Put/call parity is a captivating, noticeable reality arising from the options markets. By gaining an understanding of put/call parity, one can begin to better understand some mechanics that professional traders may use to value options, how supply and demand impacts option prices and how all option values (at all the available strikes and expirations) on the same underlying security are related. Prior to learning the relationships between call and put values, we’ll review a couple of items.

Arbitrage

Let us begin by defining arbitrage and how arbitrage opportunities serve the markets. Arbitrage is, generally speaking, the opportunity to profit arising from price variances on one security in different markets. For example, if an investor can buy XYZ in one market and simultaneously sell XYZ on another market for a higher price, the trade would result in a profit with little risk.

The selling pressure in the higher priced market will drive XYZ’s price down. Conversely, the buying of XYZ in the lower price market will drive XYZ’s price higher. The buying and selling pressure in the two markets will move the price difference between the markets towards equilibrium, quickly eliminating any opportunity for arbitrage. The “no-arbitrage principle” indicates that any rational price for a financial instrument must exclude arbitrage opportunities. That is, we can determine the value of a financial instrument if we assume arbitrage to be unavailable. Using this principle, we can value options under the assumption that no arbitrage opportunities exist.

When trying to understand arbitrage as it relates to stock and options markets, we often assume no restrictions on borrowing money, no restrictions on borrowing shares of stock, and no transactions costs. In the real world, such restrictions do exist and, of course, transaction costs are present which may reduce or eliminate any perceived arbitrage opportunity for most individual investors. For investors with access to large amounts of capital, low fee structures and few restrictions on borrowing, arbitrage may be possible at times, although these opportunities are fairly rare.

Defining Derivatives

Options are derivatives; they derive their value from other factors. In the case of stock options, the value is derived from the underlying stock, interest rates, dividends, anticipated volatility and time to expiration. There are certain factors that must hold true for options under the no arbitrage principle.

For example, a $50 call option on XYZ expiring June of the current year must be priced at the same or lower price than the September XYZ $50 call option for the current year. If the September call is less expensive, investors would buy the September call, sell the June call and guarantee a profit. Note that XYZ is a non-dividend paying stock, the options are American exercise style and interest rates are expected to be constant over the life of both options.

Here is an example of why a longer term option premium must be equal to or greater than the premium of the short term option.

Transaction 1: Buy September call for $3.00

Transaction 2: Sell June call for $3.50

Transaction 3: Assigned on June call, receive $50/share, short 100 XYZ

Transaction 4: Exercise September call, pay $50/share, flatten existing short position

Result: $0.50 per share profit

*note XYZ is a non-dividend paying stock*

In our interest free, commission free, hypothetical world, the timing of the assignment does not matter, however the exercise would only occur after an assignment. Note too that if XYZ falls below the $50 strike price, it does not impact the trade as a result of the $0.50 credit received when the positions were opened. If both options expire worthless, the net result is still a profit of $0.50.

This example shows why a $50 XYZ call option expiring this June, must trade at the same or lower premium than a $50 call option expiring the following September. If the June premium was higher (like in the example), investors would sell the June call, causing the price to decline and buy the September, causing the price of that option to rise. These trades would continue until the price of the June option was equal to or below the price of the September option.

A similar relationship can be seen between two different strike prices but the same expiration. For example, if an XYZ June $50 call was trading at $4.00 and the June $45 call was trading at $3.00, a rational investor would sell the $50 call, buy the $45 call, generating a $1 per share credit and pocket a profit.

Synthetic Relationships

With stock and options, there are six possible positions from three securities when dividends and interest rates are equal to zero – stock, calls and puts:

1. Long stock
2. Short stock
3. Long call
4. Short Call
5. Long Put
6. Short Put