Put/Call Parity.

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.

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

1. Transaction Costs: Fees and spreads may erode arbitrage profits.
2. Execution Timing: Prices need to be executed instantaneously; delays can negate profits.
3. European Options Only: The formula applies strictly to European-style options due to their fixed expiration feature.
4. 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:

1. 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

2. 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

1. Buy the call: Pay C=10C = 10C=10.
2. Sell the put: Receive P=7P = 7P=7.
3. Sell the stock: Receive S=100S = 100S=100.
4. 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.