Arbitrage Strategies and Price Relationships

When looking at an option chain, you see all the data for an underlying asset and its related options.  Between the various sections – the underlying, the call and put options, and the different expiration months – there are fundamental relationships that underlie their pricing. 

When these relationships get out of line, an arbitrage opportunity exists—buying an option(s) and selling the related option(s) for a (near) risk-free profit.  To illustrate these relationships we will use arbitrage strategies, and we will begin by discussing synthetics, which form the basis for all the different arbitrage strategies. 

Synthetic Relationships

There can be up to three different parts to any potential option strategy: The underlying asset; the Call options; and the Put options.  Most arbitrage strategies use the concept of synthetics, and they are a large part of the strategies we use here.  A synthetic strategy is one where you combine any two parts (calls, puts and/or the underlying) to create a position that looks like the third one. 

For example, if you buy both the stock and a put option, you will make money if the market goes up, but your loss is limited if the market falls.  That's exactly the same risk/reward you would get if you bought a call option – you make money if the market goes up but your loss is limited to the premium paid if the market falls.  Buying the stock and buying a put is therefore called a synthetic call.  In terms of risk and reward, it is exactly the same thing!

The various synthetic relationships may seem a little confusing, but with a little practice you will see how easy it is to understand.  An important rule to keep in mind is that the strikes and months of the calls and puts must be identical.  For synthetics that involve both the stock and options, the number of shares represented by the options must be equal to the number of shares of stock.  The table below lists the basic synthetic positions:

Synthetic Relationships
Synthetic short stock	=	Long put + short call*
Synthetic long stock	=	Long call + short put*
Synthetic long call option	=	Long underlying + long put
Synthetic short call option	=	Short underlying + short put
Synthetic long put option	=	Short underlying + long call
Synthetic short put option	=	Long underlying + short call

* Using the same strike price and expiration date

These are not new strategies. You will see them covered in many intermediate or advanced option trading books.  The point here is not to suggest new trades you should consider doing, but rather to use them to put things in perspective and show how the underlying, options, and their prices are related.

Synthetics are used in arbitrage because all basic option strategies have a synthetic equivalent.  And 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.  If this condition is violated, an opportunity for arbitrage exists.  The arbitrage strategies we will be using in this article are:

Conversions
Reversals
Box Spreads
Jelly Rolls.

Most option traders will probably never use these arbitrage strategies.  But even if you don't trade these opportunities, understanding the mechanics of arbitrage and the relationships will make you a better trader and give you a new way of looking at options.  Let's begin by looking at the first two, conversions and reversals, since these two strategies clearly show the relationship between the price of the underlying and the price of the put and call options.

The Conversion Strategy

The price of the put and call options across the same strike prices can not get very far out of line from the fair value dictated by the underlying price.  As long as the risk and reward is the same, a synthetic call should cost the same as an actual call option.  If it is not, then an arbitrage opportunity exists.  You can buy the cheap one and sell the expensive one for a risk-free return.  This is what the conversion strategy does.  A conversion involves:
• Buying 100 shares of the underlying stock 
• Buying a put option; and
• Selling a call option (at the same strike price as the put).

The Reversal Strategy

The reversal strategy is just the opposite of the conversion.  That is, you sell the underlying short and place a synthetic long position.  The actual transactions needed would be:

• Short 100 shares of the underlying stock 
• Buy a call option; and
• Sell a put option (at the same strike price as the call).

Inter-Month Relationships and Jelly Rolls

Under normal circumstances, the implied volatility of the options in the farther-out months is a little higher than the front month.  When that relationship is backwards, it may seem like a great trading opportunity.  When traders see this happen they tend to think about placing calendar spreads that involve selling front month options and buying the farther-out options.  The idea is to sell higher implied volatility and buy lower implied volatility, and then make money when the volatilities reverse themselves.

Transaction Costs and Dividends

In all the above examples, I have only touched on the effects that transaction costs and dividends can have on option pricing.  It is easy to see the effect transaction costs can have.  You simply need to make sure that the profit from any of these strategies is greater than your transaction costs.  Since is worthwhile to do the trade only if you can at least cover your transaction costs, options prices can thus be out of line (and stay that way even in the long term) by the amount of those transaction costs.