Put/Call Parity and Synthetic Positions
Jim Graham

Option traders should have a good understanding of one of the foundations of option pricing, the theory of Put/Call Parity. Put/Call parity means that the value of a call option implies a certain fair value for the corresponding put, and visa versa. To explain why this pricing relationship always holds, the entire argument relies on arbitrage. If the value of puts and calls were to diverge, arbitrageurs would step in to eliminate any departure from put call parity by making a profit on risk-free trades.
 
The relationship is strict only for European-style options but the concept works for American-style options after adjusting for dividends and interest rates. Dividends increase put values and decrease call values. If the dividend is increased, the puts expiring after the ex-dividend date will rise in value, while the calls will decrease by a similar amount. Changes in interest rates have the opposite effect on put and call values. Rising interest rates increase call values and decrease put values.
 
So what happens if the puts and calls for an asset are not in parity? There are a number of strategies that can be used for option arbitrage. The most common are the conversion and reverse conversion. The conversion involves having a long position in the stock while simultaneously buying a put and selling a call (at the same strike price). A reverse conversion (often called a reversal) means you short the stock while simultaneously selling a put and buying a call. The figure below shows a conversion in Silicon Labs:
 

ELN Position Analyzed

 
This article is not to say whether this would be a good trade or not. It is only to illustrate the basis of arbitrage – buy low and sell high for a small but fixed profit. Notice that the risk graph for this trade at expiration is a horizontal line. No matter where the price of the stock goes, from zero to infinity, you make $140. These trades are most commonly done on the floor or by market makers. The window of opportunity for these trades usually is a very short time and they are able to do the transaction in seconds with very low commissions.
 
But while arbitrage strategies may not be the best strategy for an individual trader to place, you need to know how they work to understand options a little better. With these strategies you can get a feel for how puts, calls, and the underlying stock are interrelated, and learn why the price of one can’t move far without the price of the others adjusting.
 
If you are familiar with the risk graphs of various option strategies, you should know that the risk profiles of all the basic strategies can be duplicated with more complex strategies.
 

Consider the simplest option strategy, the long call. When you buy a call, your loss is limited to the premium paid, while your possible gain is unlimited. Now consider the purchase of a put and its underlying stock. Again, your loss is limited to the premium paid for the put plus any out-of-the-money amount, and your profit potential is unlimited as the stock price increases. Below is a graph that compares these two positions:

 

MSFT Position Analyzed

 
If the two graphs appear identical, it’s because they are. A long put/long stock position is almost identical to owning the call of the same strike and month. In fact, the long put/long stock position is often called a “synthetic” long call. The main difference between the two lines is the $10 in dividends that the owner of the stock receives.
 
All basic option strategies have a synthetic equivalent. The rule for synthetics is that the strikes and months of the calls and puts must be identical. For all synthetics that involve both stock and options, the number of shares represented by the options must be equal to the number of shares of stock. Below is a list of synthetics with their equivalents:
 
PositionSynthetic Position
Long CallLong Put/Long Stock
Long PutLong Call/Short Stock
Short CallShort Stock/Short Put
Short PutLong Stock/Short Call
Long StockLong Call/Short Put
Short StockLong Put/Short Call
Long StraddleShort Stock/Long Two Calls
Short StraddleLong Stock/Short Two Calls
 
Conversion and reverse conversion strategies utilize synthetic strategies. Looking at the various positions and their equivalent synthetics, if the risks and rewards are the same (across the same strike prices) then a synthetic position should be priced the same as the actual position. That is, at the same strike prices, a synthetic call should cost the same as an actual call.
 

Put/Call parity can differ only by trivial amounts such as trading costs. If parity is violated, an opportunity for arbitrage exists. While you may never get the chance to execute an arbitrage trade, it is important to understand them and their importance in the options pricing mechanism.