The concepts outlined in this chapter form the basis for the option strategies in Part Two. These concepts expand on the basics in Chapter 3. They are not necessary for most traders who are mainly looking at option strategies to hold to expiration. The first topic in this chapter will be a quick introduction to option pricing models, particularly the Black-Scholes Model. Also discussed will be the greeks and how they affect the price of an option; probability distributions and how they affect options; option pricing models and their advantages, disadvantages, and foibles and using them. The final major topic will be the concept of delta neutral.

which is a key concept for many of the advanced strategies in this book. Which option should you buy? What if you are looking for the price of Widget futures to move from 50 to 60 over the next four months? Do you buy the option that expires in three months and roll it over near expiration? Or do you buy the six-month option and liquidate it in four months? The answer to these questions is whichever option maximizes profit for a given level of risk.

To decide on an option, you need to find the fair value and characteristics of the various options available for your preferred strategy. You need to find out which option provides the best value, which requires an ability to determine the fair value of an option and to monitor the changes in that fair value. You must be able to determine the likely future price of that option, given changes in such critical components of options prices as time, volatility, and the change in the price of the underlying instrument (UI)


Option pricing models help you answer key questions:
* What is a particular option worth?

* Is the option over- or undervalued?
* What will the option price be under different scenarios?

Option pricing models provide guidance, not certainty. The output of an option pricing model is based on the accuracy of the model itself as well as the accuracy and timeliness of the inputs. Option pricing models provide a compass to aid in evaluating an option or an option strategy. However, no option model has yet been designed that truly takes into account the totality of reality. Corners are cut, so only an approximation of reality is represented in the models. The model is not reality but only a guide to reality. Thus, the compass is slightly faulty, but having it is better than wandering blindly in the forest. Option pricing models allow the trader to deal with the complexity of options rather than be overwhelmed. 

Option pricing models provide a framework for analysis of specific options and option strategies. They give the strategist an opportunity to try out “what if” scenarios. Although option pricing models are not 100 percent accurate, they provide more than enough accuracy for nearly all option trading styles. The inability to account for the last tick in the price of an option is essentially irrelevant for nearly all traders. On the other hand, arbitrageurs, who are looking to make very small profits from a large number of trades, need to be keenly aware of the drawbacks and inaccuracies of option pricing models. They must look at every factor through a microscope. 

 One early book that was related to options pricing was Beat the Market by Sheen Kassouf and Ed Thorp. This book sold very well and outlined a method of evaluating warrants on stocks, which are essentially long-term options on stocks. However, these models that came before the Black-Scholes Model are rarely mentioned today mainly because of two factors: (1) they were not arbitrage models; and (2) options were not popular, so few traders or academics were paying attention to options pricing problems.

Arbitrage Models

An arbitrage model is a pricing model in which all the components of the model are related to each other in such a way that if you know all of the components of the model but one, you can solve for the unknown component. This applies to all of the components. It ties up all the factors relating to the pricing of an option in one tidy package. Furthermore, an arbitrage model is a model that prices the option, given certain inputs, at a price where the buyer or seller would be ambivalent between the UI and the option. 

For example, a thoroughly rational bettor would be ambivalent between being given $1 or putting up $1 with another bettor and flipping a coin to see who wins the $2. The expected return from both of these deals is $1. An arbitrage model attempts to do the same thing. The expected return from, say, owning 100 shares of Widget mania at $50 should be exactly the same as owning an option to buy the same shares. There are many different option pricing models. The most popular is the Black-Scholes Model. Other models for pricing options are:

* Cox-Ross-Rubenstein (or Binomial) Model
* Garman-Kohlhagen Model
* Jump Diffusion Model
* Whalley Model
* Value Line Model

Each model takes a look at evaluating options from a different perspective. Usually the goal of the model is to better estimate the fair value of an option. Sometimes the goal is to speed up computation of the fair value.

Black-Scholes Model

The first arbitrage model is the most famous and most popular option pricing model—the Black-Scholes Model. Professors Stanley Black and Myron Scholes were fortunate that they published their revolutionary model just as the Chicago Board Options Exchange (CBOE) was founded. The opening of the CBOE shifted the trading of options from a small over- the-counter backwater of the financial community to a huge and growing market and created a demand for greater information about options pricing. The Black-Scholes was deservedly at the right place at the right time. The initial version of the Black-Scholes Model was for European options that did not pay dividends. 

They added the dividend component soon after. Mr. Black made modifications to the model so that it could be used for options on futures. This model is often called the Black Model. Mark Garman and Steven Kohlhagen then created the Garman-Kohlhagen Model by modifying the Black-Scholes Model so that it gave more accurate pricing of options on foreign exchange. All of these versions of the Black-Scholes Model are similar enough that they are often simply described generically as the Black-Scholes Model. Another popular model is the Cox-Ross-Rubenstein, or Binomial, Model. 

This model takes a different approach to the pricing of options. However, many option traders feel that it is generally more accurate than the Black-Scholes Models. The main drawback, however, is that it is computationally more time consuming. The Black-Scholes Model is used only for pricing European options. Yet most options traded in the world are American options, which allow for early exercise. It has been found, however, that the increase in accuracy from using a true American-pricing model is usually not worth the greater cost in computational time and energy. 

This is particularly true with options on futures. Arbitrageurs will sometimes shift to an American pricing model when a stock option gets near expiration or becomes deep in-the-money. These are the circumstances when the chances of early exercise become more likely and the greater accuracy of a model that prices American-style options becomes more important. Another apparent oddity is that the Black-Scholes Model does not price put options, only calls. However, the price of a put can be found by using the model to price a call and using the put-call parity principle.

The Black-Scholes Model assumes that two positions can be con- structed that have essentially the same risk and return. The assumption is that, for a very small move in either of the two positions, the price of the other position will move in essentially the same direction and magnitude. This was called the riskless hedge and the relationship between the two positions was known as the hedge ratio. Generally speaking, the hedge ratio describes the number of the underlying instrument for each option. 

For example, a hedge ratio of 0.50 means that one half of the value of one option is needed to hedge the option. In the case of a stock option, a hedge ratio of 0.50 would mean that 50 shares of the underlying stock are needed to hedge one option. In the case of an option on a futures contract, a hedge ratio of 0.50 would mean that one half of a futures contract is needed to hedge the option. Clearly, one can-not hold only one half of a futures contract, but that is how many would be needed to theoretically hedge the option on that futures contract. 

The Black-Scholes Model assumes that the two sides of the position are equal and that an investor would be indifferent as to which one he or she wished to own. You would not care whether you owned a call or the UI if the call were theoretically correctly priced. In the same way, a put would be a substitute for a short position in the UI. This was a major intellectual breakthrough. Previously, option pricing models were based more on observing the past rather than strictly and mathematically looking at the relationship of the option to the UI. An arbitrage model relies heavily on the inputs into the model for its accuracy. Designing a model using gibberish for inputs will lead to a model that outputs gibberish. The Black-Scholes Model takes these factors into account:

* Current price of the UI
* Strike price of the option
* Current interest rates
* Expected volatility of the UI until expiration
* The possible distribution of future prices
* The number of days to expiration
* Dividends (for options on stocks and stock indexes)

Given this information, the model can be used to find the fair price of the option. But suppose the current price of the option was known, and what was wanted was the expected volatility that was implied in the price of the option. No problem. The Black-Scholes Model could be used to solve for the expected volatility. The model can be used to solve for any of the listed factors, given that the other factors are known. This is a powerful flexibility. A further advantage of the model is that the calculations are easy. The various factors in the model lend themselves to easy calculation using a sophisticated calculator or a simple computer. 

The calculations with other models, which might give better results, take so long that they have limited use. Option traders are usually willing to give up a little accuracy to obtain an answer before the option expires! The Black-Scholes Model is the standard pricing model for options. It has stood the test of time. All of the examples in this book, and virtually all other books, are derived using the Black-Scholes Model. However, the model has some drawbacks. As a result, the model is no longer the standard for options on bonds, foreign exchange, and futures, though the standard models for these three items are modifications of the original.

Assumptions of the Black-Scholes Model

Examining the assumptions of the Black-Scholes Model is not done to criticize the model but to identify its strengths and weaknesses so that the strategist does not make a wrong move based on a false assumption.

Current Price of the UI

The current price of the UI is usually known with some certainty for most option traders. They can look on the screen or call their broker and get a price for the UI. It usually does not matter if the price quote is a little wrong. However, arbitrageurs often have a problem determining exactly what the price of the UI is. They ask: How wide is the bid/ask spread? Is the last trade on the bid, in the middle, or on the ask? Has the bid/ask spread moved since the last trade? Are prices extremely volatile, and will I have a hard time executing a trade at the current bid or ask because the bids and offers are moving so much .

The Strike Price of the Option

Fortunately, this one factor is stable and does not change significantly. Strike prices for stock options do change whenever there is a stock split or a stock dividend.

Interest Rates

The Black-Scholes Model assumes that setting up the right relationship between the UI and the option will lead to a neutral preference by the investor. The value of the UI and the value of the option will be balanced because the Black-Scholes Model is an arbitrage model. The model assumes that the so-called risk-free rate is the proper rate. Traditionally, the risk-free rate is considered the rate paid on U.S. government securities, specifically, Treasury bills, notes, and bonds. To make the model work, it is assumed that interest is being paid or received on balances. It is assumed that all positions are financed, an assumption that is reasonable because there is always an opportunity cost even if the position is not financed. 

The Black-Scholes Model assumes that you would invest your money in Treasury bills if you did not invest it in an option. The term of the interest rate used in the model should be the term to expiration of the option. For example, if you are pricing an option that matures in 76 days, then you should theoretically use the interest rate corresponding to a Treasury bill that matures in 76 days. In the real world, of course, you would simply select a Treasury bill that matures close to that perfect number of days. 

The problem is that the model assumes that you both invest your money and borrow money at the risk-free rate. It is quite reasonable to assume that you will invest your money in Treasury bills in the real world. However, only the U.S. government can borrow at the Treasury-bill rate. All other borrowers must pay more, sometimes much more. As a result, some options traders assume that they invest at the Treasury-bill yield but that they borrow at the Eurodollar yield or at the prime rate. In general, the rate assumed in the model will have little effect on the price of the option. The level of interest rates mainly affects the price of multi year options.

Probability Distribution

The probability distribution is the expected future possible distribution of prices, that is, the probability that any price will occur in the future. The model basically assumes that prices are randomly distributed around the current price in roughly a bell shaped curve.

Expected or Implied Volatility

Expected volatility is the volatility of the price of the UI expected in the future by the investor or the market. Expected volatility is the width of the bell curve mentioned in the preceding paragraph.

Days to Expiration

Fortunately, the number of days to expiration of the option does not change.


The Black-Scholes Model does not take into account the effect o taxes on the pricing of options. In fact, no major model does. This is not a major problem, but it might affect some arbitrageurs. For example, it was shown that the model assumes the risk-free or T-bill rate as the interest rate, but that is not usually the case in the real world: The investor might be receiving T-bill interest, which is exempt from state and local taxes, but paying the equivalent of Eurodollar rates or even the prime rate. The investor might or might not be able to deduct the cost of the borrowing from the proceeds of the trade. Some traders will be taxed differently on the interest or dividend in- come than on the gain or loss from the option. Interest and dividend in- come are usually ordinary income, whereas gains and losses from options are capital gains and losses. Taxes are an important subject but beyond the scope of this book. Variations in taxes could have an impact on the fair price of an option for a particular trader.



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