November 11, 2025
5 mins read
At first glance, put-call parity can look like a dense formula tucked away in financial theory. Yet, it is nothing more than a relationship linking the price of put and call options on the same asset. The idea was first explained by economist Hans Stoll in 1969, and it still influences how options are priced today.
Put call parity works like a fairness check, making sure option prices stay balanced and leaving little space for arbitrage when markets run smoothly.
Put-call parity is an important concept in mathematical finance to understand how the prices of a put and a call option with the same strike price and expiration interact with each other and also how they depend on each other.
It also assumes there is (at least) a perfect market, meaning no arbitrage.
The formula is as follows:
A₀ + P₀ = C₀ + E(1 + r)⁻ᵀ
Where:
A₀ = Price of the underlying asset
P₀ = Put option premium
C₀ = Call option premium
E = Exercise (strike) price
r = Risk-free rate
T = Time to expiry
The concept is straightforward.
The price of a call option + discounted strike price equals the price of a put option + the asset.
Whenever that equilibrium is violated, it signals a mispricing of the market.
In earlier times, traders often used such gaps to make arbitrage profits.
Today, these mismatches are less common, but not impossible.
Factors like interest rates and dividends can tip the scale. Higher dividends usually push put prices up and call prices down. Rising interest rates, on the other hand, do the opposite.
Put-call parity operates to maintain that there is no free lunch through options trading. If a call and put are out of equilibrium, traders could take advantage of the disequilibrium for risk-free profit.
It is similar to two sides of a scale. On one side, a call option plus the present value of the strike. On the other, a put option plus the asset. Both must stay equal, or an imbalance creates arbitrage opportunities.
Consider an investor aiming to benefit from put-call parity while guarding against losses. Let us compare two portfolios.
At time zero, the investor buys a call option priced at C₀ with exercise price E, along with a bond redeemable at E after time T. The cost is C₀ + E(1 + r)⁻ᵀ. Profit comes when the asset price (Aᵀ) exceeds E.
Instead, the investor purchases the asset at A₀, and also buys a put option for a price of P₀ with exercise price E. The total outlay is equal to A₀ + P₀.
Both portfolios provide similar protection in that both will produce a gain if the asset increases in value and will give some protection - insulation - against loss if the asset decreases in value. This equality proves the put-call parity formula.
Example
Exercise price = $45 (expiry in 3 months)
Asset price = $55
Put option = $4.2
Risk-free rate = 4.8%
Call option (C₀) = 55 + 4.2 – 45(1.048)⁻⁰·²⁵ = $14.74
When prices slip away from parity, arbitrage strategies appear. These are called synthetic positions, created when puts and calls with the same strike and expiry are combined differently. Two common approaches are the conversion strategy and the reversal strategy.
The formula is the same: A₀ + P₀ = C₀ + E(1 + r)⁻ᵀ. If this balance is disturbed, traders can lock in risk-free profits. The actual direction depends on whether the call is overpriced or underpriced relative to the put.
In a conversion, the trader buys the underlying asset, purchases a put, and simultaneously sells a call. This locks in a near risk-free payoff. It is often called a synthetic short forward.
Formula: Portfolio cost = A₀ + P₀ – C₀
If the call is overpriced, the income from selling it offsets the cost of the put and the asset. At expiry, the investor either delivers the stock (if exercised) or holds onto it. In both cases, the payoff resembles holding a bond, effectively capturing arbitrage profit. The calculation ensures that the relationship between calls and puts stays intact by bringing the equation back into balance.
The reversal is the mirror image of conversion. Here, the trader short-sells the underlying asset, sells the put, and buys the call. It is often labelled a synthetic long forward.
Formula: Portfolio cost = C₀ – P₀ – A₀
When the put is overpriced, this strategy makes sense. The payoff, at expiry, mimics holding the asset at a discount. The trade balances out the mispricing between calls and puts.
Put-call parity sits at the heart of options pricing. It prevents free arbitrage in efficient markets and keeps option values in line. When the balance shifts, strategies like conversion or reversal step in to restore order.
For investors, it is useful as a guide — but remember, it assumes an ideal market. Real-world factors such as dividends, transaction costs, and varying interest rates mean the clean parity line is sometimes blurred. Yet, as a tool for understanding fair value, it remains essential.
Share this article: 
No result found
The put call parity formula establishes a relationship between the values of call options, put options, underlying asset, and exercise price assuming ideal market conditions.
Yes, deviations from put-call parity create arbitrage opportunities, allowing investors to profit by exploiting price differences between related call and put options in the market.
The main assumptions include the absence of arbitrage opportunities, the existence of perfect market conditions, and zero transaction costs, ensuring fair and efficient pricing in financial markets.
For American options, the parity formula is adjusted to include the effects of interest rates and dividends, which influence the option’s value and overall pricing relationship.
Deviations in put-call parity can occur due to market imperfections, illiquidity, dividend payments, taxes, and transaction costs, all of which affect option pricing and market efficiency.
If the put call parity relationship is violated, then traders can buy the underpriced option and sell the overpriced option which translates to a risk-free profit.
The Black-Scholes model uses the put-call parity relationship to accurately determine the fair value of options by linking call and put prices through mathematical equations.
Disclaimer :
The information on this website is provided on "AS IS" basis. Bajaj Broking (BFSL) does not warrant the accuracy of the information given herein, either expressly or impliedly, for any particular purpose and expressly disclaims any warranties of merchantability or suitability for any particular purpose. While BFSL strives to ensure accuracy, it does not guarantee the completeness, reliability, or timeliness of the information. Users are advised to independently verify details and stay updated with any changes.
The information provided on this website is for general informational purposes only and is subject to change without prior notice. BFSL shall not be responsible for any consequences arising from reliance on the information provided herein and shall not be held responsible for all or any actions that may subsequently result in any loss, damage and or liability. Interest rates, fees, and charges etc., are revised from time to time, for the latest details please refer to our Pricing page.
Neither the information, nor any opinion contained in this website constitutes a solicitation or offer by BFSL or its affiliates to buy or sell any securities, futures, options or other financial instruments or provide any investment advice or service.
BFSL is acting as distributor for non-broking products/ services such as IPO, Mutual Fund, Insurance, PMS, and NPS. These are not Exchange Traded Products. For more details on risk factors, terms and conditions please read the sales brochure carefully before investing.
Investments in the securities market are subject to market risk, read all related documents carefully before investing. This content is for educational purposes only. Securities quoted are exemplary and not recommendatory.
For more disclaimer, check here : https://www.bajajbroking.in/disclaimer
Level up your stock market experience: Download the Bajaj Broking App for effortless investing and trading
Open Your Free Demat Account
Enjoy low brokerage on delivery trades
|
Please Enter Mobile Number
Open Your Free Demat Account
Enjoy low brokerage on delivery trades
|
