Understanding Put-Call Parity: Complete Guide
Put-call parity is a fundamental principle in options pricing that defines the relationship between the prices of European put and call options with the same strike price, expiration date, and underlying asset. This relationship is crucial for understanding option pricing and identifying potential arbitrage opportunities in the market.
What Is Put-Call Parity?
Put-call parity states that the price of a call option implies a certain fair price for the corresponding put option with the same strike and expiration (and vice versa). If this relationship doesn't hold, it creates an arbitrage opportunity where traders can profit from the mispricing without taking any risk.
The put-call parity equation is:
Where:
C = Call option price
P = Put option price
S = Current stock (underlying) price
K = Strike price
PV(K) = Present value of strike = K × e-rT
r = Risk-free interest rate
T = Time to expiration (in years)
How Put-Call Parity Works
The put-call parity relationship arises because two portfolios with identical payoffs at expiration must have the same value today. Consider these two portfolios:
Portfolio A: Long call option + Cash equal to PV(K)
Portfolio B: Long put option + Long stock
At expiration, both portfolios have identical payoffs regardless of where the stock price ends up. Therefore, they must have the same value today, which gives us the put-call parity equation.
Example Calculation
Given the following parameters:
- Stock Price (S) = $100
- Strike Price (K) = $100
- Call Price (C) = $10
- Risk-Free Rate (r) = 5%
- Time to Expiry (T) = 1 year
Calculate the fair put price:
- PV(K) = $100 × e-0.05×1 = $100 × 0.9512 = $95.12
- P = C + PV(K) - S = $10 + $95.12 - $100 = $5.12
Important Limitations
Put-call parity has several important limitations:
- European Options Only: The formula applies only to European options, which can only be exercised at expiration. American options, which allow early exercise, do not strictly follow this relationship.
- No Dividends: The basic formula assumes no dividends are paid during the option's life. For dividend-paying stocks, the formula must be adjusted.
- Market Frictions: In practice, transaction costs, bid-ask spreads, and taxes can prevent perfect arbitrage even when parity is violated.
- Borrowing Constraints: The ability to borrow at the risk-free rate is assumed but may not be available to all investors.
Arbitrage Opportunities
When put-call parity is violated, arbitrage opportunities exist:
If C + PV(K) > P + S:
The call is overpriced relative to the put. Strategy: Sell the call, buy the put, buy the stock, and borrow PV(K). This locks in a risk-free profit.
If C + PV(K) < P + S:
The put is overpriced relative to the call. Strategy: Buy the call, sell the put, short the stock, and lend PV(K). This locks in a risk-free profit.
Put-Call Parity with Dividends
For stocks that pay dividends, the put-call parity equation is modified to account for the present value of expected dividends:
Where PV(D) = Present value of expected dividends during the option's life
Practical Applications
- Option Pricing Verification: Traders use put-call parity to check if options are fairly priced
- Synthetic Positions: Create equivalent positions using different combinations of options and stock
- Risk Management: Understand the relationship between different positions in a portfolio
- Market Efficiency Testing: Researchers use parity violations to study market efficiency