Chapter 2

In Chapter 1, we introduced the core philosophy of derivatives: they are risk-transfer tools governed by the strict law of No-Arbitrage. Now, it is time to look at the foundational building blocks of the derivatives universe: Linear Contracts.

We call forwards and futures “linear” because their payoffs are straight lines. If you buy a futures contract for $100 and the price of the asset goes up by $1, you make exactly $1. If it goes down by $1, you lose exactly $1. There is no complex curvature, no optionality, and no choice. When you enter a forward or futures contract, you are making a binding obligation to perform a transaction in the future.

In this chapter, we are going to look under the hood. How do these contracts settle? How does the exchange prevent traders from defaulting when they lose money? And most importantly, using our No-Arbitrage rule, how exactly do we calculate the mathematically “fair” price of a forward contract today?

Let’s get to work.

2.1 Mechanics of Linear Contracts: Delivery vs. Settlement

As we discussed, a forward contract is a private, Over-The-Counter (OTC) agreement, while a futures contract is a standardized agreement traded on a public exchange. But what exactly happens when that future date finally arrives? How does the contract resolve?

There are two ways to close out a linear derivative at expiration:

1. Physical Delivery

This is exactly what it sounds like. If you are long (the buyer) a crude oil futures contract, and you hold it all the way to expiration, you are legally obligated to travel to Cushing, Oklahoma, and take physical possession of 1,000 barrels of crude oil. If you are short (the seller), you are obligated to deliver those 1,000 barrels.

Physical delivery is crucial for commercial hedgers; like agricultural producers or oil refineries who actually need the raw materials. However, for most financial speculators, taking delivery of 5,000 bushels of live cattle is a logistical nightmare.

2. Cash Settlement

Because physical delivery is cumbersome, many modern futures contracts are cash-settled. Instead of delivering the actual asset, the two parties simply exchange the cash difference between the contract price and the final market price.

Furthermore, some assets cannot be physically delivered. Think about the S&P 500 stock index. You cannot physically deliver an “index.” Therefore, S&P 500 futures are purely cash-settled. If you agreed to buy the index at 5,000, and it expires at 5,100, the seller simply wires you the $100 difference (multiplied by the contract size).

Note: In reality, over 95% of all futures contracts are closed out before they ever reach expiration. A speculator who bought a contract in May will simply sell it in June, cancelling out their obligation entirely and walking away with their cash profit or loss.

2.2 The Role of the Clearinghouse and Margin

If you are trading on an exchange, you don’t have to worry about the other party running away if they owe you money. As we learned, the Clearinghouse acts as the ultimate middleman. But how does the clearinghouse protect itself from bankruptcy?

It does this through a brilliant, daily accounting system known as Margin and Mark-to-Market. Let’s break this down step-by-step.

1. Initial Margin (The Good Faith Deposit)

When you want to buy or sell a futures contract, you don’t have to pay the full value of the underlying asset. You only have to post a small deposit, usually 3% to 10% of the contract’s total value. This is your Initial Margin. It proves you have skin in the game.

2. Daily Mark-to-Market (Settling Up)

Unlike a stock, where your profit or loss exists only “on paper” until you sell, futures contracts are settled every single day at the close of trading. If the market moves in your favour today, the clearinghouse literally deposits cash into your account tonight. If the market moves against you, the clearinghouse withdraws cash from your account tonight.

3. Maintenance Margin (The Red Line)

If the market keeps moving against you, your account balance will drop. The exchange sets a minimum floor called the Maintenance Margin. If your account drops below this red line, you trigger a Margin Call. You must instantly wire more funds to the exchange to bring your balance back up to the Initial Margin level. If you fail to do so, the exchange will forcefully liquidate your position immediately to prevent further losses.

A Step-by-Step Margin Example:

• The Setup: You buy 1 Gold futures contract at $2,000. The contract size is 100 ounces (Total value = $200,000).
• The Rules: The exchange requires an Initial Margin of $10,000 and a Maintenance Margin of $8,000. You deposit $10,000.
• Day 1: Gold drops to $1,980. You lost $20 per ounce × 100 = $2,000.
  • Mark-to-Market: The clearinghouse takes $2,000 from your account.
  • New Balance: $8,000. You are exactly at the Maintenance Margin line. You are safe, barely.
• Day 2: Gold drops to $1,970. You lose another $10 per ounce × 100 = $1,000.
  • Mark-to-Market: The clearinghouse takes $1,000.
  • New Balance: $7,000.
  • Alert! You are now below the $8,000 Maintenance Margin. You receive a Margin Call. You must deposit $3,000 immediately to get your balance back up to the original $10,000.

This daily settlement mechanism is the financial plumbing that keeps the global derivatives market from collapsing. Losses are never allowed to accumulate into massive, unpayable debts.

2.3 Pricing Linear Contracts: The Cost-of-Carry Model

Now, the million-dollar question: If the spot (current) price of Gold is $2,000 today, what should the fair price be for a Forward contract to buy Gold in exactly one year?

Should we guess? Should we ask an economist to predict the future? Absolutely not.

Remember the golden rule from Chapter 1: The No-Arbitrage Principle. The forward price is a mathematical certainty dictated by the current spot price. This is known as the Cost-of-Carry Model.

Here is the intuition. There are two ways to guarantee you have Gold in one year:

• Path A: Buy a 1-year Forward contract today.
• Path B: Go to the bank, borrow $2,000 today, buy physical Gold in the spot market today, and store it in a vault for a year.

Because both Path A and Path B result in you holding Gold in exactly one year, they must cost the exact same amount.

If you choose Path B, what are your costs?

• The Spot Price: You must pay for the asset ($2,000).
• Interest: You must pay interest to the bank on the borrowed money.
• Storage Costs: You have to pay the vault to safely hold your gold.

Therefore, the mathematical formula for the Fair Forward Price is simple:

Forward Price = Spot Price + Interest + Storage Costs

If the Forward Price is trading even one penny higher than this formula, an arbitrageur will instantly execute Path B (borrow money, buy spot gold, store it) and simultaneously short-sell the overpriced Forward contract. They lock in a guaranteed, risk-free profit.

The Twist: Dividends and Convenience Yields

What if the asset pays you to hold it?

If you hold a stock index, it pays dividends. If you hold physical crude oil during a shortage, there is a massive operational benefit to having it on hand right now (we call this a Convenience Yield).

Any income or benefit you receive while physically holding the asset reduces your overall cost of carry.

The Complete Cost-of-Carry Formula:

Forward Price = Spot Price + Interest + Storage – Dividends – Convenience Yield

By understanding this, you realize that the futures price is not a prediction of the future. It is simply an arbitrage-enforced mathematical relationship with the present.

2.4 Applied Hedging and Basis Risk

Let’s apply our knowledge to a real-world corporate hedging scenario.

The Long Hedge vs. The Short Hedge

• The Long Hedge: Executed by someone who needs to buy an asset in the future and fears rising prices. (e.g., Southwest Airlines fearing rising jet fuel costs). They buy futures today to lock in their purchase price.
• The Short Hedge: Executed by someone who needs to sell an asset in the future and fears falling prices. (e.g., A corn farmer fearing a market crash). They sell futures today to lock in their selling price.

The Reality of Basis Risk

In academia, hedges work perfectly. In the real world, they are a bit messy.

Imagine you are a corn farmer in remote Iowa. You short a CBOT Corn Futures contract to lock in a price of $5.00 a bushel. Six months later, you harvest your corn. But you can’t deliver it to Chicago, it’s too far. You have to sell it to your local town elevator.

The exchange price in Chicago might be $4.50, but your local elevator is only offering $4.30 because local supply is incredibly high.

This discrepancy is called Basis.

Basis = Spot Price (Local Cash Market) – Futures Price

When you hedge with futures, you eliminate directional price risk (the risk of the whole market crashing), but you inherit Basis Risk (the risk that your local price doesn’t perfectly correlate with the exchange price).

Furthermore, what if there isn’t a futures contract for your specific asset? Southwest Airlines needs Jet Fuel, but the exchange might only trade Heating Oil futures. Because the two are chemically similar and prices move together, Southwest will use Heating Oil futures to hedge their Jet Fuel risk. This is called Cross-Hedging, and it introduces even more Basis Risk!

Hedging is an art as much as it is a science. It doesn’t eliminate all risk; it simply transforms unacceptable, massive directional risk into a much smaller, manageable basis risk.

<br>

Chapter Summary

• Settlement: Linear contracts conclude either via Physical Delivery of the actual asset or via Cash Settlement of the price difference.
• Margin: The Clearinghouse eliminates default risk using a system of Initial Margin (deposit), daily Mark-to-Market (realizing daily gains/losses), and Maintenance Margin (the minimum threshold before a margin call).
• Pricing (Cost-of-Carry): The forward price is mathematically derived from the spot price, plus the cost to finance and store the asset, minus any yields or dividends. It is enforced entirely by the threat of arbitrage.
• Hedging Realities: Hedging eliminates directional price risk but introduces Basis Risk: the mismatch between the local cash asset being hedged and the standardized futures contract used to hedge it.

<br>

Discussion Questions

1. Why does the daily “Mark-to-Market” process make the futures market significantly safer than the OTC forward market during a massive financial crisis?
2. If the Central Bank suddenly raises interest rates by 2%, what should theoretically happen to the price of a 1-year forward contract on Gold, assuming the spot price of Gold doesn’t change? (Think about the Cost-of-Carry formula).
3. If you are a jewelry maker hedging your silver costs, why might you prefer to hedge using an OTC forward contract rather than an exchange-traded futures contract, despite the credit risk?

Step-by-Step Exercise: Surviving the Margin Call

Let’s track a trader’s margin account over three highly volatile days.

The Setup:

• You sell (short) 2 Crude Oil futures contracts at $80.00 per barrel.
• Contract multiplier: 1,000 barrels per contract. (Total exposure = 2,000 barrels).
• Initial Margin required: $5,000 per contract ($10,000 total).
• Maintenance Margin required: $4,000 per contract ($8,000 total).
• You deposit exactly the $10,000 Initial Margin.

Remember: Because you sold short, you make money when the price goes DOWN, and lose money when the price goes UP.

Calculate the ending account balance and indicate if a Margin Call occurs on each day:

• Day 1 Close: Crude Oil settles at $78.00.
  • Step: Calculate your profit/loss. Adjust your balance. Margin call? (Y/N)
• Day 2 Close: Crude Oil violently spikes to $83.00.
  • Step: Calculate your profit/loss from Day 1’s close. Adjust your balance. Margin call? (Y/N). If yes, how much must you deposit?
• Day 3 Close: Crude Oil settles at $82.00.
  • Step: Assuming you met the margin call on Day 2, calculate your new balance.

(Work through the math step-by-step. Tracking these cash flows is a fundamental skill for any derivatives practitioner!)