# Determination of Futures Prices

A futures contract is nothing more than a standard forward contract. Therefore, the determinants of the value of either type of contract is the same, so the following discussion will focus on futures. When a contract is 1^{st} entered into, the price of a futures contract is determined by the spot price of the underlying asset, adjusted for time plus benefits and carrying costs accrued during the time until settlement. Even if the contract is closed out before the delivery date, these costs and benefits are taken into account in determining the price of the contract, since there may be a delivery. Benefits that accrue with ownership include dividends and interest that is paid by the underlying asset. Costs associated with ownership include storage costs, such as with oil, and the interest rate used to determine the present value of a transaction, which represents the opportunity cost of delaying the transaction. To simplify the following discussion, benefits and costs will be restricted to present value and income yield.

When a futures contract is initially agreed to, the net present value of the transaction must be equal for both parties; otherwise, there would be no agreement. The delivery price is the price agreed to in the contract. However, with time, the position of the parties will change as a spot price of the underlier changes, with the gains by one party equal to loss of the other party. As the settlement date approaches, the prices of the futures contract and its underlying asset must necessarily converge, so that on the delivery or settlement date, the futures price will equal the spot price of the underlying asset. Because futures contracts can be used to hedge positions in the underlying asset, a perfectly hedged position must necessarily yield the risk-free rate of return — otherwise, arbitrage opportunities would arise that would conform the rate of return to the risk-free rate of return.

For instance, suppose that you invest $2,600 in an ETF that tracks the NASDAQ 100 and you enter into a short position for a hypothetical futures contract for the Nasdaq 100 for a price of $2,700.

NASDAQ 100 Futures Contract Price | 2700 | ||||
---|---|---|---|---|---|

Stock Portfolio Value | 2600 | 2650 | 2700 | 2750 | 2800 |

Short Futures Position Payoff | 100 | 50 | 0 | -50 | -100 |

Dividend Income | 50 | 50 | 50 | 50 | 50 |

Total | 2750 | 2750 | 2750 | 2750 | 2750 |

While dividend payments are not entirely certain, the probability of their change is small compared to the probability of a change in stock prices.

Generally, the price of a futures contract is related to its underlying asset by the **spot-futures parity theorem**, which states that the futures price must be related to the spot price by the following formula:

Futures Price = Spot Price × (1 + Risk-Free Interest Rate − Income Yield)

Otherwise, the deviation from parity would present a risk-free arbitrage opportunity. Entering a futures position does not require a payment of cash, so the risk-free rate that can be earned from the cash is added. (Although margin must be posted, it is much less than the value of the contract, and margin can be in the form of Treasuries, which earn interest.) The income yield is subtracted because no income is earned without owning the underlying asset. Applying this formula to a stock:

Futures Price = Stock Price × (1 + Risk-Free Interest Rate − Dividend Yield)

## Example: Futures Market Arbitrage Opportunity If Spot-Futures Parity Violated

Suppose that you pay $2,600 for 1 share of a stock index exchange-traded fund (ETF) that tracks the Nasdaq 100 at the beginning of the year and that it pays $52 in dividends during the year. At the same time, you sell a futures contract short for the Nasdaq 100 that is cash-settled, requiring you to pay $2,700 at the end of the year. (Note this futures contract is hypothetical since there is no contract for just 1 share of an ETF or stock, but it simplifies the math while still illustrating the principle.) Suppose further that:

- Risk-free rate = 5%
- Dividend yield = 2%

Therefore, the futures settlement price should be:

= ETF Price × (1 + .05 − .02) = $2,600 × 1.03 = $2,678, but if it is $2,700 instead, here is what you can do:

Initial Cash Flow | Cash Flow in 1 Year | |||
---|---|---|---|---|

Borrow at 5% Interest | $2,600 | -$2,730 | = 2600 * 1.05 | |

Buy Stock Index ETF | -$2,600 | S_{t} + | $52 | Dividend Payment |

Sell Stock Index Futures Short | 0 | -S_{t} + | $2,700 | Mispriced |

Total | $0 | $22 | Risk-Free Profit | |

Cash Flow if Futures Contract is Priced according to Parity | ||||

Borrow at 5% Interest | $2,600 | -$2,730 | = 2600 * 1.05 | |

Buy Stock Index ETF | -$2,600 | S_{t} + | $52 | Dividend Payment |

Sell Stock Index Futures Short | 0 | -S_{t} + | $2,678 | Correctly Priced |

Total | $0 | $0 | No Profit |

At the end of the year, the trader with a long position pays you the settlement price of the futures contract in exchange for your ETF, so if the futures contract was overpriced, then you can earn a riskless profit by the amount of the overpricing.

Although the net initial investment is 0, the future cash flow in 1 year is a riskless positive number which is exactly equal to the mispricing of the futures contract compared to what it would be at parity. Because of this riskless arbitrage, traders would bid up the price of the stock or ETF and bid down the price of the futures contract until parity was satisfied. If the futures contract was less than the corresponding stock price, then a reversal of the arbitrage could be done to earn riskless profits, thereby bidding up the futures price and bidding down the stock price. To summarize:

F_{0} = S_{0}(1 + r_{f} − d)

- F
_{0}= Initial Futures Price - S
_{0}= Initial Stock Price - r
_{f}= Risk-Free Interest Rate - d = Dividend Yield

The parity relationship is also known as the **cost-of-carry relationship** because it asserts that the futures price is determined by the relative costs of buying a stock with deferred delivery in the futures market versus buying it in the spot market with immediate delivery and carrying it as inventory. When buying the stock, the interest that could be earned with the money used to buy the stock is forfeited for the duration of the stock ownership. However, dividend payments may be received. Thus, the net carrying cost advantage of deferring delivery of the stock is the risk-free interest rate minus the dividend per period, which is why the futures price differs from the spot price by the amount of the future-parity equation.

The parity relation must also hold for longer contract periods. Because money has time value, there must be a larger difference between the price of a longer term futures contract and the current spot price compared to a short-term contract, so for a contract maturity of t periods, the spot-futures parity equation is modified:

F_{0} = S_{0}(1 + r_{f} − d)^{t}

This is equivalent to the formula for calculating this future value of an investment, where the spot price is the initial value, the term (1+ r_{f} − d) is the interest rate, and t represents the number of compounding periods.

The spot-futures parity equation can also be applied to other futures contracts with different underlying assets by making the appropriate modifications. For instance, for bonds, the coupon payment would be equal to the dividend payment. If the underlying asset pays no dividends, such as a commodity like silver, then the dividend is simply set equal to 0, so the price of the futures contract would be equal to asset price multiplied by the risk-free interest rate. By taking the long position in the futures contract, the trader can earn the risk-free rate of interest with the money that would otherwise be used to buy the asset; ergo, the long position must agree to a higher price to compensate the short position for holding an asset that pays no interest or dividends.

## Spreads

Because the price of a futures contract is fixed relative to its underlying asset for any maturity, there must also be a relationship between futures contracts of the same underlying asset but with different maturities. If this relationship does not hold, then arbitrage opportunities will arise that will cause prices to conform to parity. If the risk-free interest rate exceeds the dividend yield, then futures prices will be higher for contracts with a longer maturity. This is usually the case, but there are times when the risk-free interest rate is actually lower than the dividend yield, in which case futures contracts with longer maturities will be cheaper than futures contracts with shorter terms. the price relationship between futures contracts of different maturities can be used by finding the relationship between the equations for each individual contract:

F(t_{1}) = S_{0}(1 + r_{f} − d)^{t}_{1}

F(t_{2}) = S_{0}(1 + r_{f} − d)^{t}_{2}

F(t_{2}) / F(t_{1}) = (1 + r_{f} − d)^{(t2 − t1)}

F(t_{2}) = F(t_{1})(1 + r_{f} − d)^{(t2 − t1)}

This equation is like the spot-futures parity equation, except that the price of the futures contract of shorter maturity is substituted for the current spot price. Delaying delivery from t_{1} to t_{2} allows the earning of risk-free interest during that time interval, but it also entails the loss of the dividend yield during that time period, hence the equation. Since both the dividend yield and the interest rate are annual yields, the time difference between 2 contracts is generally calculated as the number of months between the delivery dates divided by the 12 months of the year.

### Example: Spread Pricing

Consider a hypothetical futures contract that is priced at $100 for January delivery. The risk-free rate is 3% and the dividend yield is 1%. Therefore, the futures price for April delivery, which is 3 months later, should be:

$100 (1 + .03 − .01)^{((4 − 1)/12)} = $100 (1.02)^{(3/12) }= $100 (1.02)^{(1/4)} = $100.50

The above arguments make it apparent that futures contracts of different maturities based on the same underlying asset move in unison.

In the spot-future parity theorem, an assumption is made that the futures contract would only pay on delivery. However, futures are marked to market daily, which causes the futures price to deviate from parity and to deviate from the forward price. If interest rates are high, then marking to market will give an advantage to the long position causing the price of futures contracts to exceed the corresponding **forward contracts**. If interest rates are low, then the advantage accrues to the short positions, so that futures prices will be less than parity. Because higher interest rates favors the long position, futures traders are willing to accept a higher price on the futures contract, while a negative correlation between futures prices and interest rates will favor the short position, causing the futures price to be less than the corresponding forward price.

However, since the covariance between futures prices and interest rates is low, the price differential between forwards and futures is negligible, except for futures contracts on long-term fixed income securities, since their value highly depends on interest rates, thus causing a substantial spread between forwards and futures prices.