can trade at a premium or discount to the spot price of an underlying can indicate the expected direction in which the price of the underlying asset may move.

If the futures price is higher than the spot price of an underlying asset, traders may feel that the spot prices may go up. They usually refer to it as a “Contango Market”.

Similarly, if the futures prices are trading at a discount to the spot price, traders may feel that the spot price is anticipated to move downwards. This falling market is generally referred to as the “Backwardation Market”.