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Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

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Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Learning Objectives Coverage

LO1: Explain how the concepts of arbitrage and replication are used in pricing derivatives

Core Concept

Arbitrage and replication form the foundation of derivative pricing by establishing that identical cash flows must have identical prices. This no-arbitrage principle ensures that derivatives are priced consistently with their underlying assets through the absence of risk-free profit opportunities. Every pricing formula in the remaining topics — from forward pricing to the binomial model — rests on this foundation. exam-focus

Key Formulas formula no-arbitrage

  • No-arbitrage condition: S₀ = S_T(1 + r)^(-T)
  • Law of one price: S₀^A = S₀^B (identical assets)
  • Forward replication: F₀(T) = S₀(1 + r)
  • Replication payoff: CF_T = S_T - F₀(T) (long forward)

Examples

Arbitrage Opportunity Example:

  • Two zero-coupon bonds: EUR99 vs EUR99.15
  • Arbitrage profit: EUR0.15 per bond
  • Strategy: Sell at EUR99.15, buy at EUR99

Replication Example:

  • Gold spot: USD1,783.28
  • Risk-free rate: 2%
  • Time: 0.25 years
  • Forward price: USD1,783.28 × (1.02)^0.25 = USD1,792.13

DeFi Application defi-application

In DeFi protocols like dYdX and Perpetual Protocol, arbitrage bots maintain price parity between spot prices on DEXs (Uniswap) and perpetual futures prices. Funding rates adjust continuously to eliminate arbitrage opportunities, serving the same economic function as the no-arbitrage conditions enforced by market makers in traditional finance.

Smart contracts enforce these conditions through automated market makers (AMMs), flash loan arbitrage mechanisms, and cross-protocol price oracles (Chainlink). The speed and atomicity of on-chain transactions mean that arbitrage opportunities in DeFi can be captured and eliminated within a single block.

LO2: Explain the difference between spot and expected future price and the cost of carry

Core Concept

The cost of carry model explains why forward prices differ from spot prices based on the net costs and benefits of holding the underlying asset. This includes opportunity costs (the risk-free rate), storage costs, and income generated by the asset. The model is the quantitative backbone of all forward commitment pricing in the finance curriculum. exam-focus

Key Formulas formula pricing

  • Basic cost of carry: F₀(T) = S₀(1 + r)
  • With costs/benefits: F₀(T) = [S₀ - PV₀(I) + PV₀(C)](1 + r)
  • Continuous rate model: F₀(T) = S₀e^((r+c-i)T)
  • FX forward: F₀,f/d(T) = S₀,f/d x e^((r_f-r_d)T)

Examples

Equity with Dividends (Unilever):

  • Spot: EUR50
  • Dividend: EUR0.30 quarterly
  • Risk-free: 5%
  • Time: 6 months
  • PV(dividends): EUR0.5892
  • Forward: (EUR50 - EUR0.5892)(1.05)^0.5 = EUR50.63

Commodity with Storage (Gold):

  • Spot: USD1,783.28
  • Storage cost: USD2 at maturity
  • PV(storage): USD1.99
  • Forward: (USD1,783.28 + USD1.99)(1.02)^0.25 = USD1,794.13

DeFi Application defi-application

Synthetix implements cost of carry through funding rates for synthetic assets, staking rewards (negative carry cost), and protocol fees (positive carry cost). Meanwhile, Compound Finance demonstrates carry costs directly: supply APY represents a negative carry benefit, borrow APY represents a positive carry cost, and cToken appreciation reflects the net carry. These mechanics mirror the traditional cost-of-carry framework, with smart contract parameters replacing the risk-free rate and storage cost terms.

Core Concepts Summary (80/20 Principle)

Critical 20% to Master:

  1. No-Arbitrage Principle: Identical cash flows → identical prices
  2. Basic Forward Formula: F₀(T) = S₀(1 + r)
  3. Cost of Carry Components: Opportunity cost + Storage costs - Income benefits
  4. Replication Strategy: Long spot + Borrow cash = Long forward

Results in 80% Understanding:

  • All derivative pricing derives from no-arbitrage
  • Forward prices adjust for holding costs/benefits
  • Market efficiency eliminates arbitrage opportunities
  • Replication enables synthetic position creation

Comprehensive Formula Sheet

Arbitrage Relationships

No-arbitrage condition:     S₀ = S_T/(1 + r)^T
Law of one price:           S₀^A = S₀^B
Arbitrage profit:           π = F₀(T)_market - F₀(T)_theoretical

Forward Pricing Models

Basic (discrete):           F₀(T) = S₀(1 + r)^T
Basic (continuous):         F₀(T) = S₀e^(rT)
With costs/benefits:        F₀(T) = [S₀ - PV₀(I) + PV₀(C)](1 + r)^T
Rate-based model:           F₀(T) = S₀e^((r+c-i)T)

Asset-Specific Formulas

Equity with dividends:      F₀(T) = [S₀ - PV₀(Div)](1 + r)^T
Equity index:              F₀(T) = S₀e^((r-δ)T)        [δ = dividend yield]
Foreign exchange:          F₀,f/d(T) = S₀,f/d × e^((rf-rd)T)
Commodity with storage:    F₀(T) = [S₀ + PV₀(Storage)](1 + r)^T

Replication Formulas

Long forward replication:   Borrow S₀, Buy spot
Short forward replication:  Lend S₀, Sell spot short
Synthetic position:         Long spot + Short forward = Risk-free bond

Market Conditions

Contango:                   F₀(T) > S₀ (costs > benefits)
Backwardation:             F₀(T) < S₀ (benefits > costs)
Normal market:             F₀(T) = S₀(1 + r)^T

HP 12C Calculator Sequences

Forward Price Calculation

Basic Forward Price:
[Spot] ENTER
[1] [+] [Rate] [%] [=]
[Time] [y^x]
[×]

Example: S₀=$100, r=5%, T=0.5
100 ENTER
1.05
0.5 y^x
× 
Result: $102.47

Present Value of Costs/Benefits

PV of future payment:
[Future Amount] ENTER
[1] [+] [Rate] [%] [=]
[Time] [CHS] [y^x]
[×]

Example: $2 cost in 0.25 years, r=2%
2 ENTER
1.02
0.25 CHS y^x
×
Result: $1.99

Arbitrage Profit Calculation

Market price - Theoretical price:
[Market Forward] ENTER
[Spot] [1] [+] [Rate] [%] [=] [Time] [y^x] [×]
[-]

Example: Market=$52.50, Spot=$50, r=4%, T=0.5
52.50 ENTER
50 1.04 0.5 y^x × 
-
Result: $1.51 profit

Practice Problems

Basic Level

Problem 1: Calculate forward price

  • Spot gold: $1,800/oz
  • Risk-free rate: 3%
  • Time: 6 months

Solution: F₀(T) = 1,800 × (1.03)^0.5 = $1,826.65

Problem 2: Identify arbitrage opportunity

  • Stock spot: $75
  • 3-month forward: $77
  • Risk-free rate: 4%

Solution: Theoretical forward = 75 × (1.04)^0.25 = 77) > Theoretical (77 - 1.26

Intermediate Level

Problem 3: Forward with dividends

  • Stock price: $120
  • Quarterly dividend: $1.50
  • Risk-free rate: 6%
  • Forward maturity: 9 months

Solution: PV(dividends) = 1.50/(1.06)^0.25 + 1.50/(1.06)^0.5 + 1.50/(1.06)^0.75 PV(dividends) = 1.478 + 1.456 + 1.435 = 120.14

Problem 4: FX forward pricing

  • Spot EUR/USD: 1.1850
  • EUR rate: 0.5%
  • USD rate: 2.5%
  • Time: 1 year

Solution: F₀(T) = 1.1850 × e^((0.005-0.025)×1) = 1.1850 × e^(-0.02) = 1.1616

Advanced Level

Problem 5: Commodity with storage and convenience yield

  • Oil spot: $85/barrel
  • Storage cost rate: 3% per annum
  • Convenience yield: 5% per annum
  • Risk-free rate: 4%
  • Time: 3 months

Solution: F₀(T) = 85 × e^((0.04+0.03-0.05)×0.25) = 85 × e^(0.02×0.25) = $85.43

Problem 6: Multi-period replication strategy Create synthetic 6-month forward on index:

  • Index: 4,500
  • Dividend yield: 2%
  • Risk-free: 3%

Solution:

  1. Borrow: 4,500 at 3%
  2. Buy index receiving 2% dividends
  3. Net forward price: 4,500 × e^((0.03-0.02)×0.5) = 4,522.52
  4. Payoff at maturity: S_T - 4,522.52

DeFi Applications & Real-World Examples

Perpetual Futures (dYdX, GMX) defi-application

  • Funding rate = (Mark - Index) / Index
  • Positive funding: Longs pay shorts (contango)
  • Negative funding: Shorts pay longs (backwardation)
  • 8-hour funding periods maintain price convergence

Synthetic Assets (Synthetix)

// Simplified synthetic forward creation
function createSyntheticForward(uint256 spotPrice, uint256 rate, uint256 time) {
    uint256 forwardPrice = spotPrice * (10000 + rate) ** time / 10000 ** time;
    mint(msg.sender, forwardPrice);
}

Flash Loan Arbitrage (Aave)

// Arbitrage between spot and futures
if (futuresPrice > spotPrice * (1 + borrowRate)) {
    flashLoan(amount);
    buySpot(amount);
    sellFutures(amount);
    repayLoan(amount + fee);
    profit = futuresPrice - spotPrice - loanCost;
}

Traditional Finance Examples

Gold ETF (GLD):

  • Tracks spot gold minus storage/insurance costs
  • Forward price includes vault fees (~0.4% annually)
  • No convenience yield for financial holders

Currency Forwards (Corporate):

  • Multinational hedging EUR/USD exposure
  • 6-month forward locks in exchange rate
  • Cost of carry = interest rate differential

Agricultural Futures:

  • Wheat contango during harvest (high supply)
  • Backwardation during shortage (convenience yield)
  • Storage costs significant for physical delivery

Common Pitfalls & Exam Tips

Frequent Mistakes

  1. Forgetting time adjustment: Using annual rates without scaling for partial years
  2. Sign confusion: Adding costs when should subtract benefits
  3. Discrete vs continuous: Mixing (1+r)^T with e^(rT)
  4. PV calculation errors: Not discounting future costs/benefits

Exam Strategy

  • Quick check: F₀(T) > S₀ implies net positive carry
  • FX convention: Higher interest rate currency trades at forward discount
  • Arbitrage questions: Always calculate theoretical price first
  • Time scaling: Monthly = T/12, Quarterly = T/4

Key Relationships to Memorize

ConditionForward vs SpotMarket State
r + c > iF₀(T) > S₀Contango
r + c < iF₀(T) < S₀Backwardation
r + c = iF₀(T) = S₀Equilibrium

Key Takeaways

  1. No-arbitrage pricing is the foundation of all derivative valuation
  2. Cost of carry explains the forward-spot relationship through net holding costs
  3. Replication allows creation of synthetic positions using spot and borrowing/lending
  4. Forward prices incorporate all costs and benefits of holding the underlying
  5. Arbitrage opportunities are quickly eliminated in efficient markets
  6. DeFi protocols automate arbitrage through smart contracts and AMMs
  7. Convenience yield can cause significant backwardation in commodity markets

Cross-References & Additional Resources

  • Topic 5: Forward contract valuation (builds on cost of carry)
  • Topic 6: Futures vs forwards (daily settlement impact)
  • Topic 8: Options pricing (uses replication concepts)
  • Volume 6 Topic 9: Term structure (forward rate agreements)

DeFi Resources

Academic References

  • Hull, J. “Options, Futures, and Other Derivatives” - Chapter 5
  • Cox, Ross, Rubinstein (1979) - Binomial option pricing
  • Black-Scholes (1973) - Option pricing through replication

Review Checklist

Conceptual Understanding

  • Can explain why arbitrage opportunities cannot persist
  • Understand how replication creates synthetic positions
  • Know the components of cost of carry
  • Can identify contango vs backwardation conditions

Calculation Skills

  • Calculate forward prices with various cost/benefit structures
  • Determine arbitrage profits when prices deviate
  • Apply cost of carry to different asset classes
  • Convert between discrete and continuous compounding

Application Ability

  • Identify real-world arbitrage opportunities
  • Design replication strategies for forwards
  • Explain DeFi funding rate mechanisms
  • Analyze convenience yield in commodity markets

Formula Mastery

  • F₀(T) = S₀(1 + r)^T (basic forward)
  • F₀(T) = [S₀ - PV₀(I) + PV₀(C)](1 + r)^T (with costs/benefits)
  • F₀,f/d(T) = S₀,f/d × e^((rf-rd)T) (FX forwards)
  • All variations and applications memorized

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