Topic 6: Pricing and Valuation of Futures Contracts
Learning Objectives Coverage
LO1: Compare the value and price of forward and futures contracts
Core Concept
Futures and forward contracts are similar derivatives but differ fundamentally in their settlement mechanisms and margin requirements. While both have zero initial value, their ongoing valuation diverges because futures use daily marking-to-market whereas forwards settle only at maturity. This difference in cash flow timing, combined with the cost-of-carry framework, drives the pricing distinctions between the two instruments. exam-focus
Key Formulas
Futures Price (no costs/benefits):
f₀(T) = S₀(1 + r)ᵀ
Forward Contract MTM Value:
Vₜ(T) = Sₜ - F₀(T)(1 + r)^(-(T-t))
Futures MTM Reset Daily:
V₀(T) = 0 (resets to zero after daily settlement)
Practical Example
Gold Futures vs Forward Contract:
- Spot price: $1,770/oz
- Risk-free rate: 2%
- Time to maturity: 91 days (0.24932 years)
- Both futures and forward price: $1,778.76/oz
Day 1 Price Movement (Spot rises to $1,775):
- Forward MTM: -$498 (unrealized loss)
- Futures MTM: -$500 (realized loss, paid immediately)
DeFi Application defi-application
In DeFi perpetual futures (like dYdX or GMX), funding rates replace daily settlement, smart contracts automate margin calls, and no centralized clearinghouse is needed. For example, GMX uses dynamic funding rates that adjust every hour based on long/short imbalance — a continuous version of the daily mark-to-market mechanism in traditional futures markets.
LO2: Explain why forward and futures prices differ
Core Concept
Price differences between futures and forwards arise from the interaction between daily settlement cash flows and interest rate movements. The correlation between futures prices and interest rates determines whether futures trade at a premium or discount to forwards. This is a subtle but frequently tested distinction. exam-focus
Key Relationships
When futures prices are positively correlated with interest rates:
→ Futures Price > Forward Price
When futures prices are negatively correlated with interest rates:
→ Forward Price > Futures Price
When correlation = 0 or rates are constant:
→ Futures Price = Forward Price
Practical Example
Commodity Futures During Inflation:
- Oil futures prices rise with inflation
- Interest rates also rise with inflation
- Positive correlation → Futures > Forwards
- Daily gains can be reinvested at higher rates
- Daily losses are financed at higher rates (offset)
DeFi Application defi-application
Perpetual swaps on platforms like dYdX use funding payments instead of daily settlement. The absence of an expiry date means continuous price convergence through the funding rate mechanism: Funding rate = (Mark Price - Index Price) / Index Price. When BTC perps trade above spot, longs pay shorts every 8 hours, driving price convergence just as the daily mark-to-market does in traditional futures.
Core Concepts Summary (80/20 Principle)
The 20% You Must Know:
- Futures reset daily - MTM settlements occur every day, value returns to zero
- Convergence at maturity - Futures price must equal spot price at expiration
- Margin requirements - Initial and maintenance margins manage counterparty risk
- Interest rate correlation - Determines if futures trade above/below forwards
- Central clearing - Eliminates counterparty risk through clearinghouse
The 80% That Matters Most:
- Daily settlement creates different cash flow patterns than forwards
- Margin accounts require immediate funding for losses
- Price convergence prevents arbitrage opportunities
- Interest rate volatility affects pricing differences
- Central clearing standardizes risk management
Comprehensive Formula Sheet
Basic Futures Pricing
Initial Value:
V₀(T) = 0
Futures Price (Discrete):
f₀(T) = S₀(1 + r)ᵀ
Futures Price (Continuous):
f₀(T) = S₀e^(rT)
With Storage Costs/Income:
f₀(T) = [S₀ - PV₀(I) + PV₀(C)](1 + r)ᵀ
Interest Rate Futures
Price Convention:
fₐ,ᵦ₋ₐ = 100 - (100 × MRRₐ,ᵦ₋ₐ)
Basis Point Value:
BPV = Notional × 0.01% × Period
FRA Settlement:
Net Payment = (MRRᵦ₋ₐ - IFRₐ,ᵦ₋ₐ) × Notional × Period
Present Value Settlement:
PV = Net Payment / (1 + MRR/n)
Margin Calculations
Initial Margin Requirement:
IM = Contract Value × IM%
Maintenance Margin:
MM = Contract Value × MM%
Margin Call Trigger:
Account Balance < MM
Variation Margin:
VM = (Current Price - Previous Price) × Contract Size
HP 12C Calculator Sequences
Futures Price Calculation
Example: Gold futures, 91-day maturity
S₀ = $1,770, r = 2%
[f] [CLX]
1770 [ENTER]
1.02 [ENTER]
91 [ENTER]
365 [÷] // T = 0.24932
[y^x] // (1.02)^0.24932
[×] // Result: $1,778.76
Present Value of Storage Costs
Storage = $2 at maturity, 91 days
[f] [CLX]
2 [ENTER]
1.02 [ENTER]
91 [ENTER]
365 [÷]
[y^x]
[÷] // Result: $1.99
Interest Rate Futures BPV
Notional = $50M, 1-month period
[f] [CLX]
50000000 [ENTER]
0.0001 [×]
12 [÷] // Result: $416.67
Practice Problems
Basic Level
Problem 1: Calculate the 3-month futures price for gold:
- Spot: $1,800/oz
- Risk-free rate: 3%
- Time: 3 months
Solution:
f₀(T) = $1,800 × (1.03)^0.25
f₀(T) = $1,800 × 1.00741
f₀(T) = $1,813.34
Problem 2: Initial margin calculation:
- Contract value: $100,000
- Initial margin: 10%
- Maintenance margin: 6%
Solution:
- Initial margin required: $10,000
- Margin call trigger: < $6,000
- Replenishment amount: Back to $10,000
Intermediate Level
Problem 3: Forward vs Futures MTM comparison:
- Initial spot: $50
- Forward/Futures price: $51
- Day 1 spot: $52
- Risk-free rate: 4%
- Time to maturity: 30 days
Solution: Forward MTM:
V₁ = $52 - $51/(1.04)^(29/365)
V₁ = $52 - $50.97
V₁ = $1.03 (unrealized)
Futures MTM:
Daily gain = $52 - $50 = $2
Settled immediately = $2 (realized)
Contract resets at $52
Advanced Level
Problem 4: Interest rate futures hedging:
- Portfolio value: $10M
- Duration: 5 years
- Hedge with 3-month Eurodollar futures
- Contract notional: $1M
- Contract BPV: $25
Solution:
Portfolio BPV = $10M × 5 × 0.01% = $5,000
Contracts needed = $5,000 / $25 = 200 contracts
Position: Short 200 contracts (to hedge rate increases)
DeFi Applications & Real-World Examples
1. Perpetual Futures Protocols
dYdX Example:
- No expiry date (perpetual)
- Funding rate every hour
- Oracle price feeds from Chainlink
- Automatic liquidation engine
- Cross-margining capabilities
GMX Implementation:
- GLP liquidity pool acts as counterparty
- Dynamic fees based on utilization
- Real-time oracle pricing
- Zero price impact for large trades
2. Decentralized Clearing
Traditional Futures:
- CME Clearinghouse
- Initial margin: 5-10%
- Daily settlement through banks
DeFi Futures:
- Smart contract clearing
- Over-collateralization (often 150%+)
- Instant settlement on-chain
- No counterparty risk
3. Synthetic Assets
Synthetix Protocol:
- Creates synthetic futures (sFutures)
- Collateralized by SNX token
- Oracle-based pricing
- No order book needed
4. Real-World Arbitrage
Basis Trade Example:
Spot BTC: $40,000
3-month futures: $41,200
Implied rate: ($41,200/$40,000)^4 - 1 = 12.55% annualized
Strategy:
1. Buy spot BTC
2. Short futures
3. Lock in 3% return (3 months)
4. Risk-free if held to maturity
Common Pitfalls & Exam Tips
Common Mistakes to Avoid
-
Confusing price and value
- Price is agreed upon rate
- Value is MTM profit/loss
-
Forgetting daily settlement impact
- Futures realize gains/losses daily
- Forwards only at maturity
-
Misunderstanding convergence
- Futures price → Spot price at maturity
- Not gradual, can be volatile
-
Ignoring margin requirements
- Initial margin ≠ Maintenance margin
- Margin calls require full replenishment
Exam Strategy Tips
-
Quick identification:
- “Exchange-traded” → Futures
- “OTC” → Forwards
- “Daily settlement” → Futures
-
Calculation shortcuts:
- If rates constant → Futures = Forward price
- Positive correlation → Futures > Forwards
- At maturity → Futures = Spot
-
Time saver formulas:
- Use continuous compounding when allowed
- Remember BPV = Notional × 0.01% × Period
Key Takeaways
Must Remember:
- Futures contracts have zero initial value but require margin
- Daily marking-to-market distinguishes futures from forwards
- Convergence ensures futures price equals spot at maturity
- Interest rate correlation determines pricing relationships
- Central clearing eliminates counterparty risk
Critical Insights:
- Futures provide better liquidity than forwards
- Daily settlement affects cash management
- Margin requirements create leverage limits
- Standardization enables efficient markets
- DeFi perpetuals innovate on traditional futures
Cross-References & Additional Resources
Related Topics:
- Topic 5: Forward contract pricing mechanics
- Topic 7: Swap valuation (series of forwards)
- Topic 8: Option pricing (different settlement)
Key Readings:
- CME Group: Understanding Futures Margins
- Hull, J.: “Options, Futures, and Other Derivatives”
- DeFi Specific: dYdX Documentation on Perpetuals
Practice Resources:
- CME Institute Education Portal
- Finance Question Bank
- Binance Academy: Futures Trading
DeFi Protocols to Study:
- dYdX: Decentralized perpetuals
- GMX: Decentralized spot and perpetual exchange
- Synthetix: Synthetic futures
- Perpetual Protocol: vAMM-based futures
Review Checklist
Conceptual Understanding
- Can you explain the difference between forward and futures contracts?
- Do you understand daily marking-to-market?
- Can you describe when futures > forward prices?
- Do you know the convergence property?
Calculations
- Can you calculate futures prices with the cost-of-carry model?
- Can you compute margin requirements and margin calls?
- Can you determine MTM values for both forwards and futures?
- Can you calculate interest rate futures prices?
Applications
- Can you identify appropriate hedging strategies?
- Do you understand basis trades?
- Can you explain DeFi perpetual funding rates?
- Can you compare traditional and DeFi futures?
Exam Readiness
- Memorized key formulas
- Practiced HP 12C sequences
- Reviewed common pitfalls
- Completed practice problems
DeFi Integration
- Understand perpetual swaps mechanics
- Know major DeFi futures protocols
- Can explain on-chain settlement advantages
- Familiar with funding rate calculations