Pricing and Hedging Derivative Securities
Introduction to Futures
What Are Futures Contracts?
Futures contracts are standardized agreements to deliver an asset at a future date in exchange for payment at that date.
- Futures price: Price at which a contract trades. The price is paid at delivery.
- Delivery: Asset exchanged at expiration
- Payment: Made at delivery, not when contract is traded
- Both buyer and seller have obligations (unlike options!)
Futures vs. Options
Futures Contracts
- Both parties have obligations
- Both must post margin
- No upfront payment
- Symmetrical payoff
- Daily settlement
Options Contracts
- Only seller has obligation
- Only seller posts margin
- Buyer pays premium upfront
- Asymmetric payoff
- Settlement at expiration
Historical Context
Futures have a long history in commodity markets:
- 1851: Modern futures trading begins in the United States
- 1981: Cash-settled contracts introduced
- Today: Futures on commodities, financial assets, indices, currencies, and more
Cash-settled contracts: No physical delivery - settled based on underlying value at expiration
Long and Short Positions
Long (Buyer): Obligation to accept delivery, profits when price rises
Short (Seller): Obligation to deliver, profits when price falls
Example: Trading Corn Futures
Farmer sells futures at $5.00/bushel, spot later drops to $4.00:
Physical Delivery: Deliver corn, receive $5.00/bushel
Offset Position:
- Buy futures at $4.00 → Profit = $1.00/bushel
- Sell corn in spot market at $4.00
- Total: $5.00/bushel (same result!)
Key insight: Spot-futures convergence at expiration
Contract Specifications
Exchanges standardize all contract terms:
- Delivery location and date
- Quantity and quality (with price adjustments)
Example: CME Corn Futures - 5,000 bushels per contract, Illinois delivery locations
Cash-Settled Futures
No physical delivery - settled based on index value:
E-mini S&P 500: $50 multiplier per index point
- Buy at 6000, index expires at 6500
- Profit = $50 × 500 = $25,000
Benefits: Eliminates delivery logistics, enables index futures
Trading vs. Gambling
Key distinction: Futures must serve legitimate risk-management needs
Examples:
- Fund manager hedging equity portfolio
- Airline hedging fuel costs
- Utility hedging temperature risk
Market Structure
Like options, futures have:
- Exchange-traded: Standardized, transparent
- Clearinghouse: Central counterparty
- Open interest: Long = short positions
- Most positions closed early via offsetting trades
Margin and Marking to Market
Daily Settlement Process:
- Exchange sets settlement price each day
- All positions marked to this price
- Profits/losses immediately transferred
- Virtually eliminates counterparty risk
Daily Settlement Example
Buy 1 e-mini S&P 500 contract at 6000:
| 0 (Entry) |
6000 |
- |
- |
| 1 |
6010 |
+$500 |
+$500 |
| 2 |
5990 |
-$1,000 |
-$500 |
| 3 |
6020 |
+$1,500 |
+$1,000 |
Day 3: Close position, walk away with $1,000 gain
Calculation: (6020 - 6000) × $50 = $1,000
Margin Requirements
Two types of margin:
Initial Margin
- Required to open position
- Typically 3-12% of contract value
- Set by exchange
Maintenance Margin
- Minimum to keep position open
- Lower than initial margin
- If breached → margin call
Margin Call: Demand for additional margin to restore account to initial margin level. Failure to meet = forced liquidation.
Leverage in Futures
10% margin = 10:1 leverage
- 1% price move = 10% gain/loss on margin
- 10% price move = 100% gain/loss on margin
Safety mechanisms: Daily marking to market + margin calls + clearinghouse
The Expectations Hypothesis
Futures prices = Expected future spot prices
Logic: If \(F \neq E[S_T]\), trading for expected profit drives \(F \to E[S_T]\)
Problem: Assumes risk-neutral investors!
Reality: Risk Premia
Risk-averse investors won’t trade unlimited amounts
Result: Futures prices can deviate from expected spot prices
- Deviation depends on covariance with market risk
- From CAPM: covariance with market return
Despite this: Expectations still prime determinant of prices
Spot-Futures Parity
Unlike expectations hypothesis, parity is an arbitrage relationship:
\[F = \mathrm{e}^{(c-y)T}S\]
where:
- \(F\) = futures price
- \(S\) = current spot price
- \(c\) = cost of carry (continuously compounded)
- \(y\) = convenience yield (continuously compounded)
- \(T\) = time to expiration
Cost of Carry and Convenience Yield
Cost of Carry (\(c\))
- Foregone interest on capital
- Storage costs (commodities)
- No costs for holding futures
For financial assets: \(c = r\)
Convenience Yield (\(y\))
- Dividends/cash flows
- Benefits of physical possession
- Never negative
For financial assets: \(y\) = dividend yield
Arbitrage: Buy Spot, Sell Futures
If \(F > \mathrm{e}^{rT}S\) (assuming \(c=r\), \(y=0\)):
- Borrow \(S\), buy spot asset
- Sell futures at \(F\)
- At expiration: deliver asset, receive \(F\), repay \(\mathrm{e}^{rT}S\)
Profit: \(F - \mathrm{e}^{rT}S\) (risk-free!)
Arbitrage: Sell Spot, Buy Futures
If \(F < \mathrm{e}^{rT}S\) (assuming \(c=r\), \(y=0\)):
- Short sell spot asset for \(S\), invest proceeds
- Buy futures at \(F\)
- At expiration: pay \(F\), accept delivery, return to lender
Profit: \(\mathrm{e}^{rT}S - F\) (risk-free!)
Why Parity Holds
Institutional arbitrageurs exploit violations quickly
Extensions:
- Storage costs → add to \(c\)
- Dividends → subtract from \(c\) (or add to \(y\))
- General formula: \(F = \mathrm{e}^{(c-y)T}S\)
Forward Curves
Forward curve: Relationship between futures price and contract maturity
From spot-futures parity:
\[F = \mathrm{e}^{(c-y)T}S\]
Shape depends on \(c - y\):
- If \(c > y\): Upward sloping (contango)
- If \(c < y\): Downward sloping (backwardation)
- If \(c = y\): Flat
Gold Forward Curve
Gold: negligible storage, no dividends, no convenience yield → \(c - y = r\)
Currency Forward Curves
For currencies: \(c - y = r - r_f\) (domestic vs. foreign interest rates)
S&P 500 Forward Curve
For stock indices: \(c - y = r - \text{dividend yield}\)
Commodity Forward Curves
More complex due to storage costs and convenience yields:
- High storage costs (electricity, natural gas)
- Convenience yield varies with supply/demand
- Upward or downward sloping
- Seasonal patterns (natural gas, corn)
Bound: \(F \leq \mathrm{e}^{cT}S\) (may have little predictive power)
Crude Oil: Extreme Scenarios
Crude Oil Insights
April 2020:
- Coronavirus demand collapse
- Storage at Cushing full
- Extremely high storage costs
- Zero convenience yield
- Result: Steep upward slope
- Front month went negative!
March 2022:
- Russian invasion of Ukraine
- Future supplies uncertain
- High convenience yield
- Result: Steep downward slope
- Premium for oil now vs. later
Cyclical Commodities
Natural gas and corn show seasonal patterns:
Natural Gas Seasonality
Summer to Winter (upward): High storage costs
Winter to Summer (downward): High convenience yield (demand peak)
Overall slope: Depends on supply/demand conditions
Corn Seasonality
Pre-harvest (summer): High convenience yield (scarcity)
Post-harvest (fall): High storage costs (silos full)
Overall slope: Varies with spot market conditions
Options on Futures
A derivative on a derivative!
Call: Right to buy futures at strike
Put: Right to sell futures at strike
Benefits: Leverage, liquidity, avoids delivery/storage
Exercise of Futures Options
When exercised, creates futures position + cash transfer:
Option Holder:
- Call → Long futures position
- Put → Short futures position
- Marked to market at current futures price
- Receives cash = (Futures price - Strike)
Option Writer:
- Call → Short futures position
- Put → Long futures position
- Marked to market at current futures price
- Pays cash = (Futures price - Strike)
Futures Options Example
Buy call on corn futures, strike = $5.00, exercise at $5.50:
You (exerciser):
- Long futures at $5.50 + receive $0.50 cash
Option writer:
- Short futures at $5.50 + pay $0.50 cash
Result: Both can immediately close futures positions
Summary: Key Takeaways
Futures = Obligations for both parties (vs. options)
Daily marking to market virtually eliminates counterparty risk
Leverage through margin enables high returns/losses
Spot-futures parity is an arbitrage relationship: \(F = \mathrm{e}^{(c-y)T}S\)
Forward curves reveal cost of carry minus convenience yield
Commodity curves complex due to storage and seasonality
Futures options combine benefits of both derivatives
Next Steps
Building on this foundation:
- Hedging with futures: Optimal hedge ratios, basis risk
- Futures pricing models: Storage models, asset pricing theory
- Spread trading: Calendar spreads, inter-commodity spreads
- Arbitrage strategies: Cash-and-carry, reverse cash-and-carry
- Real-world considerations: Transaction costs, margin management, delivery options