Almost all stock options have American exercise rules. This simply means that the option can be exercised at any time. There are other options that can only be exercised on the expiration date. These are called European options. Some index options are European. Since American options can be exercised at any time, they tend to have a slightly higher value than their European counterparts.

Stock and index options expire on the 3rd Friday of the month. Futures options have different expiration dates that do not have straightforward rules. Always be sure of your expirations by asking your broker, receiving literature from the exchanges, or visiting the exchange's web sites.

Stock and index options all have multipliers of 100. If you buy a stock option for 4 1/2, you will have to pay $450. Futures options on the other hand, have no straightforward rules. Each contract is different.

Assume you are bullish on Texas Instruments and want to buy a January 110 call. The underlying instrument for this option is Texas Instruments stock. If you are online take a look at the situation on Dec 10, 1999 by clicking on the Texas Instruments link above. This link is an example of "Edit My Option Table" on the Optionetics Platinum web site.

On this date the stock was trading at "f=106 7/8". The Jan00 options expire on "01-21-00" or Jan 21, 2000. The Jan00 options have 42 days remaining before expiration. The Jan 110 call is trading for 6 3/16. Therefore, to get the call option you would have to pay the ask price which is worth: (6.1875)*100 = $619.

The question in your mind at this point should be: is this option worth $619? The purpose of option pricing models is to help determine if the option is worth $619? The model can tell us what any option should be worth in current market conditions.

The Platinum v1.0 pay site function "Create An Option Table" provides the market quote of an option and the model quote of the same option. The model quote is sometimes referred to as the fair value of the option. In the tables the model value of a call is labeled "MCall=", whereas the market value (the quote) is labeled "Call=." For the put options: "Mput=" and "Put=." are used. Mcall is an option value generated by a mathematical options pricing model to determine what an option is really worth. The option model value for TXN in the above table is 6 1/4. This is within a 1/16 of the market quote, so this option is "fairly priced."

The option pricing model used on this website is known as the Bjerksund Stensland American option model. It is a variant of the original Black-Scholes model. If you are interested in the mathematics behind the option model, a good book on the subject is Natenburg, S. Option Volatility and Pricing, 1994, Irwin Professional Publishing. A great book on options without the math is Fontanills, G. trade options online, 1999, John Wiley & Sons. An older version of Optionsanalysis is covered in a few pages in the book by Fontanills. The Bjerksund Stensland American option model is discussed in Haug, E. The Complete Guide to Option Pricing Formulas, 1998, McGraw-Hill.

Volatility is the key to trading options effectively, because volatility is a primary determinant of an option's price. What is volatility? Volatility is a measure of the degree of change in a stock's value stated as a percentage over one-year. Mathematically, volatility is a one standard deviation price change in percent at the end of a one-year period.

Simply stated, a high volatility means that the stock, index or future has a very good chance of moving big. A low volatility stock, index or future has a very good chance of not moving much at all! Let's assume a trader has bought a call option on a stock. The trader wants the stock to go higher as quickly as possible. An indication of the likelihood of the stock going higher (or lower) is the volatility. The higher the volatility the better the chance your stock will jump higher, and hence the better the chance the call option will be worth more in the future. An option usually costs more at the time of purchase if it has a good chance of being worth more in the future.

The figure below shows the highs and lows of Texas Instrument stock plotted versus days.

Each day the stock price varied between the high and low intra-day, then settled on a closing price at the market close. The key assumption in the Bjerksund Stensland American option model is that the difference in price from day to day satisfies a Gaussian distribution.

The next figure shows the stocks closing price differences plotted using a histogram. A histogram is constructed by breaking up the stock price differences into 16 bins and counting how many times a stock price difference landed in a bin. The bin counts are shown as the left axis in the plot. The 16 bins are the bars in the plot. The price differences were small most of the time. Downside and upside movement may occur now and then.

We can compute the standard deviation, std, of the price differences. The mean, m, ( ie, the average value of the price differences) is close to zero. A Gaussian distribution overlaid on top of the histogram can be drawn given the two parameters std and m. The red line in the above figure is the graph of the Gaussian distribution equation given by,

The standard deviation of the stock price difference, expressed as a percentage over one year is called the statistical volatility or SV. The SV of the stock in the above example is 37%.

The volatility value has no directional bias. Volatility expresses no information about the direction the stock will ultimately move. Volatility only tries to characterize the amount of the move in the stock price from day to day. For example, the above stock is trading at 70 and has a volatility of 37%. One year from now we expect the stock to be trading between the prices of 48 and 101. The probability the stock will fall between these two price values is 68%. The price prediction equation is:

More about web site stock price predictions are discussed in Help Home > Chapter 4 > Quick Start.

Statistical volatility is a historical indication of the amount of stock price difference variation that has occurred in the past. You can simply look back at past stock price values and see the variation in price as a function of time. Theoretically, an options value shouldn't "care" about what the past volatility has been. The stock option value is more concerned about stock price variation in the future rather than what has happened in the past. However, statistical volatility is an important number to look at when trading options and highly influences each day's option prices.

The Optionetics Platinum pay site has several rankings based on the statistical volatility (SV.) There is currently no information for SV on the free site. All of the site's stock price predictions into the future use SV, and the price prediction equation shown above.

The calculation of SV on a day to day basis is fairly straightforward. To determine the SV for any given day:

Compute the standard deviation of the stock price differences using closing prices
Loop over number of days
x = log((close[i]/close[i-1]))
sumx = sumx + x
sumx2 = sumx2 + x*x
Compute the SV annualized over 1 year
if(days>1) SV = sqrt((252./(days-1))*( sumx2 - sumx*sumx/days ))

Where:
close is the stock closing price
sqrt = square root
ln = natural log
days = market trading days

  The SV computed in the above equations is an annualized number! Note that when days are set to one year of trading, 253 days, the multiplier reaches 1.0. This means a 7 day SV would include 7 trading days rather than 7 calendar days. The day to day SV can vary widely. The following figure shows daily variations in SV for the same TXN data used in prior plots.

The Optionetics Platinum site offers the user many options about N day SVs, such as a 4-day SV and a 10-day SV. The N means the web site is using that many days to average daily SVs together in order to get an average SV. The N day SV can be calculated by exponentially averaging the daily SVs and weighting the current day the highest:

  Let:
a = 1.0
alpha = 0.92
numer = 0
denom = 0

Loop:
   numer = numer + (a*SV)
   denom = denom + a
   a = a*alpha
End Loop
average SV = numer/denom

Reference:
The Optionvue Informer, vol 9, Issue 56, July/August 1997.

The SV figure above shows the results of a 10 day and 4 day SV average. The higher the number of days used for N, the smoother the N day SV values are from day to day. Shorter values of N generate values of SV that are more representative of current stock price variation.

For any option that has a quote it is possible to determine what the buyers and sellers of the option think stock price difference volatility will be in the future. It's the volatility implicit in the price, or in other words an implied volatility. We noted previously that the volatility was an input parameter to determine the fair value. If you assume the options quote is the fair value, you can SOLVE for the volatility. This is the implied volatility made famous by the Black-Scholes option pricing model! We use the Bjerksund Stensland American option model to get better prices for American options.

Implied volatilities are the precise numbers needed in order to decide if an option is overpriced or underpriced. Just looking at the price of the option is not sufficient. An in the money option can be very expensive, but still be cheap from a volatility standpoint! In a sense the implied volatility is what the option market traders THINK the statistical volatility WILL BE before the expiration of the option! It's a very interesting number to look at and to compare with the statistical volatility.

The Bjerksund Stensland American option model assumes that the probability distribution for the percent change in a stock's price over an instant of time obeys a normal distribution. For example, one day the stock goes up 5%, the next day it's down 1%, and then it's up 10%. These changes in stock price are assumed to be normally distributed. An example using Texas Instruments stock data is shown above in the Volatility help section. A normal distribution is the familiar Bell shaped curve otherwise known as the Gaussian distribution. A normal distribution is characterized by its mean and standard deviation. A unit normal distribution, N'(x), has 0.0 mean and a 1.0 standard deviation and is given by the formula:

The probability a random sample of a stock percent change is less than x standard deviations is given by the cumulative value of N'(x).

The cumulative normal density function is defined to be N(x). It is the area under N'(x) from -infinity to x. The probability is greater than 99% than a stock percent change will fall between +/- 3 standard deviations. The normal distribution's standard deviation of the percent change is the same thing as the stocks volatility. One is mathematical terminology and the other is stock market terminology. Given the volatility V, it implies that the stock's percent change in 1 year will be less than +/- 3*V, 99% of the time. It will be between +/- V, 68.3 % of the time.

  The normal distribution assumption for the percent change in the stock means the stock price itself obeys a log normal distribution. Based on these assumptions the value of a European call option, assuming no dividends, is given by

If the implied volatility is low compared to statistical volatility, and you think that the current level of statistical volatility will indeed be present before expiration, there are option strategies that can be used to profit from this prediction!

Two types of option combinations that may be used to profit from increasing volatility are the buy straddles and backspreads. The buy straddle combo is formed by buying a put and a call with the same expiration date and the same strike price. The strike price chosen is usually close to the current stock price. The stock has to move away from the strike price in either direction before expiration to make a profit. The profit zone often requires a large stock price movement. An example of a buy straddle is shown above. A strangle is similar to a straddle except calls and puts are bought at different strikes that are not near the current stock price.

A backspread is formed by buying a call (long position) and selling a call (short position) that expire at the same time but have different strike prices. You also buy more long positions then short positions so that at the time of purchase you are delta neutral. The trade profits if the stock price moves away from a certain stock price, usually the current stock price at the time of the option purchase before expiration. An example of a backspread is shown above.

Delta is explained as one of the Greeks in the definitions section below. Delta neutral means your option price is initially insensitive to the direction of the stock price movement. You can make money if the stock moves one way or the other when you are long a delta neutral position, or you can make money if the stock does not move in either direction and you are short a delta neutral position.

If the implied volatility is high compared to statistical volatility, and you think the current level of implied volatility is as high as it will be before expiration, there are options strategies for that situation as well. Selling a straddle or strangle or a ratio spread is an attempt to profit from decreasing volatility. You do not want the stock price to move much up or down before expiration.

The ratio spread is formed by buying a call (long position) and selling a call (short position) that expire at the same time but have different strike prices. You also buy more short positions then long positions so that at the time of purchase, you are delta neutral. An example of a ratio spread is shown above.

The current implied volatilities can help you decide the best way to trade an idea. For instance, you are bullish on Texas Instruments. Implied volatilities are high. If you think the volatilities will either stay the same or drop, you do NOT want to buy a call option! Even though it's a bullish strategy, it will be hurt by falling volatility. In other words you may be right about Texas Instruments going higher, but still lose money buying a call option

Another bullish strategy that would benefit from falling volatility would be to SELL a call and buy the stock or index. This position makes money if the stock goes higher. It makes money if the stock stays the same as time passes and if the implied volatilities drop. It will lose money if implied volatilities increase or if the stock goes lower than the strategy breakeven point. You can sell an in the money call and buy the stock or index and be insulated from minor drops in the underlying. If you do this you will sacrifice profit potential in higher moves of the underlying.

Option Greeks tell us what can happen to our option position if price, time or volatility change. For example, a trader buys a call. Will it make money if the price goes up? How much? Will it lose money as time passes? How much? Will it lose money if volatility decreases? How much? The Greeks ANSWER these questions!

They are especially useful when you have an option position composed of many different strikes or months. You can still determine your risk as a function of price, time or volatility! In order to trade complex option positions you should understand the Greeks and what they are saying about your position. You cannot find implied volatility and Greek information in newspapers and data download services. \ You need analysis tools like Optionsanalysis.

Delta is the amount your option will make or lose as the price of the underlying moves from its current price. Go back to our Jan Texas Instruments 110 call by clicking on the above Texas Instruments link. Return here using the back button. In the TXN Jan 110C, the delta is 48. This means that the owner of the call has the equivalent of 48 shares of TXN stocks. If the price of TXN (the underlying) goes up 1 tick, your option will appreciate 0.48 tick. If the stock goes down 1 tick, your option will lose 0.48 tick. The delta is also viewed as the probability the option will expire in the money. In this case the delta indicates that there is a 48% probability that TXN will be above 110 at expiration.

Now look at the TXN Jan 110 PUT. Its delta is -51. This means if you own this put (you have bought the put), you have the equivalent of 51 short TXN stocks. In other words, if the price of TXN goes up 1 tick, you will lose 0.51 ticks. If TXN goes down 1 tick, you will make 0.51 ticks.

One final variation: Suppose we buy the Jan 110 Call and sell the Jan 110 Put. ALL GREEKS ARE ADDITIVE, so the delta of the combined position is (48) - (-51) = 99. We have just created one SYNTHETIC Long option contract in Texas Instruments! If you perform the same calculation on the other Greeks they should all disappear, since a long options contract has only delta. The other Greeks are zero. This synthetic position will give you a credit of 9.1875 - 6.1875 = +3. This is almost the exact intrinsic value of the put: 110 (the strike) - 107( the close). The put can be assigned at any time causing you to have to buy 100 shares of TXN stock at 110. Since the stock was at 107 when you sold the put, you will have an immediate loss of 3 points ($300). The $300 credit offsets the loss. In effect, as the stock falls below 107, it's as if you bought the stock at 107 and you are now losing money from that position. If the price of TXN rises enough to make the call in the money, you could exercise the call at any time before expiration and collect your profits. In all scenarios you are long 100 shares of TXN stock at a value of 107, the current closing price of the stock.

Gamma is the amount delta will change as the price of the stock/index/futures contract moves from its current price. It would be nice and simple if delta never changed but unfortunately it does! Delta changes as a function of underlying price

Click on the TXN link above, which shows the TXN table. Again, referring to the Jan TXN 110 call, the gamma is 2.2. This means if you have bought the call and the price of TXN moves 1.0 unit higher, the delta of the option will increase from 48 to 50.2. Note that gamma also changes as a function of price.

Vega is the amount your option will gain with a 1 percentage point rise in implied volatility. The term "trading volatility" means that you are trading Vega. If your option's Vega is positive, you are bullish on volatility. You expect it to go higher. If your Vega is negative, a decline in volatility will help you. An option position that has all the Greeks zero except for Vega would be a pure volatility trade. It is difficult to combine different options and obtain a pure volatility trade, but a trader can get close to it.

In the TXN 110 Call Vega was 0.144. This number is in terms of the options price. If the volatility increases by 1.0%, the price of the call should increase from 6.187 to 6.331. We sometimes display Vega in dollars gained instead of points gained by multiplying Vega by 100. When Vega is shown in dollars, a $ sign is placed in front of the number.

Theta is the amount your option will gain with the passage of one day. All options that are bought will experience time decay and hence will have negative thetas. In the TXN 110 Call the theta is -0.085. This means that if everything remains the same, the option price will decrease from 6.187 to 6.102 in one day.

Options sellers have a positive theta, since the probability increases that the seller will keep the option premium if things stay the same.

We sometimes display Theta in dollars gained instead of points gained by multiplying Theta by 100. When Theta is shown in dollars, a $ sign is placed in front of the number.

The web site offers users the capability to rank option strategies using different criteria. Many criteria are self explanatory such as (maximum profit/maximum risk), volume, days to expiration etc. Three criteria that need further explanation are Probability of Profit, Odds and Expected Profit. The best way to explain these criteria is with an example.

Given the statistical volatility of a stock SV, discussed above, the stock price distributions into the future can be computed using the equation (See Natenburg, Appendix B, Referenced above)

The left graph shows price projections for 1, 2 and 3 SVs for 60 days into the future from the last known stock closing price. The projections are random and have no upward or downward bias. The apparent bias in the projection is the use of the exponential in the projection equation. The most a stock price can go down to is 0.0, no matter how many SVs are used. The upward price projection is unlimited. The right graph shows the probability distribution of the stock price.

Given a future predicted stock price below the last known close price, the probability shown in the right graph is the likelihood of the stock price being at or below the future predicted stock value.

For a price above the close the probability shown in the graph is the likelihood of the stock price finishing at or above the future predicted stock value. If we predict that the stock price will be at the close price 60 days into the future, the probability is 50% that this will happen. The 50% probability at the predicted close price means that we have no idea which direction the price is going, and the price is equally likely to be above or below today's closing price 60 days from now.

If you were to sum up the area underneath the probability distribution, you would get a sum of 1.0. Projecting the stock price randomly into the future is another one of the assumptions in the Bjerksund Stensland American option pricing model.

The Quick Search option on the Optionetics Platinum site Welcome page uses the terms "goes sideways" or "does not go sideways". These terms mean that stock future predictions are computed as completely random stock price projections from the input dates stock closing price.

Given the stock price future price predictions using SV, we can introduce our knowledge of the option strategy price distribution versus stock price at xx days into the future. Knowing the option strategy type (put or call), the costs of the options, the expiration date and direction (long or short), we can compute the option strategy profit and losses versus stock price. These are known as the Risk Graphs of an option trade. The next figure shows an example, for Texas Instruments stock, using a call option.

The blue line in the right graph is the expiration risk graph of a call option. This risk line is plotted in the risk graph section of the Platinum web site.

The call costs $400 with a strike at 70. If the stock stays at 70 or below we loose the $400 costs. As the stock price rises the option value rises. If we start summing the area underneath the probability curve in the prior graph over all option strategy prices in the above graph that have a positive value (we are making money), the resulting sum is the Probability of Profit. As shown in the figure, the Probability of Profit is 39%. The Probability of Profit is the area under the probability distribution where the predicted stock price is at or above the call option break even price.

All Probabilities of Profit shown in the web site "Risk Graph" strategy tables, are computed using random stock prices projected to the expiration date of the option leg in the trade which expires first. In the next section we describe how you may arrive at the Risk Graph web page with projected option combo probabilities that use deterministic stock price prediction components. The two probabilities will not agree, because they have different underlying assumptions about the future.

Another approach to evaluating option strategies is to integrate the option prices over the projected probabilities. Multiply the probability times the price and sum over all possibilities. When we are done, we have computed the expected wins and the expected losses of the option combo. In the above figure doing the integration yields expected winnings of $360 and expected losses of $215.

If we divide the expected statistical winnings by the statistical losses we arrive at the Odds of the trade, $360/$215 = 1.67 to 1. The Odds indicate we are more likely to win money than to loose money. Odds of 1:1 are even money bets. Casinos that offer slot machines with 98% payback are offering Odds of 49/50 = 0.98:1. In other words slots have slightly less than even money payback. You should strive to trade option combos that show predicted Odds greater than 1:1 when you are making an option trade using random price projections.

If you subtract the expected losses from the expected winnings you get the Expected Profits. For example, the amount of money the trade is expected to make using randomly distributed future price projections and the input SV. In this case the Expected Profit is $145. If the web site shows that the option strategy has a positive Expected Profit, the Odds have to be better than 1:1. You should strive to trade option combos with positive Expected Profits.

Odds and Expected Profits do not yield the same rankings. Expected Profits are influenced by the stock price itself and tend to favor the higher priced stocks. An interesting use of the Expected Profits ranking is that the Expected Profit value serves as a target profit you can use to make the decision to sell the option combo.

The prior section shows Probability of Profit, Odds and Expected Profits for completely random price projections. The Optionetics Platinum web site allows the user to combine deterministic price future projections with random price projections. In Quick Search on the Platinum site Welcome page, choosing "go up", "go up big", "go down" or "go down big" can all use a combination of deterministic and random stock price projections. In Create A Search the user can turn on deterministic price projections and choose the days and amount of SV to project into the future.

The following figure shows Texas Instruments stock projected with deterministic and random components. The stock is first projected deterministically 30 days into the future using a positive 1*SV and then projected randomly another 30 days into the future thereafter. The total stock prediction time of 60 days matches the amount of time used in the example in the prior help section.

The figure shows that TXN stock is projected to go to about 84 in 30 days using 1 SV. The deterministic price projection has a dramatic effect on the Probability of Profit which is now 94%. The random price distributions have moved considerably into the call option profit regime. Almost all random ending prices are profitable. The Expected winnings are large ($1519), and the Expected Losses are small ($14). The Odds are enormous $1519/$14 = 108 to 1. The Expected Profit is $1505.

The integrations are performed assuming the stock future prices have followed your projections and the web site has assumed you are correct. This is the reason behind all your good fortune! If your deterministic stock projection is wrong 30 days from now, all web site predictions are wrong. If you are right or close to right, the web site creates rankings and chooses strategies that will maximize your winnings.

In the rankings that occur after using Quick Search or Create A Search, a column is present which shows the deterministic price you have used in your price projections. If you are not using a deterministic price projection the column will show 0 days projection and the same price as the latest closing price.

As an example, consider using the web site to decide whether to sell a straddle or buy a call. With only random price projections the sell straddle option combo will almost always win in terms of Odds and Expected Profit. However, with some deterministic positive price projections, buying a call becomes the better way to go. You have to make a decision about which way you think the stock price is going into the future. Once you decide, enter these assumptions into the web site Search Engines, and the resulting ranked lists will show you those option combos that benefit from your assumptions.

For the most part we recommend that you use 0 days and no stock price deterministic predictions if you are new to options trading. Most traders do not use deterministic predictions to pick option trades.

Volume is the total number of trades made in the option that day. OI is the Open Interest and is the number of open contracts outstanding. Every person that buys an option must have a corresponding person who sold the option, and that creates an OI of 1.