3  Futures

Futures contracts are among the oldest and most important derivative securities, with organized futures trading dating back centuries. These standardized agreements to buy or sell an asset at a predetermined price on a future date serve critical functions in modern financial markets, from price discovery and risk management to speculation and arbitrage.

A futures contract is a standardized agreement between two parties to trade a specific asset at a predetermined price on a specified future date. Unlike forward contracts, which are customized over-the-counter agreements, futures contracts are traded on organized exchanges with standardized terms, daily settlement, and clearinghouse guarantees.

3.1 Key Features of Futures Contracts

Standardization and Exchange Trading

Contract Specifications: Exchanges standardize every aspect of futures contracts: - Underlying asset: Specific grade, quality, and delivery location - Contract size: Standard quantity (e.g., 5,000 bushels of corn, 100 troy ounces of gold) - Delivery months: Specific months when contracts expire - Tick size: Minimum price movement (e.g., $0.25 per bushel for corn) - Daily price limits: Maximum allowed price changes in a single session

Example: The Chicago Mercantile Exchange (CME) E-mini S&P 500 futures contract specifies: - Underlying: S&P 500 Index - Contract size: $50 × S&P 500 Index value - Delivery months: March, June, September, December - Tick size: 0.25 index points = $12.50 per contract

Long and Short Positions

Long Position: Agreement to buy the underlying asset - Profits when futures price rises - Obligated to accept delivery (unless closed before expiration) - Initial margin required

Short Position: Agreement to sell the underlying asset
- Profits when futures price falls - Obligated to make delivery (unless closed before expiration) - Initial margin required

Unlike options, both parties to a futures contract have obligations, not rights.

Physical vs. Cash Settlement

Physical Settlement: Actual delivery of the underlying asset - Common for commodities (agricultural products, metals, energy) - Delivery occurs at exchange-designated locations - Most contracts are closed before delivery

Cash Settlement: Payment of cash difference between futures and spot prices - Used for financial futures (stock indices, interest rates) - No physical asset changes hands - Settlement price typically based on underlying index value

3.2 Marking to Market

One of the most distinctive features of futures trading is the daily settlement or “marking to market” process. This mechanism virtually eliminates counterparty risk and enables leverage while maintaining market integrity.

The Daily Settlement Process

End-of-Day Valuation: Each trading day, the exchange establishes a settlement price (typically close to the closing price) for each futures contract. All open positions are marked to this price.

Variation Margin: The daily profit or loss on each position is calculated and immediately transferred between accounts: - Profit: Credited to the trader’s margin account - Loss: Debited from the trader’s margin account

Example: Suppose you buy one S&P 500 futures contract at 4,000: - Day 1: Settlement price = 4,010 → Gain = (4,010 - 4,000) × $50 = $500 (credited to your account) - Day 2: Settlement price = 3,990 → Loss = (3,990 - 4,010) × $50 = $1,000 (debited from your account) - Day 3: Settlement price = 4,020 → Gain = (4,020 - 3,990) × $50 = $1,500 (credited to your account)

Margin Requirements

Initial Margin: Deposit required to open a futures position - Typically 3-12% of contract value (much lower than stock margin) - Provides leverage but amplifies risk - Set by exchanges based on volatility and risk assessment

Maintenance Margin: Minimum account balance required to keep position open - Usually 75-80% of initial margin - If account falls below maintenance margin, a margin call occurs

Margin Call Process: 1. Account balance falls below maintenance margin 2. Trader must deposit additional funds (variation margin) to restore account to initial margin level 3. If trader fails to meet margin call, position is liquidated

Why Marking to Market Works

Eliminates Credit Risk: Daily settlement ensures that losers pay winners immediately, preventing the accumulation of large credit exposures.

Enables Leverage: Because credit risk is minimized, exchanges can offer high leverage ratios that would be impossible in other markets.

Price Transparency: Daily marking provides continuous price discovery and ensures that futures prices reflect current market conditions.

Liquidity Enhancement: Reduced counterparty risk attracts more participants, increasing market liquidity.

3.3 Spot-Futures Parity

The relationship between spot prices and futures prices is governed by arbitrage principles similar to those in options markets. Spot-futures parity establishes the theoretical fair value of a futures contract based on the current spot price and carrying costs.

The Basic Parity Relationship

For a futures contract on a non-dividend-paying asset:

\[F = S e^{rT}\]

Where: - \(F\) = Futures price - \(S\) = Current spot price
- \(r\) = Risk-free interest rate - \(T\) = Time to expiration

This relationship reflects the cost of carry - the net cost of holding the underlying asset until the futures expiration date.

Derivation Through Arbitrage

Consider two strategies with identical payoffs at time \(T\):

Strategy A: Buy futures contract at price \(F\) - Cost today: \(0\) (ignoring margin) - Payoff at \(T\): \(S_T - F\)

Strategy B: Buy underlying asset, finance with borrowing - Cost today: \(S - S = 0\) (borrow \(S\) to buy asset worth \(S\))
- Payoff at \(T\): \(S_T - Se^{rT}\) (asset value minus loan repayment)

Since both strategies have identical payoffs, they must have equal costs: \[S_T - F = S_T - Se^{rT}\] \[F = Se^{rT}\]

Incorporating Dividends and Storage Costs

Dividend-Paying Assets: \[F = (S - PV(\text{Dividends}))e^{rT}\]

Expected dividends reduce the futures price because futures holders don’t receive dividend payments.

Commodities with Storage Costs: \[F = Se^{(r+c)T}\]

Where \(c\) = storage cost rate. Storage costs increase the futures price because they represent an additional carrying cost.

Convenience Yield: For commodities, there may be benefits to physical ownership (convenience yield \(y\)): \[F = Se^{(r+c-y)T}\]

Convenience yield can make futures prices lower than the basic cost-of-carry model predicts.

Arbitrage When Parity Is Violated

If \(F > Se^{rT}\) (Futures overpriced): 1. Sell futures contract 2. Buy underlying asset
3. Finance purchase by borrowing \(S\) 4. At expiration: Deliver asset against futures, collect \(F\), repay loan \(Se^{rT}\) 5. Risk-free profit: \(F - Se^{rT}\)

If \(F < Se^{rT}\) (Futures underpriced): 1. Buy futures contract 2. Sell underlying asset short 3. Invest proceeds \(S\) at risk-free rate 4. At expiration: Take delivery via futures, return borrowed asset, collect investment proceeds 5. Risk-free profit: \(Se^{rT} - F\)

Practical Considerations

Transaction Costs: Reduce arbitrage opportunities and create a “no-arbitrage band” around theoretical parity.

Borrowing vs. Lending Rates: Different rates for borrowing and lending create wider no-arbitrage bands.

Delivery Options: Many futures contracts give the short party choices about delivery timing, location, or grade, making exact parity calculations complex.

3.4 The Expectations Hypothesis

The expectations hypothesis proposes that futures prices represent the market’s unbiased expectation of future spot prices. This theory attempts to explain how futures prices relate to expected future spot prices, though empirical evidence provides mixed support.

The Basic Expectations Hypothesis

\[F = E[S_T]\]

Where \(E[S_T]\) is the expected spot price at the futures expiration date \(T\).

Under this hypothesis: - Futures markets are purely informational, aggregating diverse opinions about future prices - No risk premium is embedded in futures prices - Futures prices should be unbiased predictors of future spot prices

Normal Backwardation vs. Contango

Normal Backwardation: Futures prices below expected future spot prices - \(F < E[S_T]\) - Hedgers are net short, speculators are net long - Speculators earn risk premium for bearing price risk

Contango: Futures prices above expected future spot prices
- \(F > E[S_T]\) - Hedgers are net long, speculators are net short - Hedgers pay risk premium to transfer price risk

Example: Agricultural commodities often exhibit seasonal patterns: - Post-harvest: Abundant supply, low spot prices, futures in contango - Pre-harvest: Low inventory, high spot prices, futures in backwardation

The Theory of Storage

The theory of storage explains futures price patterns for storable commodities:

High Inventory Periods: - Storage is costly, convenience yield is low - Futures prices exceed spot prices (contango) - Market expects lower future prices as new supply arrives

Low Inventory Periods: - Storage provides high convenience yield - Spot prices exceed futures prices (backwardation)
- Market fears supply shortages

Modern Portfolio Theory Perspective

Modern finance theory suggests that futures prices should include risk premiums related to systematic risk:

\[F = E[S_T] - \text{Risk Premium}\]

Risk Premium Determinants: - Correlation between commodity returns and market portfolio - Systematic risk of the underlying asset - Risk aversion of market participants

Empirical Evidence: - Mixed support for pure expectations hypothesis - Some commodities show persistent backwardation or contango - Risk premiums appear to vary over time and across commodities

3.5 Options on Futures

Many exchanges offer options on futures contracts, creating a derivative on a derivative. These instruments combine the features of options (limited downside risk) with the efficiency and liquidity of futures markets.

Contract Specifications

Call Option on Futures: Right to buy a futures contract at a specified strike price - Exercise creates a long futures position at the strike price - Immediate mark-to-market profit/loss upon exercise

Put Option on Futures: Right to sell a futures contract at a specified strike price - Exercise creates a short futures position at the strike price - Immediate mark-to-market profit/loss upon exercise

Exercise and Settlement

American-Style: Most options on futures can be exercised any time before expiration

Exercise Process: 1. Option holder notifies broker of exercise decision 2. Clearinghouse assigns exercise to a random short option holder 3. Assigned option writer receives a futures position 4. Both parties receive immediate mark-to-market settlement

Example: Crude oil futures trading at $80, holder exercises $75 call: - Call holder receives long futures position at $75 - Immediate credit: $(80 - 75) × 1,000 = \(5,000\) per contract - Option writer receives short futures position at $75 - Immediate debit: $5,000 per contract

Pricing Models

Black Model (1976): Extension of Black-Scholes for options on futures:

\[C = e^{-rT}[F \Phi(d_1) - K \Phi(d_2)]\]

\[P = e^{-rT}[K \Phi(-d_2) - F \Phi(-d_1)]\]

Where: \[d_1 = \frac{\ln(F/K) + \frac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}\] \[d_2 = d_1 - \sigma\sqrt{T}\]

Key Insight: The futures price \(F\) replaces the spot price in the standard Black-Scholes formula, and the entire option value is discounted at the risk-free rate.

Strategic Applications

Hedging Futures Positions: - Long futures + long put = Limited downside, unlimited upside - Short futures + long call = Limited downside, unlimited upside

Speculation with Limited Risk: - Buy calls/puts instead of futures to limit maximum loss to premium - Leverage benefits of futures with risk management of options

Spread Trading: - Calendar spreads using options on different expiration futures - Volatility trading using straddles and strangles

Advantages Over Options on Spot Assets

Leverage: Futures provide inherent leverage, amplifying option sensitivity

Liquidity: Major futures markets often more liquid than spot markets

No Storage Costs: Options on financial futures avoid physical storage issues

Extended Trading Hours: Many futures markets trade nearly 24 hours

Mark-to-Market: Immediate settlement reduces counterparty risk

3.6 Futures Markets in the Modern Economy

Futures markets serve several crucial economic functions beyond simple speculation:

Price Discovery: Aggregating information from diverse market participants to establish forward-looking prices

Risk Management: Enabling producers, consumers, and investors to hedge price risks

Market Efficiency: Facilitating arbitrage that links spot and futures markets

Capital Efficiency: Providing leverage and liquidity that enhance capital allocation

Understanding these markets and their relationships to spot prices, expectations, and options provides essential foundation for derivatives pricing and risk management in modern finance.