Topic 9: Option Replication Using Put-Call Parity
Learning Objectives Coverage
LO1: Explain put-call parity for European options
Core Concept
Put-call parity is a fundamental no-arbitrage relationship that links the prices of European put and call options with the same strike price and expiration date. It demonstrates that a protective put (stock + put) must equal a fiduciary call (call + bond) to prevent arbitrage opportunities. This is one of the most important relationships in the entire derivatives curriculum. exam-focus
Key Formula formula pricing
Put-Call Parity:
S₀ + p₀ = c₀ + X(1 + r)^(-T)
Where:
S₀ = Current stock price
p₀ = Put option premium
c₀ = Call option premium
X = Strike price
r = Risk-free rate
T = Time to expiration
Rearranged Forms
Synthetic Put:
p₀ = c₀ + X(1 + r)^(-T) - S₀
Synthetic Call:
c₀ = S₀ + p₀ - X(1 + r)^(-T)
Synthetic Stock:
S₀ = c₀ - p₀ + X(1 + r)^(-T)
Practical Example
Apple Options Arbitrage:
- Stock price: $150
- Strike: $145
- Call premium: $10
- Risk-free rate: 3%
- Time: 3 months (0.25 years)
Calculate fair put price:
p₀ = c₀ + X(1 + r)^(-T) - S₀
p₀ = $10 + $145/(1.03)^0.25 - $150
p₀ = $10 + $143.93 - $150
p₀ = $3.93
If put trades at $5 (overpriced):
- Arbitrage profit = 3.93 = $1.07
DeFi Application defi-application
Opyn Protocol enables synthetic positions built directly from put-call parity. For example, a synthetic long ETH position is constructed by buying an ETH call (0.05 ETH premium) and selling an ETH put (0.05 ETH premium). The net cost is zero (plus collateral), and the position replicates owning ETH without holding the underlying — the same synthetic stock relationship derived from the parity equation.
LO2: Explain put-call forward parity for European options
Core Concept
Put-call forward parity extends the relationship to include forward contracts instead of the underlying asset. It shows that the difference between put and call prices equals the present value of the difference between strike price and forward price. This formulation is especially useful when the underlying cannot be easily held (commodities, currencies). exam-focus
Key Formula
Put-Call Forward Parity:
F₀(T)(1 + r)^(-T) + p₀ = c₀ + X(1 + r)^(-T)
Simplified Form:
p₀ - c₀ = [X - F₀(T)](1 + r)^(-T)
Where:
F₀(T) = Forward price at time 0 for delivery at T
Relationship to Standard Parity
Since F₀(T) = S₀(1 + r)^T for non-dividend assets:
Substituting:
S₀(1 + r)^T(1 + r)^(-T) + p₀ = c₀ + X(1 + r)^(-T)
S₀ + p₀ = c₀ + X(1 + r)^(-T)
This reduces to standard put-call parity
Practical Example
Currency Options with Forward Contracts:
- Spot EUR/USD: 1.20
- 6-month forward: 1.21
- Strike: 1.22
- Risk-free rate: 2%
- Call premium: $0.03
Calculate put premium:
p₀ - c₀ = [X - F₀(T)](1 + r)^(-T)
p₀ - $0.03 = [1.22 - 1.21](1.02)^(-0.5)
p₀ - $0.03 = 0.01 × 0.9901
p₀ = $0.03 + $0.0099 = $0.0399
DeFi Application defi-application
Perpetual Protocol enables synthetic forward exposure by combining perp funding rates with options. For example, a BTC forward position can be constructed by going long a BTC perp (at a funding rate of -0.01%/hour), then selling an ATM call and buying an ATM put. This creates a synthetic short forward that captures the funding rate differential — an elegant on-chain application of forward parity.
Core Concepts Summary (80/20 Principle)
The 20% You Must Know:
- Put-Call Parity: S₀ + p₀ = c₀ + X/(1+r)
- Protective Put = Fiduciary Call in payoff
- Arbitrage exists when parity violated
- Synthetic positions created using parity
- Forward parity uses forward price instead of spot
The 80% That Matters Most:
- Parity only holds for European options
- Same strike and expiration required
- Risk-free rate creates time value difference
- Dividends modify the relationship
- All positions can be synthetically created
Comprehensive Formula Sheet
Basic Put-Call Parity
Standard Form:
S₀ + p₀ = c₀ + X(1 + r)^(-T)
With Dividends:
S₀ - PV(Div) + p₀ = c₀ + X(1 + r)^(-T)
Synthetic Positions
Synthetic Call:
Long Stock + Long Put - Short Bond = Long Call
c₀ = S₀ + p₀ - X(1 + r)^(-T)
Synthetic Put:
Long Call + Long Bond - Short Stock = Long Put
p₀ = c₀ + X(1 + r)^(-T) - S₀
Synthetic Stock:
Long Call - Long Put + Long Bond = Long Stock
S₀ = c₀ - p₀ + X(1 + r)^(-T)
Synthetic Bond:
Long Stock + Long Put - Long Call = Long Bond
X(1 + r)^(-T) = S₀ + p₀ - c₀
Forward Parity
Basic Form:
F₀(T)(1 + r)^(-T) + p₀ = c₀ + X(1 + r)^(-T)
Difference Form:
p₀ - c₀ = [X - F₀(T)](1 + r)^(-T)
With Cost of Carry:
F₀(T) = S₀(1 + r + s - d)^T
Where s = storage cost, d = dividend yield
Arbitrage Conditions
If S₀ + p₀ > c₀ + X(1 + r)^(-T):
→ Sell stock and put, buy call and bond
If S₀ + p₀ < c₀ + X(1 + r)^(-T):
→ Buy stock and put, sell call and bond
HP 12C Calculator Sequences
Calculate Bond Present Value
Strike = $100, r = 5%, T = 0.5 years
[f] [CLX]
100 [ENTER]
1.05 [ENTER]
0.5 [y^x]
[÷] // Result: $97.59
Check Put-Call Parity
S = $50, c = $5, p = $2, X = $48, r = 4%, T = 0.25
Left side (S + p):
[f] [CLX]
50 [ENTER]
2 [+] // = $52
Right side (c + PV(X)):
48 [ENTER]
1.04 [ENTER]
0.25 [y^x]
[÷] // = $47.53
5 [+] // = $52.53
Difference: $0.53 arbitrage opportunity
Synthetic Put Calculation
c = $8, X = $100, S = $105, r = 3%, T = 0.5
[f] [CLX]
100 [ENTER]
1.03 [ENTER]
0.5 [y^x]
[÷] // PV(X) = $98.52
8 [+] // = $106.52
105 [-] // p = $1.52
Practice Problems
Basic Level
Problem 1: Verify put-call parity:
- Stock: €50
- Call: €4
- Put: €2
- Strike: €48
- Rate: 2%
- Time: 6 months
Solution:
Left side: S + p = €50 + €2 = €52
Right side: c + X/(1+r)^T
= €4 + €48/(1.02)^0.5
= €4 + €47.53
= €51.53
Difference = €0.47 (arbitrage opportunity)
Problem 2: Calculate synthetic call price:
- Stock: $75
- Put: $3
- Strike: $70
- Rate: 4%
- Time: 3 months
Solution:
c = S + p - X/(1+r)^T
c = $75 + $3 - $70/(1.04)^0.25
c = $75 + $3 - $69.32
c = $8.68
Intermediate Level
Problem 3: Arbitrage strategy:
- Stock: ¥1000
- Call: ¥80
- Put: ¥55
- Strike: ¥950
- Rate: 1%
- Time: 1 year
- Put-call parity violated
Solution:
Check parity:
S + p = ¥1000 + ¥55 = ¥1055
c + X/(1+r)^T = ¥80 + ¥950/1.01 = ¥1020.59
Since S + p > c + X/(1+r)^T:
1. Sell stock: +¥1000
2. Sell put: +¥55
3. Buy call: -¥80
4. Buy bond: -¥940.59
Net profit: ¥34.41
Advanced Level
Problem 4: Forward parity with dividends:
- Spot: $100
- Dividend: $2 in 3 months
- Forward (6-month): $99
- Strike: $95
- Call: $8
- Rate: 5%
Solution:
Adjust spot for dividend:
S_adj = $100 - $2/(1.05)^0.25 = $98.02
Forward should be:
F = $98.02 × (1.05)^0.5 = $100.43
Using forward parity:
p - c = [X - F]/(1+r)^T
p - $8 = [$95 - $100.43]/(1.05)^0.5
p - $8 = -$5.31
p = $2.69
DeFi Applications & Real-World Examples
1. DeFi Options Protocols
Lyra Protocol Arbitrage:
ETH Spot: $2000
Call (2100 strike): $50
Put (2100 strike): $140
Rate: 5%, Time: 30 days
Check parity:
S + p = $2000 + $140 = $2140
c + X/(1+r)^T = $50 + $2091.44 = $2141.44
Near perfect parity (AMM fees explain difference)
Hegic Synthetic Positions:
- Create synthetic stablecoin exposure
- Buy USDC/DAI call, sell put
- Replicates holding USDC
- Earns option premiums
2. Structured Products
Ribbon Finance Strategies:
Principal Protected Note:
1. Buy zero-coupon bond: $950
2. Buy ATM call: $50
3. Total cost: $1000
4. Minimum return: $1000 (bond maturity)
5. Upside: Unlimited via call
Jones DAO Metavaults:
- Uses put-call parity for delta hedging
- Maintains market-neutral positions
- Example: jGLP vault with options overlay
3. Credit Applications defi-application
MakerDAO as Options:
MakerDAO’s DAI minting mechanism can be understood through the lens of put-call parity. Depositing ETH and minting DAI is economically equivalent to a protective put: the user is long the underlying (ETH) with a synthetic put at the liquidation price (the 150% collateral ratio threshold). The stability fee functions as the put premium.
DAI Minting = Protective Put:
- Deposit ETH (long underlying)
- Mint DAI (synthetic put at liquidation price)
- Protected downside at 150% collateral ratio
- Stability fee = put premium
Compound Finance:
Lending on Compound resembles a covered call: the user supplies an asset (long underlying) and earns yield (analogous to call premium), with the risk that borrower default exercises the “call” on the supplied collateral.
Lending = Covered Call:
- Supply asset (long underlying)
- Earn yield (call premium)
- Risk: Borrower default (call exercise)
4. Corporate Finance Examples
Tesla Convertible Bond:
Bond Analysis using Put-Call Parity:
- Bond = Stock + Put - Call
- $1000 bond convertible at $200
- Implied volatility from bond price
- Credit spread = put premium
Apple Share Buyback:
Accelerated Repurchase = Synthetic Put:
- Company sells puts
- Strike below current price
- Premium reduces buyback cost
- Obligation to buy if exercised
Common Pitfalls & Exam Tips
Common Mistakes to Avoid
-
Using American options
- Parity only holds for European
- Early exercise breaks relationship
- American puts often worth more
-
Forgetting present value
- Strike must be discounted
- Use risk-free rate
- Time value matters
-
Wrong arbitrage direction
- If left > right: Sell left, buy right
- If left < right: Buy left, sell right
- Always check net cash flow
-
Ignoring dividends
- Reduce stock price by PV(dividends)
- Affects call values negatively
- Affects put values positively
Exam Strategy Tips
-
Quick parity check:
- Stock + Put should equal Call + Bond
- Difference indicates arbitrage
- Use calculator for PV quickly
-
Synthetic position memory:
- Want call? Buy stock, buy put, sell bond
- Want put? Buy call, buy bond, sell stock
- Want stock? Buy call, sell put, buy bond
-
Forward parity shortcut:
- Put - Call = PV(Strike - Forward)
- If forward > strike: Calls worth more
- If strike > forward: Puts worth more
Key Takeaways
Must Remember:
- S + p = c + X/(1+r)^T for European options
- Protective put = Fiduciary call payoffs
- All four positions can be synthetically created
- Arbitrage profits when parity violated
- Forward parity uses forward instead of spot
Critical Insights:
- Foundation for option pricing models
- Enables synthetic position creation
- Risk-free arbitrage when violated
- DeFi protocols automate arbitrage
- Credit spreads relate to put premiums
Cross-References & Additional Resources
Related Topics:
- Topic 8: Option valuation factors
- Topic 10: Binomial pricing model
- Fixed Income: Convertible bonds
- Corporate Finance: Capital structure (options view)
Key Readings:
- Stoll (1969): “The Relationship Between Put and Call Prices”
- Merton (1973): “Theory of Rational Option Pricing”
- Hull: Options, Futures, and Other Derivatives Ch. 11
Practice Resources:
- CBOE Put-Call Parity Calculator
- OptionStrat: Visual parity demonstrations
- Khan Academy: Put-Call Parity videos
DeFi Protocols to Study:
- Lyra: Full options suite with parity
- Opyn: Gamma protocol for options
- Hegic: On-chain options with parity
- Ribbon: Structured products using parity
- Dopex: Atlantic options with parity
Review Checklist
Conceptual Understanding
- Can you state put-call parity formula?
- Do you understand protective put vs fiduciary call?
- Can you identify arbitrage opportunities?
- Do you know forward parity relationship?
Calculations
- Can you calculate synthetic option prices?
- Can you determine arbitrage profits?
- Can you apply forward parity?
- Can you adjust for dividends?
Applications
- Can you create synthetic positions?
- Do you understand credit spread interpretation?
- Can you apply to convertible bonds?
- Can you identify DeFi applications?
Exam Readiness
- Memorized parity formula
- Practiced arbitrage identification
- Reviewed synthetic positions
- Completed practice problems
DeFi Integration
- Understand on-chain arbitrage
- Know synthetic position protocols
- Can explain automated market making
- Familiar with structured products