Put Call Parity: Overview, Equation, Example, Arbitrage, Making Profit 

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Written by Arjun Remesh | Reviewed by Shivam Gaba | Updated on 11 November 2024

Put-call parity is a fundamental concept in options pricing that defines the relationship between the prices of European calls and put options with the same strike price, expiration date, and underlying asset. Put-call parity was formally introduced by economist Hans Stoll in 1969 and has since become a cornerstone of modern financial theory, particularly in the study of derivatives. The principle ensures that the prices of these options are consistent with one another, preventing arbitrage opportunities that could lead to risk-free profits.

Suppose the combined price of a call option and the present value of its strike price exceeds that of a put option plus the underlying asset’s current price, an arbitrageur could sell (short) the call option, buy (long) the put option, and buy (long) the underlying asset. This would lock in a risk-free profit as prices realign to restore equilibrium.

Put-call parity plays an essential role in ensuring market efficiency. Prices deviating from this relationship means arbitrageurs quickly step in to correct them, ensuring that no persistent pricing inefficiencies exist. It also provides traders with insights into how options are priced relative to one another and helps them construct synthetic positions to replicate other financial instruments.

What is Put-Call Parity?

Put-call parity is a key principle in options pricing that establishes a relationship between the prices of European calls and put options with the same strike price and expiration date.

It states that the price of a call option implies a fair price for the corresponding put option, and vice versa, ensuring no arbitrage opportunities exist. Traders exploit mispricing through arbitrage by buying the underpriced option and selling the overpriced one if this relationship does not hold.

What is the Equation of Put Call Parity?

The equation for put-call parity is

C + PV(X) = P + S

Where,

• C is the price of the call option,
• P is the price of the put option,
• S is the current stock price,
• PV(X) is the present value of the strike price discounted at the risk-free rate.

For example, Suppose we have the following information

• Current stock price (S) = Rs. 50
• Strike price (X) = Rs. 55
• Price of a call option (C) = Rs. 3
• Risk-free interest rate = 5% (for six months)

First, calculate the present value of the strike price:

PV(X) = 55 / (1 + 0.05)^0.5 ≈ Rs. 53.66

Now, using the put-call parity equation to find the price of the put option (P):

3 + 53.66 = P + 50

Solving for P:

P = Rs. 6.67

Thus, according to put-call parity, the fair price of the put option should be approximately Rs. 6.67.

How Put-Call Parity Works in Practice?

Put-call parity works by demonstrating the relationship between the price of European put and call options and the underlying asset, ensuring there is no arbitrage opportunity. According to put-call parity, the premium of the call option + PV(strike price) = Premium of the put option + Current price of the underlying asset.

This equation establishes that if simultaneously buying a call option and selling a put option with the same strike price and expiry yield the same payoff as simply owning the underlying asset. So an arbitrageur can create a riskless hedge by exploiting pricing discrepancies between puts, calls and the underlying based on put-call parity.

For example, assume stock is trading at Rs.50. The call strike price is Rs.55 and call premium is Rs.3. The put strike price is also Rs.55 while its premium is Rs.5. Plugging this into parity equation:

Call premium + PV(strike) = Put premium + Stock price Rs.3 + Rs.55/1.06 = Rs.5 + Rs.50 Rs.3 + Rs.51.89 = Rs.5 + Rs.50 Rs.54.89 = Rs.55

This shows the equation balances. Now suppose the put premium rises to Rs.7 while everything else remains constant. The new equation is:

Rs.3 + Rs.51.89 = Rs.7 + Rs.50 Rs.54.89 ≠ Rs.57

Now there is an arbitrage opportunity since the left side does not equal the right per put-call parity. To capitalize, the trader should –

1. Sell the overvalued put for Rs.7
2. Buy the stock for Rs.50
3. Buy back the stock via the call option for Rs.55 when it expires.

This series of trades locks in a Rs.2 risk-free arbitrage gain and restores parity without any outlay of cash. Traders execute advanced options spreads using put-call parity to hedge directional risk. Delta-neutral trading strategies particularly rely on put-call parity math. So it remains an integral principle for options trading in equities, forex and derivatives.

What is an Example of Put Call Parity?

Let’s walk through an example of put-call parity.

Scenario

• Stock Price (S): Rs. 50
• Strike Price (X): Rs. 55
• Price of a European Call Option (C): Rs. 3
• Price of a European Put Option (P): Rs. 7.46
• Risk-Free Interest Rate: 5% per year
• Time to Expiration: 6 months

Step 1- Calculate the Present Value of the Strike Price

The present value of the strike price is calculated by discounting it at the risk-free interest rate over the time to expiration. Since the risk-free rate is 5% annually, and we are dealing with a six-month period (0.5 years), we can calculate the present value of the strike price as follows

PV(X) = X / (1 + r)^t

Where

• X = Rs. 55 (strike price),
• r = 0.05 (risk-free rate),
• t = 0.5 years (time to expiration).

PV(X) = 55 / (1 + 0.05)^0.5 ≈ Rs. 53.66

Step 2 – Apply the Put-Call Parity Equation

The put-call parity equation is

C + PV(X) = P + S

Now, plug in the values we have:

3 + 53.66 = P + 50

Simplifying

56.66 = P + 50

Solving for P

P = 56.66 – 50 = Rs. 6.66

Step 3 –  Compare with Actual Put Price

In this example, the calculated price of the put option based on the put-call parity equation is Rs. 6.66, but in our scenario, the actual market price of the put option is Rs. 7.46.

Step 4 –  Arbitrage Opportunity

Since there is a discrepancy between the calculated put price (Rs. 6.66) and the actual market price (Rs. 7.46), an arbitrage opportunity exists.

Here’s how an arbitrageur could exploit this mispricing

1. Sell (short) a put option for Rs. 7.46.
2. Buy a call option for Rs. 3.
3. Invest in cash equal to the present value of the strike price (Rs. 53.66).
4. Borrow money or sell short one share of XYZ stock for Rs. 50.

At expiration

• Stock price above Rs. 55 –  both options expire worthless, but you profit from owning one share of stock bought at Rs. 53.66.
• Stock price below Rs. 55 – you exercise your put option and sell your share at Rs. 55, locking in a profit.

In either case, this strategy allows you to make a risk-free profit by exploiting the difference between the theoretical and actual prices of options.

What Happens When Put-Call Parity is Violated?

There emerges an arbitrage opportunity due to mispricing when put-call parity is violated. For instance, call options become overvalued compared to puts and stock as per the parity equation, then traders execute certain trades to lock in guaranteed profits. 

Firstly, they sell the expensive calls and simultaneously buy the undervalued puts to capitalize on the premium divergence. Secondly, they sell the puts, buy the underlying stock and buy back the stock later via the call options at expiry. In both cases, the trades hedge directional risk and generate arbitrage gains resulting from the parity violation.

This actually helps restore market equilibrium. As more arbitrageurs trade against the discrepancy, demand-supply forces typically bring back option prices in line until parity is re-established. Therefore, parity violations tend to be short-lived in efficiently traded markets.

How to Use Arbitrage using Put-Call Parity?

To use arbitrage using put-call parity, traders look for situations where the put-call parity equation is imbalanced. This signals that either puts or calls are mispriced relative to the other instruments. Traders can lock in risk-free profits by executing trades that exploit this divergence. For instance, if put premiums spike higher compared to calls and the stock price as per the relationship, arbitrageurs can sell the puts and simultaneously buy out-of-the-money calls.

The upside profit using the calls offsets any losses from being short the puts. When parity is restored, the combined position is unwound at no loss/profit, capturing the original premium divergence. Alternately, traders can go long stock and long call options, while shorting the puts to capitalize on parity violations.

How to Make Profit from Put Call Parity?

To make a profit from put-call parity, traders look for mispricing between options and stocks that violate the put-call parity relationship. Parity is broken, and they execute combination trades that take advantage of the divergence to lock in riskless arbitrage gains.

For example, if put options become overvalued compared to calls and the stock price, arbitrageurs can sell the puts and buy the calls. The profit locked in by exploiting the price difference persists even when parity is restored in the market. Staying alert and acting fast before the trading opportunity window shuts helps traders earn low-risk profits using put-call parity arbitrage strategies.

What Factors Affect Put Call Parity?

Two key factors that affect put-call parity are interest rates and dividends, along with the time to expiration of the options.

• Interest Rates

Interest rates play a significant role in determining the present value of the strike price in the put-call parity equation. The present value of the strike price is discounted at the risk-free interest rate, which directly impacts the pricing of both calls and put options.

• Impact on Call Options: The present value of future cash flows (such as the strike price) decreases because money today is worth more than money in the future when the interest rate drops. As a result, call options tend to become more valuable when interest rates increase. This is because holding a call option allows investors to defer payment for purchasing the underlying asset until expiration, benefiting from higher interest rates during this period. In other words, an investor can invest their money at a higher interest rate while still having exposure to potential gains in the underlying stock through the call option.
• Impact on Put Options: Conversely, when interest rates rise, put options tend to lose value. This is because holding a put option requires paying a premium upfront, and higher interest rates make it more expensive to hold cash (or borrow money) to purchase puts. Additionally, higher interest rates make short-selling stocks more attractive than buying put options since short sellers can earn interest on their proceeds.

• Dividends

Dividends also have a significant impact on put-call parity because they affect the underlying stock price and subsequently influence both call and put option prices.

• Impact on Call Options: Its stock price typically drops by an amount equal to the dividend on the ex-dividend date when a company gives a dividend. This reduction in stock price negatively impacts call options since they derive their value from potential increases in stock prices. Therefore, when dividends are expected during an option’s life span, call options become less valuable because investors anticipate that part of their potential gain will be “lost” to dividend payments.
• Impact on Put Options: On the other hand, dividends increase the value of put options. Since put options benefit from declines in stock prices, any reduction in stock price due to dividend payments makes puts more valuable. As dividends increase, so does the likelihood that a stock’s price will fall below its strike price, making puts more attractive.

To account for dividends in put-call parity calculations, we adjust the standard formula:

C + PV(X) = P + S – D

Where

• D represents expected dividend payments.

• Time to Expiration

The time remaining until an option’s expiration also plays a crucial role in determining whether put-call parity holds.

• Time Value of Options: Both call and put options contain intrinsic value (the difference between the current stock price and strike price) as well as extrinsic value (also known as time value). The extrinsic value reflects how much time remains until expiration and how much uncertainty or volatility exists regarding future stock movements. As time passes and approaches expiration, this extrinsic value diminishes—this process is known as “time decay.”
• Effect on Parity: As time decay occurs, both call and put options lose their extrinsic value at different rates depending on market conditions such as volatility or changes in interest rates. Near expiration or for deep-in-the-money options (where intrinsic value dominates), the effect of time decay becomes minimal, and options trade closer to their intrinsic values Therefore, deviations from put-call parity caused by time decay are more likely earlier in an option’s life span when extrinsic value is higher.

What are the Misconceptions About Put-Call Parity?

There are five common misconceptions about put-call parity. Below are more details.

• Put-Call Parity Always Holds Perfectly: One common misconception is that put-call parity always holds perfectly in real-world markets. In reality, factors such as transaction costs, taxes, and market frictions can cause deviations from the theoretical relationship, making arbitrage opportunities difficult to exploit effectively.
• Applies to American Options: Another misconception is that put-call parity applies to both European and American options. In fact, put-call parity is strictly applicable to European options, which can only be exercised at expiration. American options, which can be exercised at any time before expiration, do not follow this relationship as closely.
• It Provides a “Missing Price”“: Some believe that put-call parity provides a “missing price” or an absolute fair value for options. However, it merely defines a relationship between calls and puts; it does not predict future prices or guarantee that current market prices will align perfectly with the theoretical model.
• Dividends Are Ignored: A common misunderstanding is that dividends do not affect put-call parity. In reality, dividends reduce the stock price and impact both call and put values. Adjustments must be made in the parity equation to account for expected dividend payments.
• No Arbitrage Opportunities Exist in Practice: While put-call parity suggests arbitrage opportunities when the relationship breaks down, another misconception is that these opportunities are easily exploitable. In practice, transaction costs, liquidity issues, and market inefficiencies often prevent traders from capturing these arbitrage profits.

Understanding these misconceptions is important for traders to trade effectively. 

What is the put-call parity at maturity?

At maturity, the put-call parity equation simplifies to C + X = P + S, where C is the value of the call option, P is the value of the put option, S is the stock price, and X is the strike price. At this point, there is no time value left, so the options are valued purely based on their intrinsic value.

This intrinsic value depends directly on whether the option is in or out of the money. If the stock price is above the strike price, the call option will be exercised, and its value will be the difference between the stock price and strike price, while the put option will expire worthless.

Conversely, if the stock price is below the strike price, the put option will be exercised, and the call option will expire worthless. This ensures that at expiration, a portfolio consisting of a long call and cash equal to the strike price will have an equivalent payoff to a portfolio consisting of a long put and one share of stock, maintaining the balance dictated by put-call parity.

What is forward put-call parity?

Forward put-call parity is an extension of the traditional put-call parity that applies to forward contracts instead of the underlying asset. It establishes a relationship between the prices of European-style put and call options and a forward contract with the same strike price and expiration date. The formula is p₀ – c₀ = (X – F₀(T)) / (1 + r)ᵀ, where p₀ is the put price, c₀ is the call price, X is the strike price, F₀(T) is the forward price of the underlying asset, r is the risk-free rate, and T is the time to expiration.

This equation shows that a portfolio consisting of a long put and short call (with identical strikes and expirations) is equivalent to holding a forward contract on the underlying asset. The formula relies on the assumption of no-arbitrage conditions If this relationship breaks down, arbitrage opportunities may arise, allowing traders to profit from discrepancies between option and forward prices. For example, if p₀ – c₀ deviates from the right side of the equation, traders could buy undervalued assets and sell overvalued ones to capture arbitrage profits.

Does Put-Call Parity apply to all options?

No, put-call parity does not apply to all options. It specifically applies to European-style options, which can only be exercised at expiration. The relationship defined by put-call parity holds because European options have a fixed exercise date, ensuring that the pricing of corresponding call and put options with the same strike price and expiration date remains balanced.

However, American-style options, which can be exercised at any time before expiration, do not strictly follow put-call parity because the possibility of early exercise introduces additional complexities that affect their pricing Therefore, while put-call parity is a fundamental principle for European options, it does not hold as rigidly for American options due to the flexibility of early exercise

Why Put-Call Parity Doesn’t Strictly Apply to American Options?

Put-call parity doesn’t strictly apply to American options because of the flexibility they offer in terms of early exercise. Unlike European options, which can only be exercised at expiration, American options allow holders to exercise their options at any time before the expiration date.

This early exercise feature introduces complexities that disrupt the strict relationship defined by put-call parity. For instance, an American call option is deep in the money and the underlying asset pays dividends, the holder might choose to exercise the option early to capture the dividend, which would not be possible with a European option.

This potential for early exercise creates an “early exercise premium” for American options, which alters their pricing and causes deviations from the theoretical parity equation. Additionally, dividends and interest rates can further impact the decision to exercise early, with higher rates generally increasing the early exercise premium. The flexibility to exercise early thus making it difficult to maintain the same no-arbitrage relationship that put-call parity ensures for European options.