Futures Markets

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Definitions

A derivative is an instrument whose value depends on, or is derived from, the value of another asset.
Futures and forwards are derivatives that allow traders to fix the price at which an asset will trade at a given date in the future.
Trading:
- Futures: Exchanges such as the Chicago Board Options Exchange
- Forwards: Over-the-counter (OTC) markets where customers contact sell-side traders directly.
The futures price is the delivery price that makes the value of the contract zero.

Like a forward contract, it’s an agreement to buy or sell an asset for a certain price at a certain time.
Whereas a forward contract is traded OTC, a futures contract is traded on an exchange such as CME, CBOT, COMEX, NYMEX, etc.
Available on a wide range of assets such as stock indices, commodities, interest rates, and currencies.
Contracts are standardized specifying quantity and quality, location, and delivery dates.
Settled daily

In futures markets, the spot price is defined as the closest-to-maturity futures price.
For many commodities, the spot price is close but not the same as the cash price.
- The delivery method of a futures contract might be different from the typical delivery method of the physical commodity.
More formally, if we denote by F(t, T) the futures price at time t of a contract expiring at time T, the spot price is defined as: S_{t} = F(t, t)
The futures price converges over time to the spot price.

A margin account consists in cash or marketable securities deposited by an investor with his/her broker.
The margin account balance is adjusted daily to account for daily gains or losses.
Note that futures exchanges require the margin account to be at all times above a certain minimum.
If the margin account goes below the minimum margin requirement the trader will receive a margin call.
Margins minimize potential losses that might occur because of a default event.

The E-mini S&P 500 futures contract is one of the most liquid and actively traded futures in the world.
The contract value is defined as $50 \times the value of the S&P 500 Index.
The way the margin works on this contract is as follows:

Day	Futures Price	Gain/Loss	Margin Account
0	4,645.00		12,000.00
1	4,656.75	587.50	12,587.50
2	4,652.25	-225.00	12,362.50
3	4,658.50	312.50	12,675.00

If interest rates are constant, forward and futures prices are the same.
When interest rates are uncertain, futures and forwards are in theory not exactly the same.
A strong positive correlation between interest rates and the asset price implies the futures price is higher than the forward price as would be the case for Eurodollar futures.
A strong negative correlation implies the reverse.

A stock index can be viewed as an investment asset paying a dividend yield q.
The futures-spot price relationship is F = S e^{(r - \delta) T} where \delta is the dividend yield of the portfolio tracking the index.
In this relationship, S closely tracks the level of the index as long as it is possible to trade its constituents.

Consider an index tracking a portfolio of stocks that pays a dividend yield of 3% per year with continuous compounding.
The index is currently at 4,300. The risk-free rate for all maturities is 1% per year continuously-compounded.
What should be the 6-month futures price of the index?
If we denote by F the futures price, then we have that: F = 4300 e^{(0.01-0.03) \times 6/12} = 4257.21
Note that because the dividend yield is higher than the risk-free rate, the futures price is less than the current spot price.

Index arbitrage involves simultaneous trades in futures and many different stocks.
Very often a computer is used to generate the trades.
When F > S e^{(r - \delta) T} an arbitrageur buys the stocks underlying the index and sells futures.
When F < S e^{(r - \delta) T} an arbitrageur buys futures and shorts or sells the stocks underlying the index.
Occasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F and S does not hold.

For commodities, the convenience yield y represents the net dividend paid by the commodity and the futures price is computed as: F = S e^{(r - y) T}
The cost of carry, c = r - y, is the storage cost plus the interest costs less the income earned, so that for an investment or consumption asset we have that: F = S e^{c T}

Suppose that \mu is the expected return required by investors in an asset.
We can invest F e^{-r T} at the risk-free rate and enter into a long futures contract to create a cash inflow of S_{T} at maturity.
This shows that F e^{-r T} = \operatorname{E}(S_{T}) e^{-\mu T} or F = \operatorname{E}(S_{T}) e^{(r - \mu) T}