Introduction

Put-Call Parity is a fundamental concept in the world of options trading and valuation. It establishes a relationship between the prices of European call and put options with the same strike price and expiration date, as well as the underlying stock and its associated cash flows. Understanding put-call parity is crucial for accurately valuing options and making informed trading decisions.

Understanding Put-Call Parity

Put-Call Parity states that there is an equilibrium relationship between the prices of call and put options. It arises from the concept of arbitrage, where traders can exploit price discrepancies to make risk-free profits. In essence, if there were a violation of put-call parity, it would create an opportunity for arbitrageurs to capitalize on the mispricing.

Application to European Stock Options

In the case of European options, which can only be exercised at expiration, put-call parity can be expressed as follows:

$$C+PV(X)=P+S$$

Where:

• $C$ is the price of the European call option
• $P$ is the price of the European put option
• $X$ is the strike price of the options
• $S$ is the current stock price
• $PV(X)$ is the present value of the strike price

Application to American Stock Options

For American options, which can be exercised at any point before expiration, the put-call parity equation takes the following form:

$$C+PV(X) \geq P+S$$

In this case, the inequality arises from the fact that early exercise of the American call option might result in potential dividend payments.

Put-Call Parity with and without Dividends

Dividends play a significant role in the valuation of options. When dividends are involved, the put-call parity equation accounts for the present value of expected dividends and their impact on option prices. This adjustment ensures that the equation remains valid even when dividends are distributed during the option’s lifespan.

Expressing Put-Call Parity in Terms of Forward Prices

Put-call parity can also be expressed in terms of forward prices. The forward price is the expected future price of an asset, and this formulation of put-call parity helps establish a link between options and forward contracts.