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Expected Value Finish

By Craig Russell on
May 7, 2014 at 5:54:37 PM
∙ Comments

Welcome back to the third and last in a series of posts from our good friend Craig Hilsenrath at Option Workbench. In the first two Craig introduced us to the concept of Expected Value and why it is important to options trading and helped us understand the math behind the scenes. In this final post he gives us the rest of the equation so to speak. We here at Tradier believe the more you know the better off you are. Look for more exciting educational posts to come.

**Expected Value and Option Strategies**

Recalling the definition of expected value, the weighted average value of a variable, we

first need to determine which variable we want to model. Since the goal of placing a

trade is to earn a profit we want to determine what the expected profit of the trade is before putting it on. If the expected profit is positive then, on average, we can expect to

make a profit on this trade. If the expected profit is negative then we should NOT do the

trade.

As a simple example consider an at-the-money long $50 strike call position with ninety

days to expiration put on for a $2.50 debit. Since the value of an option is partially

derived from the price of the underlying, the expected profit can be calculated by using

the underlying price as the random variable. The role of volatility, another important

input into the calculation of the value of an option, will be discussed later. Table 1 shows

how the expected profit at expiration can be calculated. The table shows nine possible

values of the underlying stock at expiration, the probability that the stock will be at that

price in ninety days, the value of the call at expiration at the underlying price and the

profit/loss of the position. The final column shows the product of the profit and

probability columns. At the bottom of the Probable Profit column is the weighted

average sum for an expected profit of $1.85.

Analysis of the data in Table 1 reveals some important properties of the expected profit

calculation. Two principal differences between the gaming examples above are evident.

First, in practical terms, “all the possible values” of the variable cannot be known. Since

the price of the stock can take on any value between zero and infinity, a range of prices

will need to be selected that will give the trader enough confidence in the resulting

expected profit. In statistics this range is called a confidence interval. ! !

In any trading endeavor a prediction about the future value of some variable must be

made. An option trader is typically attempting to make a profit over a range of the

underlying price. Predicting this range depends on an expectation of the volatility of the

underlying price over time. Since volatility measures the standard deviation of price

returns, standard deviations can be used to determine the range of prices for the desired confidence interval. Professional risk managers typically use an interval from

-3.0 to +3.0 standard deviations because that range corresponds to a 99.7%

confidence. From a statistical perspective this means that the underlying price should

fall within that range for 99.7% of the observations. Equation 1 shows how to calculate

what the underlying price is for a given number of standard deviations.

Equation 1

where:

P0 = The current price of the underlying.

e = The natural exponential constant.

σ = The volatility for the period.

n = The number of standard deviations.

Assuming an annualized volatility of 25% --14.94% for the 90-day period using the square

root of time rule, σT = σ√(90/252) -- and a currentprice of $50 the price ranges from $31.94 to

$78.28 for the 99.7% confidence interval. The next question that arises is how to pick the

prices in between the end points. UsingEquation 1 Pn can be determined for any number of standard deviations. So the objective is to pick some increment of n that will result in enough prices to obtain an acceptable sampling but not so many as to make the calculation too time consuming. As it turns out this is closely related to how the probability of the underlying price moving to that level is determined.

Determining the probabilities is the second difference between the gaming examples

and the expected profit calculation. Whereas the probabilities for the casino games are

known and constant, the probabilities used to calculate expected profit are dependent

upon an assumption about the price volatility of the underlying asset.

**Determining the Probabilities**

In 1973 Fisher Black and Myron Scholes published a paper defining the now famous

Black-Scholes option pricing model. One of the characteristics of the Black-Scholes

model is that it assumes that the underlying asset’s daily returns follow a log-normal

distribution. The reason for this assumption is so that the model can estimate the

probability that the option will expire in the money and earn a profit for the option buyer.

There are two concepts that need to be understood in order to comprehend how the

probabilities are calculated. The first, the probability density function or PDF, is defined

as a function that describes the relative likelihood for a random variable to take on a

given value. The figure below shows a typical normal probability density function. The xaxis

shows the change of the variable, the underlying price return in our case, in

standard deviations. The Greek letter μ is used to represent the mean or average

change.

In the context of the Black-Scholes pricing model this means that the distribution of the

natural logarithm of daily returns conforms roughly to the normal distribution. What the

PDF tells us is the frequency at which we can expect the log return to change by a

certain number of standard deviations. However, this is not the probability. As indicated

in the diagram, the probability is represented by the area under the PDF. Taking the sum

of the numbers representing the area between -3 and +3 standard deviations results in

the 99.7% confidence interval.

To calculate the probabilities we need to know the area under the PDF curve for the

number of standard deviations represented by the price change. This is accomplished

by using the Cumulative Density Function or CDF. Figure 2 below shows the CDF for

the normal distribution depicted in Figure 1.

It is important to note that a point on the CDF designates the probability of the variable

being less than or equal to the corresponding number of standard deviations. Therefore,

the probability of the price moving zero or less standard deviations is 50%. Likewise, the

probability of the price moving more than zero standard deviations is 50%.

How does this help us to derive the probabilities we need to calculate the expected

profit? Another property of the CDF is that it can be used to calculate the probability of

the variable falling between two points on the curve. This is accomplished by

subtracting the probability at the lower end of the interval from the probability at the

upper end of the interval. In mathematical terms:

Equation 2

F(xL < x ≤ xU) = F(xU) - F(xL)

Referring back to Table 1, the probabilities stated in the second column were actually

determined by using the formula above and a $5 interval. So the 23.82% probability of

the underlying being at $55 is actually the probability of the underlying being between

$50 and $55. For that simple example a $5 interval was sufficient. However for actual

trading a much smaller interval is needed.

Before deciding on what interval is appropriate there is a tradeoff that needs to be

considered. In the example in Table 1 nine values for the underlying price were

evaluated. In each step the probability, the value of the call, the theoretical profit and the probable profit were calculated. The probable profit for each row is the probability times

the profit from the weighted average sum formula above. Computing the probability and

the value of the call are both complex, multi-step calculations. As such, performing

these calculations too many times can consume a great deal of computational resources.

Since a stock price cannot move less than one penny it may be tempting to pick a small

interval around a penny. For example, the interval p - 0.005 < p ≤ p + 0.005 will yield a

very precise probability. In the previous example the range from $31.94 to $78.28 will

result in 4,634 prices to consider. If the volatility assumption is increased from 25% to

50% annualized the number of prices increases to 10,214. So changing the volatility

assumption can make a big difference in the number of calculations that are performed.

When contemplating the wide dispersion of possible underlying prices and the variety of

volatility scenarios, it is easy to see that using the price-based interval can lead to vastly

different consumption of computational resources.

As it turns out the differences in the probabilities using the $0.01 range are so

minuscule that the extra computations are not worth the effort. A better solution that

results in more than adequate results is to divide the range into equal size chunks. The

objective is to choose a chunk size that results in a small enough difference in the

probabilities without incurring undue computational expense.

Given that the endpoints of the price range are defined in terms of standard deviations,

it makes sense to use an increment of standard deviations to divide the range. For

example, using 1/128 of a standard deviation (a nice round computer number), divides

the six standard deviation range into 768 chunks. Table 2 below shows a revision to the

long call example above. A new column showing the number of standard deviations has

been added. As the table shows this long call still has a positive expectation. If a trader

could buy a call like this many times, on average, a profit of $76.34 per contract would

be expected.

**Fat Tail Distributions**

The preceding examples assume that the price returns of the underlying asset follow a

log normal distribution. Very few, if any, equities or instruments of any asset class

actually exhibit this behavior. In the real world price fluctuations are subject to shocks

caused by corporate actions, natural disasters, political turmoil and other exogenous

events. Single day price moves of four or more standard deviations are not uncommon.

This means that the probability of extreme changes is higher than would be predicted by

the normal distribution, leading to the understatement of the probable profit at the

extremes or tails of the distribution.

This fact calls into question the value of using the standard statistical techniques

outlined above. In fact, there are many techniques employed by professional risk

managers that apply adjustments to the normal distribution. These so-called parametric

approaches attempt to account for extreme return fluctuations in a generalized way.

There is also a non-parametric or empirical approach that can be used. This method

uses historical data to construct the probability density and cumulative density curve.

The empirical approach is not generally used by professional risk managers because

they are typically dealing with large portfolios of diversified assets and a more

generalized approach is needed. However, for the purposes of calculating the expected

profit for an option strategy, using empirical data is a reasonable technique.

**The Empirical Probability Density**

In the normal PDF exhibited above the tails of the distribution decline smoothly and

approach zero as the number of standard deviations approach negative and positive

infinity. In a fat tail probability density the extremes of the distribution are elevated to

take into account periodic extreme moves in asset returns. The chart below is an

example of an empirical, fat tail probability density for the S&P-500 index. The daily

return data were collected for 10-year period ending in June of 2013. In the histogram

the eight standard deviation range from -4.0 to +4.0 is divided into 128 buckets. The

height of the bars indicates what percentage of the return data falls into each bucket.

The histogram is superimposed over a normal distribution to illustrate the differences in

the tails. From Equation 1, the number of standard deviations is calculated using

Equation 3.

Equation 3

where:

Pt = The closing price for the current day

Pt-1 = The closing price for the previous day

σ = The standard deviation of the returns.

Each day’s closing price is compared to the previous day’s price to determine the return

in terms of standard deviations. Each bucket encompasses a 0.0625 standard deviation

range, the eight standard deviation total range divided by 128 buckets. To determine in

which bucket a reading belongs, subtract the minimum value from the reading and

divide by the bucket range. Changes greater than or equal to 4 standard deviations are

counted in the highest bin. Changes of more than -4 standard deviations are placed in

the lowest bin.

**The Empirical Cumulative Density Curve**

Using the empirical probability density a cumulative density curve can be constructed.

The following chart shows the cumulative density curve derived from the probability

density above. The red line depicts the empirical cumulative density curve and the blue

line shows the log normal CDF for comparison. One difference between the two that is

immediately apparent is that the log normal CDF is smooth and continuous whereas the

empirical curve neither smooth nor continuous. The $SPX probability density used to

construct the cumulative density curve is missing some bars in some of the upper end

buckets. Even if all the buckets contained data the empirical cumulative density curve

would still not be continuous. Recall that to calculate the expected profit a very small

increment of standard deviations was used. Since the log normal CDF is a continuous

function a value for any number of standard deviations can be computed. This is not the

case for the empirical cumulative density curve. For values that are not points on the

empirical cumulative density curve curve an interpolation method will need to be used to

determine the probabilities.

One of the tradeoffs between using an empirical cumulative density curve versus the log

normal CDF is that the former offers an accurate approximation of “tail risk” while the

latter offers a better approximation of the probabilities. Another point to consider is what

data are used to construct the empirical probability density. If the $SPX probability

density were constructed using only 3 years of daily data the 2008 financial crisis would

not have been included. There are valid reasons for selecting both time frames

depending circumstances. One could also choose to use weekly or monthly returns to

construct the probability density. All of these choices will affect the shape of the

empirical cumulative density curve.

In summary, a fat tail distribution can offer a better estimate of expected profit but there

is not one, correct fat tail distribution. An option trader who would like to use a fat tail

distribution to calculate expected profit should be aware of what assumptions were

made in constructing the empirical probability density.

**Expected Return**

While having a positive expected profit is preferable it is not always sufficient. A trade

with a positive expectation can still result in a loss so attention must be paid to risk.

Expected return, the expected profit divided by the risk, tells an option strategist

whether the expected profit is worth the risk.

In the long call example above the risk is $2.50, the cost of the call. So the expected

return is approximately 30.54%. Suppose now that the cost of the call was actually

$3.75. That would reduce the expected profit to approximately $0.14 for an expected

return of 3.63%. Even though the trade still has a positive expectation, a return of 3.63%

does not seem worth the risk. This conclusion is subjective and has to be weighed

against alternative uses of the trading capital. Perhaps 3.63% is not a bad return given

general market conditions. This type of comparison highlights another use of expected

return.

The expected return of positions with different strikes, expiration dates and even

different underlying assets can be compared using expected return. The final step is to

normalize the return. Suppose a trader wants to compare the long call analyzed above

with a 60 day call for a price of $1.85. The expected profit of the new call position is

approximately $0.39 for an expected return of 21.26%. The question is would the trader

prefer a return of 21.26% over 60 days or 30.54% over 90 days. The reason this needs

to be considered is that the proceeds from the 60 day trade can be reinvested for the

thirty days between the two expiration dates. Equation 4 shows how to calculate the

compound annualized return.

Equation 4

where:

re = The expected return.

t = The number of days over which the expected return is realized.

Using equation 4, the original 90 day trade yields 194.69% on an annualized basis and

the 60 day trade yields 223.01%. This shows that the 60 day call has an edge over the

90 day call and the trader should prefer the new position.

**Conclusion**

Using expected value can help people to predict possible outcomes in many different

areas. Casinos use probability and the expected value of the games that gamblers play

to give the house an edge. In the same way option strategists can use expected return

to establish an edge in their portfolios. By carefully establishing and maintaining

positions with positive expected returns, option traders can use probability and statistics

to enhance portfolio returns, and in a sense, be the house.

Many thanks and stay tuned for more from other guest contributors.

Craig Russell