Options Pricing: Put/Call Parity

In this section of the tutorial, we'll show a basic put/call parity example. Put/call parity is an options pricing concept first identified by economist Hans R. Stoll in his Dec. 1969 paper "The Relationship Between Put and Call Option Prices," published in The Journal of Finance. It defines the relationship that must exist between European put and call options with the same underlying asset, expiration and strike prices. (It doesn't apply to American-style options because they can be exercised any time up to expiration.) 

Put/call parity states that the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration (and vice versa). Support for this pricing relationship is based on the argument that arbitrage opportunities would exist whenever put and call prices diverged. 

Arbitrage

When the prices of put and call options diverge, a short-lived arbitrage opportunity may exist. Arbitrage is the opportunity to profit from price variances of identical or similar financial instruments, in different markets or in different forms. For example, an arbitrage opportunity would exist if an investor could buy stock ABC in one market for $45 while simultaneously selling stock ABC in a different market for $50. The synchronized trades would offer the opportunity to profit with little to no risk. In options trading, arbitrage traders would be able to make profitable trades, theoretically free of risk, until put/call parity returned.

When prices diverge, as is the case with arbitrage opportunities, the selling pressure in the higher-priced market drives price down. At the same time, the buying pressure in the lower-priced market drives price up. The buying and selling pressure in the two markets quickly bring prices back together (i.e., parity), eliminating any opportunity for arbitrage. The reason? The market is generally smart enough not to give away free money. 

Put/Call Parity Example

The most simple formula for put/call parity is Call – Put = Stock – Strike. So, for example, if stock XYZ is trading at $60 and you checked option prices at the $55 strike, you might see the call at $7 and the put at $2 ($7 – $2 = $60 – $55). That's an example of put/call parity. If the call were trading higher, you could sell the call, buy the put, buy the stock and lock in a risk-free profit. It should be noted, however, that these arbitrage opportunities are extremely rare and it's very difficult for individual investors to capitalize on them, even when they do exist. Part of the reason is that individual investors would simply be too slow to respond to such a short-lived opportunity. But the main reason is that the market participants generally prevent these opportunities from existing in the first place. 

Synthetic Relationships

If you understand the put/call parity relationship, you can connect the value between a call option, put option and the underlying stock. The three are related in that a combination of any two will yield the same profit/loss profile as the remaining component. For example, to replicate the gain/loss features of a long stock position, an investor could simultaneously hold a long call and a short put (with the same strike price and expiration). Similarly, a short stock position could be replicated with a short call plus a long put, and so on. The six possibilities are:

Analysis

Some options trading platforms provide charting for put/call parity. Figure 7 shows an example of the relationship between a long stock/long put position (shown in red) and a long call (in blue) with the same expiration and strike price. The difference in the lines is the result of the assumed dividend that would be paid during the option's life. If no dividend was assumed, the lines would overlap. 

Figure 7 An example of a put/call parity chart created with an analysis platform.