What Is Put Call Parity

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by Gavin in Blog
March 18, 2020 0 comments

The key joy of trading different assets, whether it be stocks, bonds or options, is the potential to spot opportunities which can be profited from by an alert and knowledgeable trader.

With options trading there are many moving parts which can sometimes get out of whack.

Things like implied volatility or the delicate supply and demand balance can cause nominal option prices or option premiums to move higher or lower and sometimes traditional relationships between different elements can temporarily break down.

It’s when an established, traditional relationship breaks down that some of the greatest opportunities for profit present themselves as these can be virtually risk free.

One of these is centred around what’s called the put call parity principle.

Put call parity is a fundamental principle that every options trader should know. At its core, put call parity defines the three way relationship between puts, calls and their underlying asset.

First identified by Hans Stoll in 1969, the put call parity principle says that provided you are considering options with the same strike price, underlying and expiry, then the premium (price) of a call option will direct what the fair price is for an equivalent put option.

Put simply, it postulates that for puts and calls with the same strike, expiration and underlying, the prices will be highly correlated.

If a trader knows the price of a call option, they will be able to quickly calculate the price of a corresponding put option and vice-versa.

As a result, if for some reason this relationship breaks down and the parity principle is violated, then a trader has a strong arbitrage opportunity on their hands which they can exploit for profit.

An Important Word On Option Styles

In the world of options, there are two main styles. These are referred to as American and European options respectively. It’s important to understand the difference between the two, as this will impact the viability of the put call parity principle.

American options can be exercised at any time during the life of the option.

European options are exercised only on an option’s expiration date.

As a result of this dynamic, the put call parity principle generally only works perfectly with European options, so bear this in mind when using it in your trading.

Calculating Put Call Parity

Economist Hans Stoll first introduced the put call parity principle in his 1969 paper titled “The Relationship Between Put and Call Option Prices.”

In it, he expressed the formula as follows:

C + PV(x) = P + S


C = The price of a European call option

PV(x) = The present value (PV) of the strike price (x), which uses the risk free rate to discount from the value on the expiration date.

P = The price of a European put option

S = The spot price of the underlying asset (i.e. the current market value)

The formula is theoretical and assumes a perfectly efficient market, so don’t be surprised to see some disparities.

Recall that a call option is a contract that gives the owner the right, but not the obligation, to buy an underlying asset, while a put option gives the owner the right, but not the obligation, to sell an underlying price.

In both cases this is done at a predetermined price (called the strike price) and can be done at any time up until contract expiration.

Let’s see how the put parity principle works using an example.

Say you purchased a European call option for ABC stock, with an expiration date 12 months from now. The option has a strike price of $20 and the current risk free rate is 2%, while ABC stock itself is trading at $16 a share.

Ignoring any transaction fees, commissions and assuming ABC doesn’t pay a dividend, if we apply the put call parity formula we get:

C + PV(x) = P + S

C + (20 / 1.02) = P + 16

C + 19.61 = P + 16

Rearranging we get,

P – C = 3.61

In this scenario, ABC puts should be trading at a $3.61 premium to their related call (i.e. same strike, expiration and expiry).

Say however, you notice that for some reason puts are trading at $13 while calls are trading at $8.

Adding these values into our original formula we get,

C + 19.61 = P + 16

8 + 19.61 = 13 + 16

27.61 = 29

Basic logic shows us that 27.61 is not equal to 29, so the output of the formula has violated the put call parity principle. In this case, the call price of 27.61 is less than the put price of 29.

Whenever one side of the put call parity equation doesn’t balance with the other, it means that there is an arbitrage opportunity.

In the example we just worked through, since the put price is higher than the call price, a trader can make an almost risk-free profit by selling the more expensive side (the put) and buying the cheaper side (the call).

In reality however, these sort of opportunities tend to be short lived and very difficult to find, so if you do spot one be sure to move quickly.


The put call parity principle demonstrates the relationship between the price of complementary put and call options.

For the principle to hold, the options should be in the European style and have the same expiration, strike price and underlying asset.

The put call principle states that when the principle holds, the price of one option, such as a call option, will correlate with the price of a complementary put option and vice versa.

If for any reason the price diverges, it presents traders with a short lived, but virtually risk free profit opportunity.

Finally, to calculate the relationship, apply the formula C + PV(x) = P + S.

Trade Safe!

Disclaimer: The information above is for educational purposes only and should not be treated as investment advice. The strategy presented would not be suitable for investors who are not familiar with exchange traded options. Any readers interested in this strategy should do their own research and seek advice from a licensed financial adviser.

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Options Trading 101 - The Ultimate Beginners Guide To Options

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