What Is Put-Call Parity?

Delve into the intricacies of Put-Call Parity — discover its definition, mathematical formula, underlying mechanism, and practical examples to understand its application in financial markets.

Definition

Put-Call Parity is a fundamental principle in options pricing theory that defines a specific relationship between the prices of European put and call options that share the same underlying asset, strike price, and expiration date. Essentially, this relationship establishes that the price of a call option (C) combined with the present value of the strike price (PV(K)) should equal the price of a put option (P) plus the current price of the underlying asset (S). Formally, it is expressed as:

$$ C + PV(K) = P + S $$

Formula

The formula for Put-Call Parity can be derived as follows:

$$ C - P = S - PV(K) $$

Where:

• \( C \): Price of the European call option
• \( P \): Price of the European put option
• \( S \): Spot price of the underlying asset
• \( PV(K) \): Present value of the strike price, which is \( K \) discounted at the risk-free interest rate \( r \)

$$ PV(K) = \frac{K}{(1+r)^t} $$

Rewriting, the parity relationship stands as:

$$ C = P + S - \frac{K}{(1+r)^t} $$

Mechanism

Put-Call Parity illustrates the equilibrium state in which no arbitrage opportunities exist. If the equality does not hold, arbitrageurs would exploit the price discrepancies to make risk-free profits until the prices correct and restore the equilibrium:

• No-Arbitrage Condition: If \( C + PV(K) \neq P + S \), discrepancies represent arbitrage opportunities.
• Arbitrage Strategy: Traders could simultaneously buy the undervalued securities and sell the overvalued ones to gain risk-free profits.

Examples

Example 1: Calculating Put-Call Parity

• Suppose:
  • \( S = $100 \)
  • \( K = $100 \)
  • Risk-free rate \( r = 0.05 \)
  • Time until expiration \( t = 1 \) year
  • Call option price \( C = $10 \)
  • Put option price \( P = $5 \)

First, calculate \( PV(K) \):

$$ PV(K) = \frac{K}{(1 + r)^t} = \frac{100}{(1 + 0.05)^1} = 95.24 $$

Now verify Put-Call Parity:

$$ C + PV(K) = 10 + 95.24 = 105.24 $$

$$ P + S = 5 + 100 = 105 $$

In this case, the tiny discrepancy might be ignored due to rounding, implying the market is close to equilibrium.

Historical Context

Put-Call Parity dates back to the late 20th century when modern option pricing theories were developing. The concept was first rigorously defined by Hans R. Stoll in 1969. Since then, it has become a cornerstone of financial theory, providing foundations for more advanced pricing models.

Applications in Financial Markets

Put-Call Parity is invaluable for:

• Hedging Strategies: Investors use it to create synthetic positions, mitigating risk.
• Options Arbitrage: It helps identify mispricings in the options market.
• Validation Tool: Traders validate option pricing models and market efficiency via this relationship.

Related Terms

• European Option: A financial derivative that can only be exercised at expiration, unlike American options which can be exercised anytime before expiration.
• Synthetic Positions: Financial instruments or combinations that mimic the payoff of other securities.
• Arbitrage: The simultaneous purchase and sale of an asset in different markets to profit from price differences.

FAQs

What happens if Put-Call Parity does not hold?

Discrepancies lead to arbitrage opportunities, allowing traders to make risk-free profits until the parity is restored.

Can Put-Call Parity be applied to American Options?

No, it applies only to European options due to their nature of exercise only at expiration.

How is Present Value of Strike Price calculated?

\( PV(K) = \frac{K}{(1 + r)^t} \), where \( K \) is the strike price, \( r \) is the risk-free interest rate, and \( t \) is the time to expiration.

Summary

Put-Call Parity is an essential financial theory that establishes a concrete relationship between European put and call options. It ensures market efficiency by precluding arbitrage opportunities and aids traders in hedging strategies and validating option prices. By mastering this concept, investors can navigate the options market with enhanced precision.

References

• Hans R. Stoll, “The Relationship between Put and Call Option Prices,” Journal of Finance, American Finance Association, 1969.
• John C. Hull, “Options, Futures, and Other Derivatives,” 9th Edition, Pearson, 2014.
• Robert E. Whaley, “Derivatives on Market Volatility,” Financial Analysts Journal, 2000.