to be a good option trader, one must first be a good trader. Sometimes the particular complexities of managing an option position can blind us to this fact. In order to be successful, we have to put ourselves in situations where we can buy low and sell high. Humans have been trading forever. Despite this, there is no consensus about what good trading practices are. Perhaps when it comes to specifics this is to be expected. By its nature successful trading can destroy the anomalies that make it possible. However, at a more general level we can state the essential characteristics that all successful trades must have. Further, it is almost certainly a better idea to improve upon a method that others have found to be successful than to try to find something completely new.

I want to emphasize the place of this chapter. This is meant to complement innate trading skill, which I certainly believe exists. I do think that a solid grasp of the quantitative analysis of probability and strategies can benefit any trader, but knowing all of these things is not enough to make a good trader. Traders already do a lot of quantitative analyses, often subconsciously. Good traders can read patterns and prices in ways that statistical analysis still finds too hard. As I have stated several times, measurements in markets are very context-dependent, and a good trader often has keen insight into what variables are currently important.

Too many quantitatively driven traders dismiss the decisions of intuitive traders as merely arbitrary and little more than guesswork. But at least some intuitive traders make very thorough analyses of the situations they see. That this is subjective is generally because it is the best or only way to quickly amalgamate the data, place it into context and produce a conclusion. Granted, many more traders think they have this skill than actually do, but we should never totally dismiss the idea. Some intuitive traders, like some statisticians or some scientists, are not very good. They are irrational or even just stupid. But for those who are not, I hope this chapter helps them a little. This chapter is about general trading principles. We may use option examples in places, but the concepts are more broadly applicable. The specifics of trading options will be covered in later chapters.


You must have a definite source of edge. This is something that gives your trades positive expected value. Very loosely speaking, for something to be a source of edge it must be correct and not widely known. This second part is often forgotten. If you know only what others know, this is valueless. It will already be priced into the market. This is why, even if I was so inclined, it would be impossible for me to give you a recipe for profitable trades. The publishing of the recipe would render the trades generated from it almost immediately useless.

Looking for positive expected value is not the same as making only trades that we expect to win. This flawed thinking leads to the nonsense, “You only have to win more often than you lose.” This is simply wrong. The winning percentage of a trader is not enough to know if he has been, let alone will be, successful. A similar error is perpetrated by those who refuse to buy options, on the basis that most expire out-of-the-money. This is a mistake made by even the most experienced traders.

You must have some idea why a particular trade has positive expected value. Back-testing an idea will show you that something has worked in the past, but there are an infinite number of combinations of trade ideas, parameters, and products. Obviously some of these will have produced successful trades in the past. We need to know why they will do so in the future. This gives us confidence, but also tells us when to stop a particular trade. If you have no knowledge of why something worked in the first place, it is very tough to know that it has stopped working. Evaluating its diminishing effectiveness just from watching the results can be expensive and is also often difficult, as all trades will go through bad stretches.


A hedge is a trade that we enter in order to somehow offset an existing position. For example, imagine that we buy the 100 call because its implied volatility is cheap. However, this makes us synthetically long the underlying, so we sell some of the underlying short in order to mitigate our directional exposure. This is a hedge. We see in Chapter 4 that this idea is at the core of option pricing, but when should we hedge trades in general? We need to have an understanding of hedging concepts, so we can deal with the complexities that will occur in real trading situations. Ideally, we hedge all of the risks except those that we explicitly want to be exposed to. In practice this is seldom possible. Even most trades that we think of as arbitrages will have some minor difference in contact specifications. This exposes us to risk. Also common is the situation where a hedging instrument is available, but we have to weigh the benefits of hedging a risk against the costs of executing and managing another instrument.

There are a few situations where we should always at least consider hedging.

* We enter a trade that has multiple sources of risk and we only have positive expectation with respect to one of these. Directionally hedging an option trade to isolate the volatility exposure is an example of this situation.

* Our exposure has grown too big. This could happen as a result of losses in other parts of your portfolio, or it could occur when one position performs unexpectedly well. If you originally wanted a position to ac- count for a certain percentage of your risk, you should consider rebalancing when it significantly exceeds this.

* We can enter a hedge for no cost. For example, in times of market turmoil, it has sometimes been possible to enter long put or call spreads for zero cost or even a credit. This is a great situation. These spreads will probably be a long way out of the money. They have a low probability of making money. But they could pay off, and they cost nothing. It is like being given free lottery tickets. Such situations happen less often than they used to, but they still occur, particularly in pit-traded products.

* The hedge was part of the original trade plan. Sometimes we do certain trades because we are reasonably confident that an offsetting trade will present itself before expiration. In 1997, around 11 A . M . London time, a customer would buy several thousand 20 delta DAX puts in clips of one hundred. Once we had noticed this pattern we would have a few hours each morning to work at buying puts on the bid, knowing that our hedge would appear later. These situations occur frequently, and give market makers a significant source of profit.

All hedging decisions need to be made on the basis of a risk-reward tradeoff. And as soon as risk is introduced to a situation, personal preferences matter. What is seen as an unwanted risk by some traders will be welcomed by others. For a market maker, inventory is a source of risk that he will pay to remove, but for a position trader inventory is necessary to make money. But sometimes we just have a bad position. Part of the art of trading is recognizing this case and knowing when it is time to start again. In some types of trading this is all that position management is: merely a matter of knowing when to take profits and when to stop ourselves out of a bad position. Option positions are far more complex as they present us with multiple risks.

In fact the flexibility they offer us in being able to tailor our position to exact views of the market (for example, we might forecast that the underlying will rally slowly, both implied and realized volatilities will decrease, interest rates will remain stable, but a special dividend will be declared) can mean that we are complicit in getting ourselves into overly complex situations. Keeping things as simple as possible is a good idea in most situations. Option traders in particular have a tendency to fall into the trap of over complication. So, no matter what our views are and how certain we are in their correctness, we need always to be absolutely clear what it is we are trying to achieve, when we will admit we are wrong and what we will look to do to adjust our position as circumstances change. By definition, unforeseen circumstances cannot be predicted, however some contingency planning is always possible and the more it is done, the more effective it will become.




We will examine four market outlooks with trading strategies corresponding to each:

Bullish -  The expectation of an increase in price. This category has two subcategories:

* Moderately bullish-Although the outlook is for higher prices, the increase is not likely to be dramatic.

* Extremely bullish  - Expecting a dramatic, explosive increase in price (generally anticipated to occur in the short term)

Bearish -  The expectation of a decrease in price. This category has two subcategories :

* Moderately bearish - Although the outlook is for lower prices, the decrease is not likely to be dramatic.

* Extremely bearish - Expecting a dramatic sell-off in the stock (generally anticipated to occur in the short term)

* Neutral (front spread) - Expecting little price movement over a given time period. Neutral strategies enable the trader to make money in a market where prices remain the same or move little.

* Volatile (back spread) - The anticipation that prices will move dramatically, but the direction of that move is not clear.

Our goal in separating our discussion of strategies into the four general categories of bullish, bearish, neutral, and volatile is to provide our readers (after determining your market outlook) with a reference for the potential strategies. This chapter will also discuss risk-reduction strategies for managing an existing stock portfolio.


A number of these strategies involve option spreads. You construct a spread by being long an option(s) and being short an option(s) of the same type in the same underlying asset. For example, buying a call and selling another call with a different strike or a different expiration is a spread. Buying a put and selling another put with either a different strike or a different expiration is also a spread. In contrast, buying a call and either buying or selling a put is not a spread. 

Spreads offer the investor an array of strategies for attempting to benefit from almost any anticipated market condition while reducing risk. For example, you can use a spread to take a bull position or a bear position, for selling high volatility and buying low volatility, or to finance the purchase of other options. The degree of risk reduction varies among the different types of spreads. While some spreads have limited risk, others have risks that are comparable to buying the underlying security outright. There are several different types of spreads:

1. Calendar spread (in other words, a time or horizontal spread)-With this type of spread, all options are of the same type and have the same strike price and underlying asset, yet they have different expiration dates. The purchase (sale) of one option has a different expiration date from the sale (purchase) of another. Buying one XYZ March 85 call, for example, and selling one XYZ February 85 call would be a calendar spread.

2. Diagonal spreads-This kind of spread is similar to the time spread in that the options are of the same type and underlying asset; however, the expiration date and the strike prices are different. This time spread uses different strike prices. Buying one XYZ March 90 call and selling one XYZ February 85 call is an example of a diagonal spread.

3. Vertical spread-A vertical spread consists of options of the same type, on the same underlying asset, and with the same expiration date, but these options have different strike prices. Buying one XYZ May 90 put and selling one XYZ May 85 put is an example of a vertical spread.

4. Ratio spreads-A ratio spread is any of these types of spreads in which the number of options purchased differs from the number of options sold. Buying one XYZ July 90 call and selling two XYZ July 95 calls is an example of a ratio spread.


Several strategies involve three or more options strikes. As a practical matter, you cannot put on these positions simultaneously at reasonable prices. In order to achieve these positions at prices that produce an acceptable risk/reward profile, you must put on the positions in a series of separate trades. This process is called legging. Although the analysis of these specific positions includes a discussion of how to approach legging, we should give you some general comments concerning legging at this point. Until a position is fully legged into, your ability to complete the position at an acceptable price is subject to the risk of changing prices in the positions that you have not yet executed. The key to legging is knowing which order to execute the trades in order to minimize that risk.

To minimize this risk, we look at the supply and demand factor of each building block in the options strategy. Once we determine the supply and demand for each building block, the trader can leg into the position first from the building block that has the most demand and finish the implementation with the building block that has the most supply. This procedure is called legging the hard side first. The hard side is the trade that is the most difficult to put on. If the stock is rising rapidly in price and the trader wishes to purchase the stock, because the stock is rising, this side is considered the demand side.

The demand side refers to what the majority of traders are doing, whether buying or selling. If a stock is rising quickly, we would say that there is demand for the stock-hence, there would be more buyers than sellers. Selling the stock would be easy; because there are many buyers. We would then call this side the supply side. Because buying the stock in a rising market situation is difficult (getting a good price is difficult because of the high demand), we call this side the hard side.Consider the following example of legging the hard side

first. Assume that you are legging into a covered call in which the stock price is rising. Realizing that it will be harder to purchase the stock at a good price than it will be to sell the call, you should buy stock as the first part of the leg. Selling the call would be the easier of the two sides to fill, because the rising price of the stock should increase the demand for the call. If the trader decides to sell the call first in a rising market, he or she is taking the chance that he or she might not be filled on the stock at his or her price.


When a trader puts on a leg and cannot complete the rest of the position because the price for remaining legs has become unacceptable, the trader is said to be legged out. He or she now has a position that has gone against him or her, and it will be hard to close it without incurring a loss. Some of the option positions that we cover in this book can only be legged into. Do not even bother calling your broker with any fancy spread terminology such as a butterfly or iron butterfly: The market makers on the trading floor will just laugh your broker right out of the trading pit. There is no market maker in the world who will hand over free money, especially to a customer.

Bullish Strategies

Bullish strategies are among the most common strategies that individual investors use, probably resulting from the general view of the market that we acquire through the media and elsewhere is that rising stock prices are good, and falling stock prices are bad. In actuality, your position relative to that market movement-not the movement itself-is either good or bad for you. For example, if you position benefits from a declining market and the market does decline, that is good, while if instead it rallied, that would be bad. Most investors, then, are programmed to buy low and sell high. 

These are bullish investors who want to gain a profit from a rise in value or stock price. In fact, when investors tend to think of bullish strategies, the only thing that typically pops into their head is to purchase stock. To be sure, this strategy is great when the stock rises in price, but when a hefty sum of the investor's capital is committed to the position, this endeavor can be risky: In other words, while long stock purchase is not necessarily the wrong idea, it can be capital intensive and can create risk parameters that the individual investor might not totally understand. 

In this topic, we will show alternatives to purchasing stock, learn how to reduce market directional risk and capital exposure, and discuss the relevance of leverage. The first bullish strategy we will consider is long stock. Because long stock is the most commonly employed strategy and the one with which most traders are familiar, it will offer a good comparison study against the other bullish strategies described in this topic.




Options allow the investor to sculpt the returns in their portfolio. When you buy a stock and the price rises $1, you make $1. You lose $1 if the price declines $1. Your profits are linear and directly related to only the change in the price of the stock. Interest and dividends will make a slight change to the outcome though these factors are also linear. Options blow apart this linearity. Options are called convex instruments because the returns are not linear but curved. We saw that in the previous chapters. You can literally create millions of possible returns through the use of options. You can mix and match options to create just about any return possible.

Selecting a strategy is a multi step process. You should go through a systematic process before initiating a trade. Each step should lead to further refinement of the strategy. It can be very dangerous to your bank account to disregard some or all of the major factors that affect options prices. The most important factor that affects option prices is the price of the underlying instrument. But that is usually not the only thing that most investors look at. Only looking at the underlying instrument price can lead to significant losses for the investor. This strategy assumes that the edge that the investor has in stock selection is so superior that he can withstand a lot of headwinds caused by trading an option or options that have a lot of edges against him.

For example, what if the investor is buying a near dated call on U.S. Widget? But what if the options is overvalued and there is little gamma and the time decay is large. Here are three strikes against the investor. I have seen situations where the investor got the direction of the underlying instrument correct but all the other factors wrong and lost money on the trade. I am reminded of the old admonishment—don’t try this at home, kids. Options have a tremendous amount of power but also a lot of risk. So the design of your strategy should be the most important thing in your arsenal. You need to develop a particular frame of mind to trade options. You need to think multidimensional when you trade options. You must now think about time because options expire and the returns change over time.

 You need to think in terms of distance. By this I mean you must now consider how far the underlying instrument will move. For example, you may buy an out-of-the-money call that expires in three weeks. This means that you must expect the UI to rally at least up to the break-even point by expiration. This is very different from just owning the UI where you are expecting the UI to rally but you don’t need to put a time limit on it. Options require you to consider not only the fact that the underlying instrument will rally but how much and how quickly that rally will occur.

This chapter contains tables that show the main strategies that are the most suitable. One problem with a book like this is that it must, by necessity, simplify. For example, long straddles are usually considered neutral strategies, but they can actually be constructed with a market bias. The tables in this chapter generally refer to strategies as they are usually considered.


The strategies in this book are generally presented in their plain vanilla form. Yet the very nature of options gives greater scope to the creative strategist. For example, one of the interesting aspects of options is that you can combine strategies to create even more attractive opportunities. You could write a straddle and buy an underlying instrument to create a lower break even than by holding the instrument alone or to create greater profits if prices stagnate, but give up some of the upside potential. You should be able to examine a myriad of fascinating strategies after reading this book.

 Another feature of options is the ability to twist the expiration and strike prices to fit your outlook. For example, a straddle is constructed by buying a put and a call with the same strike price. That is the plain vanilla. But you can change the strike prices by, say, buying an out-of-the-money put and an out-of-the-money call and create what is called a strangle. Or why not buy the call for nearby expiration but the put for far expiration? The net effect is that you have a tremendous tool in options for creating exciting trading opportunities. Do not get stuck in the ordinary.


Of course, the selection of any strategy involves trade offs. For every one factor that you gain, you will likely give up another. The choice of one strategy over another largely depends on your personal expectations of the future of the market. For example, you may believe that implied volatility is going to go higher. Any strategy that is long implied volatility is going to be hurt by time decay. You are assuming that implied volatility will increase quickly and strongly enough to offset the drain on your position due to time decay.



There are three main ways to construct a strategy:

1. Use software to filter for different strategies using different criteria.
2. Use a building blocks approach.
3. Use tables such as the ones in this chapter.
We will focus on the latter two. However, we will need to use software to build our strategies using the building blocks approach. The table approach is a rule of thumb or back of the envelope approach.


There are two major techniques to identifying an appropriate strategy:

1. Identify your ideas on the major factors that affect options prices, that is, the greeks. You will need to look at such factors as market opinion, volatility, and time decay. You will then be able to make a statement like, “I think that Widgets will move slightly higher in price, volatility will decline, and time premium will decay rapidly because we are approaching expiration.” You can then start to build the strategy.

2. Systematically rank various option strategies. This technique can easily be used in conjuction with the first. For example, you may have decided that covered call writing fits your outlook. You now want to rank the covered calls on Widget International by their various risk/reward characteristics. For example, you could rank them by expected return or perhaps by the ratio of the return if unchanged to the downside break-even point. The main problem with the use of rankings is that you will need a computer to do all the possible mathematical manipulations.

Once again, the basic way to construct a position is to make a decision on the future of the key greeks and the underlying instrument. This will nearly always lead to a final position that meets your scenario. What this means is that you must have an opinion on the future direction of the UI and on the direction and level of the implied volatility. It is best if you also have an opinion on the other greeks since, although they are usually not as important, sometimes they rise to the highest level of importance. Further, it is advantageous to have an opinion on how quickly these expected changes will occur. 

For example, suppose you are bullish on Widget Life Insurance. You look for the price of the stock to move from its current $50 per share to $60 per share over the coming three months. This means that you should only look at bullish strategies. Suppose you also believe that the options are cheap from the perspective of implied volatility. Maybe you are very bullish, expecting the price to move higher very quickly. You, therefore, should only focus on very bullish strategies where you are a net buyer of calls. This suggests that you should likely buy a call that is out-of-the-money.

Now suppose that all the same conditions apply, but that you are bearish on implied volatility. This means that you should construct a position that is neutral or bearish on volatility. You might want to consider selling a put or buying a bull spread. The point is that your outlook on a given stock, its future price behavior, and the future behavior of the greeks will all have an impact on your construction of a strategy. There are six building blocks that we can use. We can be long or short a call, a put, or the underlying instrument. We can construct any strategy with combinations of those six positions.





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.





Purchasing a stock has an obvious risk/reward profile. If the stock goes up, you make money: If it goes down, you lose money. The reverse is true if you sell a stock short. We refer to this loss exposure as directional risk (refer to Graph ) . Furthermore, the amount of the profit or loss is easy to anticipate. If you purchase 100 shares of XYZ, for each $1 increase in price the position increases in value by $100, and for each $1 reduction in price, the position loses $100 in value. If you sell 100 shares of stock short, for each $1 decrease in price you will make $100, and you will lose $100 for each $1 increase in price. By contrast, determining the risk/reward profile of an option position is much more complicated. As we have seen, an option's value can be affected by a change in anyone of these five variables :

* Stock price

* Time until expiration

* Volatility

* Interest rate

* Amount and timing of dividends

When two or more of these inputs changes, the changes can either act to offset each other in whole or in part or can work in concert to magnify either the increase or decrease in price. The situation is further complicated when options are used in combination.

Fortunately; there are analytical tools available to simplify the analysis of option positions. These tools are commonly referred to collectively as the Greeks and individually as delta, theta, gamma, rho, vega, and omega. Individually; they each measure some aspect of an option position's market risk/reward profile. This statement is true whether the position is a simple one involving one or perhaps a few different options or an extremely complicated position (such as a professional floor trader who has scores of different option contracts-some of which are long and some of which are short-and who may have a long or short position in the underlying asset).

Collectively; the Greeks provide the practiced trader with a comprehensive assessment of a position's risk/reward profile. Not only will the trader have an accurate picture of which market conditions will enhance the value of the position and which will subject it to a loss of value, but the trader will be in a position to determine what adjustments are appropriate in order to reflect the trader's current expectations concerning the stock. The trader can also reduce exposure to one or more aspects of market risk.

The Greeks

The various Greeks, the variables with which they are associated, and a short introductory definition of each are summarized in . This chapter not only explores how the Greeks affect option pricing individually but also describes how they affect option pricing in combination. With the availability of option-analytical software, it is neither

necessary nor particularly useful for you to learn the mathematical formulas involved. What is important, however, is that you grasp conceptually the insight that these tools provide. This knowledge will help you identify your risks and respond to them appropriately: Let's start our inquiry by examining the impact of price movement of  the underlying asset on an option's theoretical value. We will isolate the impact of change in price of the underlying asset by keeping the other inputs constant while varying the price of the stock. Consider Stock ABC
that has a volatility of 50. Using an interest rate of 5 percent, assume that the company does not issue a dividend. A table of the theoretical values of the 60-level calls and puts with 30 days to go until their expiration is shown in  This table highlights two important aspects of the relationship between the price of an option and the price of the underlying stock:

1.  As the price of ABC increases, the price of the 60-level call also increases-while the price of the corresponding put decreases. With the stock at $50, the theoretical values of the call and put are $.41 and $10.25, respectively: By the time the price of the stock reaches $55, for example, the theoretical value of the call has risen to $1.46.  The put's theoretical value has declined to $6.23.

2.  As the price of ABC increases, the price of the call increases while the price of the put decreases. With the stock at $50, a $1 increase in the stock theoretically produces a $.14 increase in the price of the 60- level call and a $.87 decrease in the price of the corresponding put. The call increases in price by $.29 when the stock increases from $54 to $55 and by $.80 when the stock increases from $66 to $67. Correspondingly, the put declines by $.72 when the stock goes from $54 ton $55 but only by $.21 as the stock increases from $66 to $67.

The relationship between the change in the price of a stock and the corresponding change in the price of an option is referred to as the option's delta.


The delta is the most widely known of the Greeks and is an extremely important gauge of any option strategy. Delta measures how sensitive an option's price is to change in the value of the underlying stock. There are two particularly useful ways to look at delta: the measure of how much the option's price will change compared to a change in the price of the underlying asset, and the approximate probability that the option will finish ITM. We will explore both of these perspectives in some detail.

Related Change

An option's delta is the ratio of the change in its the or etical value to a small change in the price of the underlying stock. More commonly (but marginally less precise), delta is defined as how much an option's price changes for every $1 change in the price of the stock. A positive delta means that an option's price moves in the same direction as price movement in the underlying asset. An option that has a positive delta will increase in value as the underlying asset increases in value and will decrease in value as the price of the underlying asset decreases.

Conversely; a negative delta reflects the fact that an option's price moves in the opposite direction from the price movement in the underlying asset.
An option that has a negative delta will decrease in value as the underlying asset increases and will increase in value as the underlying asset decreases. Because a long call and short put increase in value as the underlying asset increases (in other words, their value rises and falls along with the underlying asset), they have a positive delta. Conversely, both a long put and a short call have negative deltas, because their values decrease as the underlying asset increases in price and increase when the

stock price declines. Going back to the table presented earlier in this chapter, with ABC stock trading at $50, the fact that the 60-level call's price would theoretically increase by $.13 when the stock price increased $1 to reach $51 indicates a delta of .13. In comparison, the delta of the 60- level put with the stock trading at $50 would be - .87. Because one equity option contract generally represents options on 100 shares of the underlying asset, the delta is most commonly expressed as the aggregate change in price of the option contract for a $1 change in the underlying asset. In other words, the delta for one option is multiplied by 100 (the number of shares represented by one contract). Under this approach, the 60-level call with ABC trading at $50 would have a delta of 13, not .13. From now on, we will refer to option deltas by using this aggregate designation.

Many option strategies and positions include holdings in the underlying asset, which can be long or short. When you are determining the delta for a complicated position that includes a combination of options and/or positions in the underlying asset, the aggregate delta of the entire position is most relevant. Using the same ratio definition of the delta as for options, the delta of one share of long stock is + 1, while the delta of one share of short stock is -1. Therefore, if a position included 1,000 shares of stock, this stock would contribute 1,000 deltas to the aggregate position delta. On the other hand, if the position included a short of 1,000 shares, this short position would contribute -1,000 deltas to the aggregate position delta.

If the holding is:                                It will contribute the following:

Long underlying security                                 Positive delta

Short underlying security                                Negative delta                                                                                                                         
Long call                                                          Positive delta

Short call                                                         Negative delta

Long put                                                          Negative delta

Short put                                                         Positive delta

Although you can calculate an option's delta precisely by using an option-pricing model, many experienced option traders approximate option deltas by using the following rules of thumb:

* An ATM option typically has a delta of about +50 for calls and -50 for puts.

* With the stock trading at or near a strike price, give the nearest ITM options a delta of 75 and the next-closest ITM options a delta of 90. Then, estimate the deltas of all other ITM options to be 100.

* With the stock trading at or near a strike price, give the nearest OTM options a delta of 25 and the next-closest OTM options a delta of 10. All other OTM options should receive a zero delta.

Determining Options Deltas

 Option Strike Price           Call Delta          Put Delta
            45                             +100                   0
            50                             +90                   -10
            55                             +75                   -25
            60                             +50                   -50
            65                             +25                   -75
            70                             +10                   -90
            75                                0                     -100

For example, if XYZ were trading at $60, we would approximate option deltas as shown in Table.

As you learned in the previous chapter, being long a call and short a put with the same strike and expiration is the equivalent of being long 100 shares of the underlying asset (synthetic long stock). Because long 100 shares always represent an aggregate 100 deltas, combining the deltas of the long call and the short put with the same strike price and expiration date must also always equal 100. Similarly; the aggregate delta of a short call, long put position with the same strike price and expiration date will be -100, the equivalent of being short 100 shares of the underlying asset. This example highlights an important relationship between the deltas of puts and calls with the same strike price and expiration date. If you know one, you can easily determine the other. For example, if the March 60 call has a delta of +38, the delta of the March 60 put is -62.

Probability of an ITM Finish upon Expiration

An easy way to think of delta is as the probability that the option will finish ITM upon expiration. The ATM calls and puts each have a 50-50 (or 50 percent) chance of finishing ITM upon expiration and both carry an approximate delta of 50. The ITM calls and puts have a much greater chance of finishing ITM upon expiration than their ATM counterparts with a deeper ITM call or put having a higher likelihood of finishing ITM than one that is less ITM. Their deltas reflect those respective probabilities. The OTM options have the least percentage chance of finishing ITM and not surprisingly; their small deltas reflect this decreased probability.

Delta and Time

The time until expiration impacts an option's delta. Considering delta as the probability of finishing ITM makes it easier to understand this effect. This thought leads to the following relationship:

* The delta for an ITM option will move towards 100 as time expiration decreases. The likelihood of the option staying ITM increases as

* the time until expiration decreases. The delta of an ATM option will remain at 50, because it still has a 50-50 chance of finishing ITM.

* The delta of an OTM option will move towards zero as expiration approaches, because the likelihood of the option finishing ITM decreases as the time until expiration decreases.

examples of delta and time until expiration. As we can see from these examples, strike prices at differing expiration dates have different deltas. Hence, as the time until expiration increases, the probability that the underlying asset will move towards.





Having reviewed the basics of option characteristics, we are now aware that we have a number of investment choices: stock, calls on the stock, and puts on the stock. For each of these, we can initiate a position either by purchase (referred to as a long position) or by sale (referred to as a short position), thus giving us six different initiating strategies :

* Long stock

* Short stock
* Long call
* Short call
* Long put
* Short put

These tools are what we can use to construct all option-based strategies. By combining these building blocks, the individual investor can create strategies ranging from basic to complex. Mastery of each of the individual building blocks is essential for understanding how they work in combination. Therefore, we will now look at each of these six alternatives in some detail. Retail investors are already familiar with one of these building blocks: long stock. With an understanding of the other five building blocks, the investor will have the ability to create positions that are best suited to capitalize off any market outlook. The only limit to creating positions is the investor's creativity. Once you master the six building blocks, the possibilities are endless.

Long Stock

Long stock is the most common position among investors. After analyzing the fundamentals of a company and deducing that the company's product, its revenue model, and current market conditions reflect the likelihood of positive growth, the retail investor purchases the stock as an investment in that company: Over a period of time, if the investor's analysis proves correct, the stock value increases-rendering a profit. In this case, the investor who purchases stock with his or her own capital is said to be long stock.

Here is an example. An investor who has no position in XYZ purchases 100 shares of XYZ at $50 per share. The investor is now long 100 shares of XYZ.  shows the profit and loss associated with the outright ownership of stock.

The profit in this case is unlimited. This position will profit as the stock increases in value. Each $1 increase in the market price of the stock will result in $100 worth of profit. The loss in this example is limited. This position will generate a loss as the stock declines in price. The risk is limited only because the stock can only decrease to zero. Each $1 decrease in the market price of the stock will result in $100 worth of loss. The outlook on this stock is bullish.

Short Stock

Being long stock is a bullish strategy, meaning that you believe there will be a rise in the market price. What if the stock is not performing positively; however? What if it is declining in price? Or, what if the investor believes that the stock is highly overvalued and is ready for a significant price pullback? Is the investor simply out of luck? Stock that is in a downward trend (decreasing in value) is referred to as behaving bearishly. Similarly, an investor who has a pessimistic out-look on a stock is referred to as being bearish.

The bearish trader can take advantage of an anticipated declining market by selling a stock short. In other words, he or she will sell a stock that he or she does not currently own. In this case, the brokerage firm lends the investor a certain number of stock shares at a particular price under the condition that the investor has capital in his or her account in order to cover the cost of the stock.

being borrowed. With the stock in hand, the investor now has the ability to capitalize off what he or she is speculating to be a downward move in the stock. The investor sells the borrowed stock in the marketplace at its existing price and waits until the price decreases. Once the stock price declines, the investor buys the stock back in the open market at the lower price. He or she is then able to return the stock to the brokerage firm while capturing the profit. To be sure, by selling stock short, a retail iIlvestor can take advantage of a declining market. There is always a risk that the stock that has been sold short will increase in price, however. This situation could force the investor to purchase the stock back at a higher price, resulting in a loss. If the investor is correct, however, the stock will decrease in price and he or she can buy the stock back from the open market in order to capture profit. 

 Here is an example. An investor who has no position in XYZ borrows 100 shares from his broker and sells it for $50 per share.  the profits and losses associated with the short sale of a security. In this case, the profit is limited to the amount collected for the stock and risk is unlimited. The profit in this situation is limited. This position will profit as the stock declines in value. Each $1 decrease in the market price of the stock will result in $100 worth of profit. The loss in this case is unlimited. This position will lose money as the stock rises in value. Each $1 increase in the market price of the stock will result in $100 worth of loss. There is also the risk of stock being demanded back by the brokerage firm. The outlook on this stock is bearish.

Long Call

The buyer (holder) of a call has as much profit potential as the owner of the underlying stock but has significantly limited the risk of loss. Because of the limited capital used in controlling a large interest, the long call position is a leveraged position. The risk involved is the total amount paid for the call. A long call position is used when the trader is bullish on the underlying security and is an alternative to long stock.

Here is an example. XYZ stock is trading at $50 per share, and the XYZ July 50 call is trading at $2. An investor purchases one XYZ July 50 call for $200. Figure 3-3 shows the profits and losses associated with ownership of a call option. The profit potential to the upside is similar to that of long stock, whereas the risk is limited to the purchase price of the option. The profit in this case is unlimited. When measured upon expiration of the option, each $1 increase in the price of the stock higher than $52 ($50 strike price of the option plus $2 paid for the option) results in $100 worth of profit.

 The profit from a long call will always be less than the profit from the same amount of long stock as represented by the option because of the time premium paid for the option. In this case, the profit from long stock would be $200 higher than the profit derived from the option. The loss in this situation is limited to the amount paid to purchase the option. When measured upon expiration of the option, if the stock is trading at $50 or lower, the option will expire worthless-and the entire. investment in the option will be lost. There will be a partial loss if the stock is trading between $50 and $52 upon expiration of the option. The outlook on this stock is bullish.

Short Call

The seller (writer) of a short call has as much loss potential as the short seller of stock but faces much less potential for gain. The retail investor will need to meet capital requirements in order to transact this position. He or she will also need to have cash in the account in order to cover the short call position or to own the underlying security. The short call is frequently combined with long stock. This strategy is called a covered call or a buy-write and is covered in detail in  "Trading Strategies." Here is an example. XYZ is trading at $50 per share, and the XYZ July 50 call is trading at $2. One XYZ July 50 call is sold for $200. Figure 3-4 illustrates the risks and losses associated with the sale of a call option. The profit potential is limited to the amount collected for the sale of the option and the risk is unlimited. 

The profit in this situation is limited. The total profit to the option seller is the $2, or $2 X 100 = $200. When measured upon expiration of the option, if the stock is trading at $50 or lower, the option will expire worthless and the entire premium received is retained. There will be a lesser gain if the stock is trading between $50 and $52 upon expiration of the option. The loss here is unlimited. When measured upon expiration of the option, each $1 increase in the price of the stock higher than $52 ($50 strike price of the option plus the $2 received for the option) will result in $100 worth of loss. The outlook on this stock is neutral/bearish.

Long Put

A long put position is used when the trader is bearish on the underlying security. The buyer/holder of a put carries the profit potential of a short stock position but has a significantly limited risk of loss. There are no margin requirements for a long put position, and the risk is the total amount purchased for the put. Also, for those who do not trade on margin, selling stock short is not an available option. Therefore, a long put is the exclusive vehicle for speculating on a decline in the price of the underlying stock.

Here is an example. XYZ is trading at $50 per share, and the XYZ July 50 put is trading at $2. One XYZ July 50 put is purchased for $200. The profit and loss associated with the purchase of a put is illustrated in . The profit potential is considered to be unlimited, although the underlying can only go to zero, and the loss is limited to the purchase price of the option. The profit in this instance is limited. The profit potential of the long put is limited only because the value of the underlying stock can only decline to zero. When measured upon expiration of the option, each $1 decrease in the price of the stock lower than $48 ($50 strike price of the option minus the $2 paid for the option) will result in $100 worth of profit.

The profit from a long put will always be less than the profit that would have been earned by selling the same amount of stock short, covered by the option contract due to the time premium paid for the long put. The loss here is limited to the amount paid in order to purchase the option. When measured upon expiration of the option, if the stock is trading at $50 or higher, the option will expire worthless-and the entire investment in the option will be lost. There will be a smaller loss if the stock is trading between $48 and $50 upon expiration of the option. The outlook on this stock is bearish. The buyer (holder) of a put has much of the profit potential of a short stock position but has significantly limited the risk of loss. No margin is required for this position. The risk is the total amount purchased for the put.

Short Put

The seller (writer) of a put has as much of the same loss potential as a long stock position but has much less potential for gain. The retail trader or investor is required to meet capital requirements for this transaction (in other words, he or she will need to have cash in the account as security against the short put). This requirement is referred to as a cash-covered put, which is similar to a covered call (discussed in the strategy section of this book). Cash in the account covers the short put.

Here is an example. XYZ is trading at $50 per share, and the XYZ July 50 put is trading at $2. One XYZ July 50 put is sold for $200. Figure 3-6 illustrates the profits and losses associated with the sale of a put. As with the outright sale of any option, the profit potential is limited to the amount collected for the sale of the option and the risk is unlimited, or in this case limited as the underlying can only go to zero. The profit here is limited to the amount received upon sale of the

When measured upon expiration of the option, if the stock is trading at $50 or higher, the option will expire worthless and the entire premium received will be retained. There will be a lesser gain if the stock is trading between $48 and $50 upon expiration of the option. The loss here is unlimited. When measured upon expiration of the option, each $1 decrease in the price of the stock lower than $48 ($50 strike price of the option minus the $2 received for the option) will result  in $100 worth of loss. There will be a smaller loss if the stock is trading between $48 and $50 upon expiration of the option. The outlook on this stock is neutrall bullish.