Put-call parity

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  1. redirect Put-call parity

Introduction

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Put-Call Parity

Put-call parity is a fundamental principle in options pricing theory that defines a relationship between the prices of European put and call options with the same strike price and expiration date. It's a no-arbitrage condition, meaning that if the parity doesn't hold, an arbitrage opportunity exists where a riskless profit can be made. Understanding put-call parity is crucial for options traders, portfolio managers, and anyone involved in derivative pricing. This article will provide a detailed explanation of the concept, its derivation, applications, limitations, and real-world considerations.

Core Concept

At its heart, put-call parity states that the cost of a portfolio consisting of a long call option and short put option (both with the same strike price and expiration date) is equivalent to the cost of buying the underlying asset and borrowing money at the risk-free rate. Mathematically, it's expressed as:

C + PV(K) = P + S

Where:

  • C = Price of the European call option
  • P = Price of the European put option
  • S = Current price of the underlying asset
  • K = Strike price of the options
  • PV(K) = Present value of the strike price, calculated as K / (1 + r)^T, where 'r' is the risk-free interest rate and 'T' is the time to expiration (expressed in years).

Derivation of Put-Call Parity

The derivation of put-call parity rests on the principle of no-arbitrage. Let's consider two portfolios:

  • Portfolio A: Long one European call option (C) and short one European put option (P), both with strike price K and expiration date T.
  • Portfolio B: Long one share of the underlying asset (S) and borrow an amount of money equal to the present value of the strike price (PV(K)).

Let's analyze the payoffs of each portfolio at expiration (time T):

  • Portfolio A Payoff:
   *   If S(T) > K (the asset price at expiration is greater than the strike price): The call option is exercised, yielding a payoff of S(T) - K. The put option expires worthless.  Total payoff: S(T) - K.
   *   If S(T) ≤ K (the asset price at expiration is less than or equal to the strike price): The call option expires worthless. The put option is exercised, yielding a payoff of K - S(T). Total payoff: K - S(T).
  • Portfolio B Payoff:
   *   Regardless of the asset price at expiration, Portfolio B always has a payoff of S(T) - PV(K) = S(T) - K / (1 + r)^T.  This is because you hold the stock and have a liability equal to the future value of the borrowed amount, which is K.

Notice that the payoffs of Portfolio A and Portfolio B are identical, regardless of the asset price at expiration. Since both portfolios have the same payoff, they must have the same cost today to prevent arbitrage. Therefore:

Cost(Portfolio A) = Cost(Portfolio B)

C - P + PV(K) = S

Rearranging the equation, we arrive at the put-call parity formula:

C + PV(K) = P + S

Arbitrage Opportunities

If the put-call parity relationship does not hold, an arbitrage opportunity arises. Let's explore two scenarios:

  • Scenario 1: C + PV(K) > P + S (The left side of the equation is more expensive)
   *   **Arbitrage Strategy:**
       1.  Sell the overpriced portfolio (long call, short put).
       2.  Buy the underpriced portfolio (long stock, short borrow).
       3.  This generates an immediate risk-free profit equal to the difference.
   *   **Details:** Selling the call and put generates cash.  Buying the stock and borrowing cash requires cash. The net effect is a positive cash flow. At expiration, the payoffs will offset each other perfectly.
  • Scenario 2: C + PV(K) < P + S (The right side of the equation is more expensive)
   *   **Arbitrage Strategy:**
       1.  Buy the underpriced portfolio (long call, short put).
       2.  Sell the overpriced portfolio (long stock, short borrow).
       3.  This generates an immediate risk-free profit equal to the difference.
   *   **Details:** Buying the call and put requires cash. Selling the stock and borrowing cash generates cash. The net effect is a positive cash flow. At expiration, the payoffs will offset each other perfectly.

In reality, arbitrage opportunities are quickly eliminated by market participants, ensuring that put-call parity generally holds true. Transaction costs and bid-ask spreads, however, can make exploiting small discrepancies unprofitable. Algorithmic trading often plays a role in identifying and exploiting these fleeting opportunities.

Applications of Put-Call Parity

Put-call parity has several practical applications:

  • **Pricing Options:** It can be used to determine the fair price of one option if the price of the other is known. For example, if the call price is observed, the put price can be calculated using the formula.
  • **Implied Volatility Consistency:** It helps ensure consistency in implied volatility between put and call options. Any significant difference suggests a potential mispricing.
  • **Synthetic Positions:** It allows for the creation of synthetic positions. For example:
   *   **Synthetic Stock:** C - P + PV(K) = S. This means a call option minus a put option plus the present value of the strike price is equivalent to holding the underlying stock.
   *   **Synthetic Call:** S - PV(K) + P = C.  This means buying the stock, borrowing cash (present value of the strike price), and buying a put option is equivalent to holding a call option.
   *   **Synthetic Put:** S - PV(K) + C = P. This means buying the stock, borrowing cash (present value of the strike price), and buying a call option is equivalent to holding a put option.
  • **Risk Management:** Understanding the relationships between options can help in managing risk, particularly in hedging strategies.

Limitations and Assumptions

Put-call parity relies on several assumptions that may not always hold in the real world:

  • **European Options:** The formula applies strictly to European options, which can only be exercised at expiration. American options, which can be exercised at any time, introduce early exercise risk and invalidate the simple parity relationship. Modifications are needed to accommodate early exercise, such as using the Black-Scholes model with adjustments.
  • **No Dividends:** The basic formula assumes that the underlying asset pays no dividends during the option's life. If dividends are expected, the formula must be adjusted to account for the present value of the expected dividends.
  • **Risk-Free Rate:** The accuracy of the calculation depends on using a reliable and constant risk-free interest rate. In practice, the risk-free rate can fluctuate.
  • **Transaction Costs & Taxes:** The model ignores transaction costs (brokerage fees, commissions) and taxes, which can impact arbitrage opportunities.
  • **Continuous Trading:** The model assumes continuous trading is possible, allowing for immediate execution of arbitrage trades. In reality, trading may be limited by liquidity or market closures.
  • **Identical Strike Price & Expiration:** The options *must* have the same strike price and expiration date.

Adjustments for Dividends

When the underlying asset pays dividends, the put-call parity formula needs to be adjusted. The adjustment subtracts the present value of the expected dividends from the stock price:

C + PV(K) = P + S - PV(Dividends)

Where:

  • PV(Dividends) = Present value of the expected dividends to be paid during the option's life.

Real-World Considerations & Deviations

While put-call parity generally holds true, small deviations can occur in practice due to:

  • **Bid-Ask Spreads:** The difference between the buying and selling prices of options can create a slight discrepancy.
  • **Market Imperfections:** Liquidity constraints, trading restrictions, and information asymmetry can contribute to deviations.
  • **Interest Rate Risk:** Fluctuations in interest rates can affect the present value calculation.
  • **Early Exercise (American Options):** As mentioned earlier, American options introduce early exercise risk, leading to deviations from the parity relationship. Models like the Barone-Adesi and Whaley model are used to price American options and account for this.
  • **Supply and Demand:** Temporary imbalances in supply and demand for options can cause price distortions.

Relationship to Other Option Concepts

Put-call parity is closely related to several other important option concepts:

  • **Delta Hedging:** Understanding the relationship between calls and puts is crucial for delta hedging, a strategy to neutralize the risk associated with option positions. Delta measures the sensitivity of an option's price to changes in the underlying asset's price.
  • **Gamma:** Gamma measures the rate of change of an option’s delta. Put-call parity is indirectly linked to gamma as it affects the dynamics of hedging.
  • **Theta:** Theta represents the time decay of an option’s value. Put-call parity doesn't directly calculate theta but influences how theta impacts the relationship.
  • **Vega:** Vega measures the sensitivity of an option’s price to changes in implied volatility. Put-call parity can be used to assess consistency in vega across calls and puts.
  • **Option Greeks:** Collectively, the Option Greeks (Delta, Gamma, Theta, Vega, Rho) are used to understand and manage the risks associated with option positions, and put-call parity provides a foundational understanding for their application.
  • **Volatility Skew and Smile:** Deviations from put-call parity can also indicate the presence of volatility skew (where out-of-the-money puts are more expensive than out-of-the-money calls) or a volatility smile (where options further from the money are more expensive).
  • **Arbitrage Pricing Theory (APT):** Put-call parity is a specific example of an arbitrage pricing relationship, falling under the broader framework of APT.
  • **Black-Scholes Model:** The Black-Scholes model is a widely used option pricing model that implicitly incorporates put-call parity.
  • **Binomial Option Pricing Model:** The Binomial Option Pricing Model can also be used to demonstrate put-call parity in a discrete-time framework.
  • **Straddles and Strangles:** Understanding put-call parity is essential for analyzing and trading strategies like straddles and strangles, which involve combinations of calls and puts.
  • **Covered Calls and Protective Puts:** Put-call parity informs the understanding of strategies like covered calls and protective puts.
  • **Collar Strategy:** The Collar strategy also leverages the principles of put-call parity.
  • **Iron Condor and Iron Butterfly:** More complex strategies, such as Iron Condor and Iron Butterfly, rely on a deep understanding of the relationship.

Resources for Further Learning

  • Investopedia: [1]
  • Corporate Finance Institute: [2]
  • OptionsPlay: [3]
  • Khan Academy: [4]
  • Derivatives Strategy: [5]
  • TradingView: [6]
  • Babypips: [7]
  • The Options Industry Council: [8]

Options trading Derivatives Financial modeling Risk management Arbitrage Black-Scholes model Option Greeks European options American options Implied volatility

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