Put Call Parity- Formula, Equation, Arbitrage,Theorem (Full Guide)

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What Is Put Call Parity?

There exists a connection between the European call options and the European put options prices, and this relationship is defined by the Put call parity. Though, the security, the strike price, and the ending month should be the same for the securities to establish the relation.

Put call parity states that holding up of the long European call with the short European put simultaneously will yield out the same return when you will be holding up a forward contract having the identical basic asset, as well as the expiry date. And here the forward price will be equivalent to the option’s strike amount.

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Put call parity equation

C + PV(x) = P + S

Where in the above put call parity equation:

  • C = the European call options price
  • PV(x) = the current value of the strike price (x), which is reduced from the price on the end date at the risk-free amount
  • P = the European put options or security price
  • S = the present market value of the underlying asset or the spot price

Need for Put Call Parity

The need for Put-Call Parity arises to compute the current worth of the cash element, that exists with an appropriate risk permitted interest rate.

For Example: Take two portfolio A and portfolio B, where Portfolio A has a European call decision and cash which is equivalent to the total shares enclosed by the call option that is being grown up by the call’s striking price. And taking portfolio B which has a European put option as well as the underlying asset. So, we get the options as follows:

  • Portfolio A (having options as) = Call + Cash, (wherever the Cash is equal to the Call Strike Price)
  • Portfolio B (having options as) = Put + Underlying Asset

The Portfolio A and Portfolio B having Call, put, cash and asset option is depicted in the above figure. And from the above figure of Portfolio A having call option and cash, and the portfolio B having put option and asset. we observe that:

Call + Cash = Put + Underlying Asset

For example: Sept 20 Call + $2500 = Sept 20 Put + 100 ABC Stock

Thus, in order to calculate the current value of the cash component in the above equation we need the put call parity equation which is as: C + PV(x) = P + S

Important Terminologies used in put call Options

  • S0 = Stock price existing today,
  • X = the Strike price
  • T = Time to expiration of the securities
  • r = Risk-free rate of return
  • C0 = the European call option premium
  • P0 = the European put option premium

Put call parity arbitrage:

The put call parity arbitrage defines the opportunity to yield out profit from the price variances that exists in a different market of a financial security. So, the put call parity arbitrage exits where the call put option does not apply at all. Or where we see that one side of the put call equation is greater than the other, or there exists some variation in the put call equation, there the put call parity arbitrage exists.

Put call parity American options:

American put call parity options allow early exercise. Therefore, in the American put call parity option the put-call parity does not applies unless the options or securities are detained to the termination.

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What is put call parity theorem?

The put call theory was first recognized by Hans Stoll in the year1969.

Put Call parity theorem tells us that price of a call opportunity suggests a confident impartial price for consistent put options, though there should be same strike price, as well as same underlying and expiry of the option. Moreover, the three sided relationship of the call, and the put as well as the underlying security is also stated and defined by the put call parity theorem.


With the help of put call parity one develops a clear understanding about the prices of the options and why the price of one option affects the price of other and there is not much change in the price of one option without the change in price of the corresponding option.


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