A Refresher on Put Call Parity

One of the initial and crucial pitstops in your journey of the world of (European) options to fuel your understanding better is the concept of Put-Call parity. There are several webpages and links that explain the idea and its application in their own formats, as the parity equation itself can be re-arranged in several ways – think of it as corollaries of the principle proof. And that’s precisely why understanding the core concept is important to later pick up on all formats of explanation with ease.

Put Call parity is a zero arbitrage equilibrium for European options  which states that the value of a portfolio that’s long a call and short a put option on the same underlying, at the same strike and expiry is same as the value of a portfolio that’s long a forward contract on the same underlying at a pre-agreed forward rate (same as the option strike) with tenor equal to the option expiry. Consider the equation below:

P_c – P_p = X – K*e^-rt

Where,

P_c and P_P are call and put option premium respectively on the same underlying X, at strike K and for an option tenor ‘t’. 

K*e^-rt is the present value at time t of the strike K, can be denoted as PV_t (K)

LHS is the difference between the cost of having a long European call option and a short European put option for the same underlying X, at strike K and expiry T. At all points in time until and including at expiry this equation must hold. Intuitively, the value of LHS would be the same as RHS, for instance at time period ‘i’ if the underlying X_i<PV_i(K) the call option would be worthless and the short put would be out of the money to the tune of the loss on the forward contract. Similarly if X_i>PV_i(K), call option value would be the same as the gains on the forward contract while the put option would be worthless. 

RHS of the equation above is also often explained as being long the underlying at its current market value and being short a zero coupon bond with a face value of K, as a synthetic way of creating a forward contract on the underlying.

Similarly, one can rearrange the equation to understand the parity as below:

P_c + K*e^-rt = X + P_p

LHS (long call + long strike or long zero coupon bond with face value K) now becomes a fiduciary call while RHS (long underlying + long put that limits the downside on the underlying) is a protective put. Whatever be the rearrangement, the equation should fundamentally hold or any deviation/risk-free profit can be squeezed out by arbitrageurs to revert it to its equilibrium. Frictionless market with sufficient liquidity are ideal conditions for this equation to hold, hence select FX markets in developed market currencies and major stock markets (barring periods of turbulence/sub-par liquidity) almost always allow for the parity to hold. 

A further extension/application of the put call parity is the put call forward parity, where the spot price of the underlying (X in the equation above) is replaced by its actively traded forward contract F.This is a more generalized explanation of the concept that includes options on income-earning underlying assets, where forward values just like the strike K are discounted using a risk-free rate r. Consider the equation below:

P_c – P_p = (F_t – K )*e^-rt,

Where,

F_t is the forward price of the underlying for tenor ‘t’ while the long forward contract position is on the pre-agreed forward rate K.

We have used continuous discounting – denoted by e^-rt - at the risk-free rate ‘r’ to arrive at the present value of the long forward position. Discounted forward price would bring us back to current market price X which aligns with the initial put call parity equation.

Description of the parity and application of several such market concepts can be explored further on pandemonium.sg, a warehouse of market insights and ideas intuitively woven together with references to real market events.

About the Author:

Pandemonium is a leading platform dedicated to sharing practical experiences and insights on financial markets and its products. Spearheaded by Varda Pandey, a seasoned financial practitioner with over 15 years of markets experience, we break down all jargon and offer logical and intuitive understanding of market concepts. The uniqueness of our content comes from references to real world market events to better learn practical application of the material.