We have prepared this put-call parity calculator for you to understand the relationship between a call and put option. It will also help you to understand how options are valued according to the no-arbitrage rule.
Accompanying this calculator, we have also written this article to help you understand what is put-call parity and how to calculate it using the put-call parity formula. We will also describe the concept in more detail by demonstrating some examples of applying this relationship.
What is the put-call parity?
Simply put, the put-call parity assumes that investors should be indifferent between going long on a call contract and holding a forward contract with the same striking price and expiration date, and a protective put, equivalent to buying a stock and longing a European put option simultaneously.
The put-call parity equation states that if one of the asset prices deviates from the relationship, an arbitrage opportunity will exist. This allows traders to exploit the opportunity by buying the underpriced asset and selling the overpriced asset.
We will explain this relationship in detail by calculating the put-call parity in the following example.
How to calculate put-call parity?
Put-call parity is as much of an equation as a relationship. Hence, the easiest way to understand the put-call parity calculation is to understand what the relationship means in different forms.
The put-call parity equation can be displayed as follows:
C + PV(x) = P + S
C– Price of a European call option of strike price
PV(x)– Present value of the strike price of the options, discounted to the present date using the risk-free rate;
P– Price of a European put option of strike price
S– Spot price of the underlying asset.
Before we demonstrate the put-call parity example, let's look at a short example of how to calculate the
PV(x). This can be calculated using the formula below:
PV(x) = strike price / ((1 + risk-free rate)(years to expiry))
So, if the
strike price is
years to expiry is
2 years and the
risk-free rate is
PV(x) will equal to
12 / (1.03)² = $11.31.
Now, we can calculate the price of 4 financial instruments using the put-call parity formula:
- Calculate the price of a European call option.
This can be achieved by using the equation as follow:
C = P + S - PV(x)
By looking at the equation, we can tell that this relationship assumes that
Cis equal to the value of a protective put (
P + S) with a short position in
- Calculate the present value of the strike price.
The equation for this calculation is:
PV(x) = P + S - C
Hence, the present value of the strike price is assumed to be the combined value of a protective put and shorting a European call option.
- Calculate the price of a European put option.
This can be achieved by using the equation as follow:
P = C + PV(x) - S
This implies that the value of a European put option is equivalent to a long position in a European call option, a long position in the present value of the strike price, and a shorting the underlying asset.
- Calculate the price of the underlying asset.
This relationship also allows you to value the underlying asset, with the equation:
S = C + PV(x) - P
This means that the price of the underlying asset is equivalent to a long position in a European call option, a long position in the present value of the strike price, and a shorting a European put option with strike price
advanced mode of our put-call parity calculator, you can change the years to expiry and risk-free values.
If you want to calculate the price of a European put-call option using the Black Scholes model, you can check out our Black Scholes calculator.
Limitations of the put-call parity
Now, let's talk about some of the limitations of the put-call parity formula.
Firstly, this equation only applied to European options but not American options. The European options only allow the buyers to execute their right to buy or sell the asset at the pre-determined exercise date. On the other hand, investors can execute their right to buy or sell an asset at any time before the exercise date when investing in an American option.
The put-call parity equation only holds when there are no market frictions. We define market frictions as implicit costs that are incurred when a trade is executed. The most common ones being taxes, brokers' commissions, and bid-ask spread.
Does the put-call parity apply to all options?
No, the put-call parity only applies to the European options, but not the American options. The European option can only be exercised on a pre-determined date, whereas American options can be exercised at any date before a pre-determined date.
What is an abitrage?
Arbitrage is a trading strategy that focuses on earning profit by simultaneously buying and selling assets. This strategy revolves around exploiting short-term mispricing in different assets.
What are market frictions?
Market frictions are the implicit costs involved in executing a trade. These costs are normally not obvious when a trade is executed. Some examples are taxes, brokerage costs, bid-ask spread, etc.
What is an option?
An option is an agreement between two parties that gives the buyer the right, but not the obligation, to buy or sell an asset at a pre-determined price at or before the exercise date. The buyer of the option has to pay a premium to the writer of the option.