# FD/U2 Topic 9 Future Pricing-Theories, Cost of Carry Model

### Future Pricing-Theories

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.

### Cost of Carry Model

Futures pricing depends on the characteristics of the underlying asset. There is no single way to price futures contracts because different assets have different demand and supply patterns, different characteristics and cash flow patterns.

Cash and Carry model for Futures Pricing:

It is also known as arbitrage modelThe assumption of the model is that in an efficient market, arbitrage opportunities cannot exist.

In other words, the moment there is an opportunity to make money due to mispricing in asset prices across different prices, arbitrageurs will start trading to profit from this mispricing thereby eliminating these opportunities.

Lets understand with an example. Practically, a futures position in a stock can be created in following manners:

• Enter into a futures contract
• Create a synthetic futures positionby buying in the cash market and carrying the asset to future date.

The price of acquiring the asset as on future date in both the cases should be the same, i.e. cost of synthetic futures contract (spot price + cost of carrying the asset from today to the future date) should be equal to the price of present futures contract.
If prices are not the same, then it will trigger arbitrage and will continue until prices in both the markets are aligned.

The cost of creating a synthetic futures position is a fair price of futures contract. Fair price of futures contract adds up the spot price of underlying asset and cost of carrying the asset  from today until delivery. Cost of carrying a financial asset from today to the future date would entail different costs like transaction cost, custodial charges, financing cost etc.