Put/Call Parity is a fundamental principle in options pricing that establishes a relationship between the prices of European call options and put options with the same strike price and expiration. It helps ensure that no arbitrage opportunities exist in efficient markets.
Contents
Put/Call Parity Formula
The relationship is expressed as:C−P=S−K⋅e−rTC – P = S – K \cdot e^{-rT}C−P=S−K⋅e−rT
Where:
- CCC: Price of the European call option
- PPP: Price of the European put option
- SSS: Current price of the underlying asset
- KKK: Strike price of the options
- rrr: Risk-free interest rate (annualized)
- TTT: Time to expiration (in years)
- e−rTe^{-rT}e−rT: Present value factor of the strike price
Rearranged, it ensures that:C+K⋅e−rT=P+SC + K \cdot e^{-rT} = P + SC+K⋅e−rT=P+S
This means the value of a call plus the discounted strike price equals the value of a put plus the underlying asset.
Exploiting Discrepancies for Arbitrage
If the parity relationship does not hold, there is an opportunity for risk-free arbitrage by creating synthetic positions.
1. If C+K⋅e−rT>P+SC + K \cdot e^{-rT} > P + SC+K⋅e−rT>P+S:
- Action:
- Sell the call (receive CCC).
- Buy the put (pay PPP).
- Borrow K⋅e−rTK \cdot e^{-rT}K⋅e−rT at the risk-free rate (pay K⋅e−rTK \cdot e^{-rT}K⋅e−rT).
- Buy the underlying (pay SSS).
- Outcome: At expiration:
- If the stock price ST>KS_T > KST>K: Exercise the call obligation.
- If ST≤KS_T \leq KST≤K: Exercise the put.
- Arbitrage profit: The initial cash inflow exceeds the cost of unwinding.
2. If C+K⋅e−rT<P+SC + K \cdot e^{-rT} < P + SC+K⋅e−rT<P+S:
- Action:
- Buy the call (pay CCC).
- Sell the put (receive PPP).
- Sell the stock (receive SSS).
- Lend K⋅e−rTK \cdot e^{-rT}K⋅e−rT at the risk-free rate.
- Outcome: At expiration:
- If ST>KS_T > KST>K: Exercise the call.
- If ST≤KS_T \leq KST≤K: Obligation under the put.
- Arbitrage profit: Initial cash inflow exceeds the cost of obligations.
Limitations
- Transaction Costs: Fees and spreads may erode arbitrage profits.
- Execution Timing: Prices need to be executed instantaneously; delays can negate profits.
- European Options Only: The formula applies strictly to European-style options due to their fixed expiration feature.
- Market Efficiency: Discrepancies are rare in highly liquid and efficient markets.
Let’s work through a numerical example to illustrate arbitrage opportunities using the put/call parity formula.
Scenario
- Current stock price (SSS): $100
- Strike price (KKK): $100
- Call option price (CCC): $10
- Put option price (PPP): $7
- Risk-free rate (rrr): 5% per year (0.05)
- Time to expiration (TTT): 1 year
We’ll first check if the put/call parity holds.
Step 1: Calculating Theoretical Relationship
Using the put/call parity formula:C−P=S−K⋅e−rTC – P = S – K \cdot e^{-rT}C−P=S−K⋅e−rT
Calculate the present value of the strike price:K⋅e−rT=100⋅e−0.05⋅1=100⋅0.9512=95.12K \cdot e^{-rT} = 100 \cdot e^{-0.05 \cdot 1} = 100 \cdot 0.9512 = 95.12K⋅e−rT=100⋅e−0.05⋅1=100⋅0.9512=95.12
Substitute the values:10−7=100−95.1210 – 7 = 100 – 95.1210−7=100−95.123≠4.883 \neq 4.883=4.88
The parity does not hold, so there is an arbitrage opportunity.
Step 2: Identifying the Arbitrage
The left-hand side (C+K⋅e−rTC + K \cdot e^{-rT}C+K⋅e−rT) and the right-hand side (P+SP + SP+S) are not equal. Let’s calculate both sides:
- Left-Hand Side:
C+K⋅e−rT=10+95.12=105.12C + K \cdot e^{-rT} = 10 + 95.12 = 105.12C+K⋅e−rT=10+95.12=105.12
- Right-Hand Side:
P+S=7+100=107P + S = 7 + 100 = 107P+S=7+100=107
Since LHS < RHS, we perform the second arbitrage strategy.
Step 3: Arbitrage Actions
- Buy the call: Pay C=10C = 10C=10.
- Sell the put: Receive P=7P = 7P=7.
- Sell the stock: Receive S=100S = 100S=100.
- Lend the present value of strike price (K⋅e−rTK \cdot e^{-rT}K⋅e−rT): Lend $95.12 at the risk-free rate.
Step 4: Outcomes at Expiration
Case 1: Stock price at expiration (STS_TST) > KKK:
- Call option is exercised. Pay K=100K = 100K=100 and receive the stock.
- You had already sold the stock at S=100S = 100S=100, so there’s no net position.
- The money lent at the risk-free rate grows to K=100K = 100K=100.
- Profit: Initial cash inflow exceeds the cost.
Case 2: Stock price at expiration (STS_TST) ≤ KKK:
- Put option is exercised by the buyer. Buy the stock at K=100K = 100K=100 (but you had sold it earlier for S=100S = 100S=100).
- The money lent at the risk-free rate grows to K=100K = 100K=100.
- Profit: Again, initial cash inflow exceeds the cost.
Step 5: Arbitrage Profit
Initial inflow:P+S=7+100=107P + S = 7 + 100 = 107P+S=7+100=107
Initial outflow:C+K⋅e−rT=10+95.12=105.12C + K \cdot e^{-rT} = 10 + 95.12 = 105.12C+K⋅e−rT=10+95.12=105.12
Net arbitrage profit:107−105.12=1.88107 – 105.12 = 1.88107−105.12=1.88
This is a risk-free profit of $1.88 per share.
