05/27/2010 12:01 am EST

Focus: OPTIONS

Dan Passarelli

Founder, Market Taker Mentoring, Inc.

In order to understand more complex spread strategies involving two or more options, it is essential to understand the arbitrage relationship of the put-call pair. Puts and calls of the same month and strike on the same underlying have prices that are defined in a mathematical relationship.

They also have distinctly related “Greeks” of vega, gamma, theta, and delta. This article will show how the metrics of these options are interrelated. It will also explore synthetics and the idea that by adding stock to a position, a trader may trade with indifference either a call or a put to the same effect.

Before the creation of the Black-Scholes model, option pricing was hardly an exact science. Traders had only a few mathematical tools available to compare the relative prices of options. One such tool, put-call parity, stems from the fact that puts and calls on the same class sharing the same month and strike can have the same functionality when stock is introduced.

Married Put vs. Long Call

For example, traders wanting to own a stock with limited risk can buy a married put—long stock and a long put on a share-for-share basis. The traders have infinite profit potential and the risk of the position is limited below the strike price of the option. Conceptually, long calls have the same risk/reward profile—unlimited profit potential and limited risk below the strike.

FIGURE 6.1 is an overview of the at-expiration diagrams of a married put and a long call.

Figure 6.1   Long Call vs. Long Stock + Long Put (Married Put)

Click to Enlarge

Married puts and long calls sharing the same month and strike on the same security have at-expiration diagrams with the same shape. They have the same volatility value and should trade around the same implied volatility. Strategically, these two positions provide the same service to a trader, but depending on margin requirements, the married put may require more capital to establish because the trader must buy not just the option, but also the stock.

The stock component of the married put could be purchased on margin. Buying stock on margin is borrowing capital to finance a stock purchase. This means the trader has to pay interest on these borrowed funds. Even if the stock is purchased without borrowing, there is opportunity cost associated with the cash used to pay for the stock. The capital is tied up. If the trader wants to use funds to buy another asset, he will have to borrow money, which will incur an interest obligation.

Furthermore, if the trader doesn’t invest capital in the stock, the capital will rest in an interest-bearing account. The trader foregoes that interest when he buys a stock. However the trader finances the purchase, there is an interest cost associated with the transaction.

Both of these positions, the long call and the married put, give a trader exposure to stock-price advances above the strike price. The important difference between the two trades is the value of the stock below the strike price—the part of the trade that is not at risk in either the long call or the married put. On this portion of the invested capital, the trader pays interest with the married put (whether actually or in the form of opportunity cost). This interest component is a pricing consideration that adds cost to the married put and not the long call.

So if the married put is a more expensive endeavor than the long call because of the interest paid on the investment portion that is below the strike, why would anyone buy a married put? Wouldn’t traders instead buy the less expensive, less-capital-intensive long call?

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Given the additional interest expense, they would rather buy the call. This relates to the concept of arbitrage. Given two effectively identical choices, rational traders will choose to buy the less-expensive alternative. The market as a whole would buy the calls, creating demand that would cause upward price pressure on the call. The price of the call would rise until its interest advantage over the married put was gone.

In a robust market with many savvy traders, arbitrage opportunities don’t exist for very long. It is possible to mathematically state the equilibrium point toward which the market forces the prices of call and put options by use of the put-call parity. The put-call parity equation states:

• c + PV(x) = p + s

Where:

• c is the call premium
• PV(x) is the present value of the strike price
• p is the put premium
• s is the stock price

Another less academic and more trader-friendly way of stating this equation is:

• Call + Strike – Interest = Put + Stock

Where interest is calculated as:

• Interest = Strike x Interest Rate x (Days to Expiration/365)

The two versions of the put-call parity stated here hold true for European options on non–dividend-paying stocks.

Dividends

Another difference between call and married-put values is dividends. A call option does not extend to its owner the right to receive a dividend payment. Traders, however, who are long a put and long stock are entitled to a dividend if it is the corporation’s policy to distribute dividends to its shareholders.

An adjustment must be made to the put-call parity to account for the possibility of a dividend payment. The equation must be adjusted to account for the absence of dividends paid to call holders. For a dividend-paying stock, the put-call parity states:

•   Call + Strike – Interest + Dividend = Put + Stock

The interest advantage and dividend disadvantage of owning a call is removed from the market by arbitrageurs. Ultimately, that is what is expressed in the put-call parity. It’s a way to measure the point at which the arbitrage opportunity ceases to exist. When interest and dividends are factored in, a long call is an equal position to a long put paired with long stock. In options nomenclature, a long put with long stock is a synthetic long call. Algebraically rearranging the above equation:

•   Call = Put + Stock – Strike + Interest – Dividend

The interest and dividend variables in this equation are often referred to as the basis. From this equation, other synthetic relationships can be algebraically derived, like the synthetic long put.

• Put = Call – Stock + Strike – Interest + Dividend

A synthetic long put is created by buying a call and selling (short) stock. The at-expiration diagrams in FIGURE 6.2 show identical payouts for these two trades.

Figure 6.2   Long Put vs. Long Call + Short Stock

Click to Enlarge

Synthetics

The concept of synthetics can become more approachable when studied from the perspective of delta as well. Take the 50-strike put and call listed on a \$50 stock. A general rule of thumb in the put-call pair is that the call delta plus the put delta equals 1.00 when the signs are ignored. If the 50 put in this example has a –0.45 delta, the 50 call will have a 0.55 delta. By combining the long call (0.55 delta) with short stock (–1.00 delta), we get a synthetic long put with a –0.45 delta, just like the actual put.

The directional risk is the same for the synthetic put and the actual put. A synthetic short put can be created by selling a call of the same month and strike and buying stock on a share-for-share basis. This is indicated mathematically by multiplying both sides of the put-call parity equation by –1:

• –Put = –Call + Stock –Strike + Interest –Dividend
 More tomorrow in Part 2… Read Part 2

By Dan Passarelli of Market Taker Mentoring LLC