Welcome to the World of Futures!
Hello there! Today, we are diving into one of the most practical and exciting parts of the CFA Level I Derivatives curriculum: Pricing and Valuation of Futures Contracts. If you’ve already looked at Forward Contracts, you’re halfway there! Futures are very similar, but they have some unique "rules of the game" because they trade on an exchange.
By the end of this note, you’ll understand how we determine the "fair price" for a future and why the "value" of a future is actually much simpler than it looks. Don't worry if derivatives feel a bit abstract at first—we'll use plenty of everyday analogies to keep things grounded.
1. The Core Principle: No-Arbitrage Pricing
Before we look at formulas, let’s understand the "Why." Why is a futures price what it is? We use a concept called No-Arbitrage Pricing. This is based on the Law of One Price, which basically says: If two things are identical, they should cost the same amount.
Imagine you want to own a gold bar six months from now. You have two choices:
1. Buy it today (Spot) and pay to store it in a vault for six months.
2. Enter a futures contract to buy it in six months.
In a fair market, both paths should cost you the exact same amount of money. If one was cheaper, everyone would rush to buy it that way, and the prices would balance out again. That "fair" price where no one can make a "free" profit is our Futures Price.
Key Terms to Know:
Spot Price (\( S_0 \)): The price of the asset if you bought it right now "on the spot."
Futures Price (\( F_0 \)): The price agreed upon today for a transaction that will happen in the future.
Quick Review: The Basic Formula
In its simplest form (with no costs or benefits), the formula is:
\( F_0 = S_0 \times (1 + r)^T \)
Where:
\( r \) = Risk-free interest rate
\( T \) = Time until the contract expires
2. The "Cost of Carry" Model
In the real world, holding an asset isn't free, but sometimes it pays you back. We call the net balance of these costs and benefits the Cost of Carry.
Costs: Storage and Interest
If you buy a physical asset (like wheat or oil) today, you have to pay to store it. Also, because you spent money today instead of keeping it in the bank, you lose out on interest. These costs make the futures price higher than the spot price.
Benefits: Dividends and Convenience Yield
Some assets pay you while you hold them. Stocks pay dividends; bonds pay interest (coupons). Additionally, for commodities, there is something called a Convenience Yield—the "non-monetary benefit" of actually having the physical oil or wheat on hand in case of a shortage. These benefits make the futures price lower.
The Full Pricing Formula:
\( F_0 = S_0 + \text{PV}(\text{Storage Costs}) - \text{PV}(\text{Benefits/Dividends}) \times (1 + r)^T \)
Analogy: Think of a futures contract like ordering a pizza for delivery in an hour. The "Spot" price is the pizza price now. The "Future" price includes the "Cost of Carry" (the delivery fee) but might be reduced by a "Benefit" (a coupon you have for a future discount).
Key Takeaway: Costs (storage, interest) increase the futures price. Benefits (dividends, convenience yield) decrease the futures price.
3. Price vs. Value: The "Daily Reset"
This is where many students get tripped up! In the CFA curriculum, Price and Value are two different things.
Futures Price: This is the number written in the contract (e.g., "I will buy gold at \$2,000").
Futures Value: This is how much the contract is "worth" to you as the market moves.
The Mark-to-Market Process
Unlike Forwards (where value builds up over time and is settled at the very end), Futures are Marked-to-Market every single day. At the end of every trading day, the exchange looks at the price movement:
1. If the price went up, the buyer’s account is credited with cash, and the seller’s account is debited.
2. The contract is essentially "re-written" at the new price.
Important Point: Because the gains and losses are paid out in cash every day, the Value of a futures contract at the start of every day is zero. You've already collected your profit or paid your loss from yesterday!
Did You Know?
This daily settlement is why futures are considered to have lower default risk than forwards. You never "owe" a massive amount at the end; you pay as you go!
4. Contango and Backwardation
These are fancy words for a simple concept: the relationship between the Spot Price and the Futures Price as time goes by.
Contango
When the Futures Price is HIGHER than the Spot Price (\( F_0 > S_0 \)).
This usually happens when storage costs and interest are high. The price curve slopes upward.
Backwardation
When the Futures Price is LOWER than the Spot Price (\( F_0 < S_0 \)).
This happens when there is a high Convenience Yield (people really want the asset now because of a shortage). The price curve slopes downward.
Memory Aid:
Contango = Carrying costs are high.
Backwardation = Benefits (like convenience) are high.
5. Common Pitfalls to Avoid
1. Confusing Forwards and Futures: Remember, Forwards have a value that changes over the life of the contract. Futures have their value reset to zero daily due to marking-to-market.
2. Mixing up Costs and Benefits: Always ask yourself: "Does this make it more expensive to hold the asset?" If yes (like storage), add it to the price. If it's a "thank you" for holding it (like a dividend), subtract it.
3. Time Units: Ensure the interest rate (\( r \)) and the time (\( T \)) match. If the rate is annual, \( T \) must be in years (e.g., 90 days is \( 90/360 \) or \( 90/365 \)).
Summary Table
Factor | Effect on Futures Price
Increase in Spot Price | Increase
Increase in Interest Rates | Increase
Increase in Storage Costs | Increase
Increase in Dividends/Benefits | Decrease
Increase in Convenience Yield | Decrease
Don't worry if this seems tricky at first! The math is usually just addition, subtraction, and basic compounding. Focus on the logic of "Cost of Carry," and the formulas will start to make perfect sense. Happy studying!