Volatility Project

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Volatility Project - Group 4

Exploring Types of Volatility and Its Use in Investments

December 14, 2007

Derek Light Chiao Liu

Elizabeth Martin Yuliya Ostrovska

Roger Obourn Tiffany Perkins

Brian Peterson Todd Peterson

This report was completed as a group assignment for Finance 622 with Dr. Arlyn Rubash as part

of the MBA Program at Bradley University.

LEGAL DISCLAIMER*WARNING* - This information should not be construed as adviceor recommendations for any investing or trading activity. Investment and trading activities are

risky by nature, and investors should seek knowledgeable and reliable counsel before engaging

in any type of trading activity.

FoldTable of Contents

IntroductionBlack-Scholes Option Pricing ModelTypes of VolatilityHistorical Volatility - HVImplied Volatility - IVVolatility IndexesVIX - CBOE Volatility IndexVXN - CBOE NASDAQ-100 Volatility Index

VXD - CBOE DJIA Volatility IndexVIX, VXN , VXD Option StrategiesConclusionBibliography
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Introduction

Today, many people understand the basics of investments and have the ability to follow financial

news to stay informed on market moves and stock price changes. Since financial information is

readily available all the time through websites such as Yahoo! Finance and the Wall StreetJournal Online, accessing this news is now easier than ever. Through these sites, anyone can find

the price histories of securities such as bonds, stocks, Treasury securities and various derivatives.

This report provides a look into the practice of using Historical Volatility data and Implied

Volatility data to measure volatility in order to predict performance. We will analyze these

different methodologies for measuring volatility and determine which method more accurately

forecasts actual volatility under certain conditions.

Black-Scholes Option Pricing Model

Accurately valuing options is difficult. The varying price of the underlying stock, as well as the

timing of when the option is exercised, all affect the value. While it is known that these factors

affect the value, the difficult part is determining by how much. "'We know it is worth less than a

share of stock, and more than zero,' says Tim Lucas, former research director at the Financial

Accounting Standards Board,"1

One of the standard finance models for predicting a stock price change can be described with the

Black-Scholes model. In 1973 Fisher Black and Myron Scholes developed what is now coined

the Black-Scholes model, which is a very common derivative-based model used to value optionsand is the foundation of option pricing. An option is a contract, which gives the buyer the right,

but not an obligation, to buy or sell an underlying asset at a specific price on or before a certain

date. An option to buy is known as a call and an option to sell is a put. "Indeed, some bankers

argue that the Black-Scholes theory, which provides an easy way to value options, has had as

much impact on finance as the discovery of DNA has had on medicine."2The Black-Scholes

method is based on the early works of Louis Bachelier and the Geometric Brownian Motion

Model. The graph below illustrates the model of stock price as a function of time:

(1)

$dS/S$=$$dt+$$dz

where

dS/S= the instantaneous rate of return on a stock dt$ = the change in the stock's return over time dt and dz$ = describes the uncertainty, volatility and drift3


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Figure 1: Stock Price as a Function of Time

4Geometric Brownian Motion Model

The model requires five components to effectively value an option:

Price of the underlying stock Strike price Time in years until expiration Risk-free interest rate Standard deviation5

Of the five components of the formula, T or time until maturity is expressed in terms of years

and must be scaled at an annualized basis. For example, a maturity of 9 months is the equivalent

of 9/12 or .75 and is the value used in the formula to accurately account for time when valuing an

option. To understand whether an option is an ideal agreement to enter, both the present value of

the strike price and current price of the underlying asset are required. This supports the notion

that you must know what you have and where you have been in order to understand where you

are going. The same logic applies to the current price and strike price. The strike price, or

exercise price, is the agreed upon price for a call or put at the expiration of the option. The

standard deviation, or volatility component, is a measurement of variability over time. The term

risk-free interest rate is described as the offered interest rate minus non-environmental risk.

Environmental risk must be factored in because businesses operate in an uncertain environment

and are part of an open system.


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Black-Scholes formula:

(2)

c0=S0N(d1)PV(K)N(d1T)

where

(3)

d1=ln(S0PV(K))T+T2

and

C0 = call premium

S0 = current stock price

N() = normal distributionPV= present value

K= strike or exercise price

= standard deviation

The formula provides a lot of information about a stock and helps identify the outcome of any

fluctuations in the five components of the equation. Without solving for the full equation the left

and right-hand portions of the formula provide critical information to an investor, while the full

equation calculates the cost of the tracking portfolio. The left-hand side of the equation yields the

number of shares of stock and the right side represents the number of dollars borrowed at the

risk-free rate.6

The Black-Scholes model is a continuous-time valuation model that allows you to calculate an

infinite number of option prices at any point in time.7In addition to following the Geometric

Brownian Motion Model as noted earlier, the Black-Scholes model assumes that the market is

frictionless, which among other things means:

Continuous trading is possible No transaction costs The purchase of fractional shares is possible Continuous borrowing and lending at the risk-free interest rate is possible Short selling is possible No chance for arbitrage The stock does not pay a dividend8

While 80% of companies use the Black-Scholes method, it is surprising that the number of large

companies using the binomial method for valuing options is as large as 20%. A 2006 study


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by Compliance Weekillustrated that 80% of companies offering employee stock options and

reporting sales over $20 billion use the Black-Scholes method. The other 20% use the binomial

method, which makes sense since the big players are the only ones that can afford the costs

associated with this complicated method. Of the 20% using the binomial method, half of them

actually switched from the Black-Scholes. These companies were Caterpillar, Citigroup, Metlife,United Technologies and Wellpoint.9

What makes the Black-Scholes model the cornerstone of option valuation models is its ability to

figure a number of option prices in a short amount of time. However, one of the limitations of the

Black-Scholes model is that it values the option at only one point in timeat expiration.Therefore, this method is unable to calculate those options that have the option of early

exercise.10With that being said, the Black-Scholes model can be tweaked in an attempt tovalue American options. The Fischer Black Pseudo-American model can value these dividend-

paying options, but is less reliable for puts.11

Types of Volatility

As mentioned in the section above, the Black-Scholes model uses several parameters for

calculating the "fair" price of the option. Volatility (measured as a standard deviation) is the only

criterion that is not easily observed. Volatility represents the amount of uncertainty or risk that

lays in the underlying security.12It is one of the most common measures of risk. It measures

upside, as well as the downside, risk of investing in a particular financial instrument. Higher

volatility means that the price of the underlying security can rapidly change in either direction:

positive or negative.

Volatility can be used as a tool for trading. The greater the volatility, the more money making

opportunities exist. Short-term market players attempt to use volatile markets to make money, as

opposed to buying and holding strategies of the traditional" investors. Nowadays, volatility canbe traded directly, through derivative securities such as options and variance swaps.13

Four types of volatility exist: future, historical, implied, and seasonal.14The majority of traders

are interested in Future Volatility. Since no one can predict the future, most practitioners use

different methods for estimating volatility for the option pricing models by using Historical orImplied Volatility and adjusting for Seasonal Volatility for commodity trades.

Volatility can be estimated in several different ways. The most common measure of volatility is

the standard deviation of a return from the mean estimate. Using historical data, a trader can

calculate a Historical Volatility. Another method for assessing volatility incorporates various

option prices to estimate the stocks standard deviation and is calledImplied Volatility.


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The Black-Scholes model assumes constant volatility throughout the life of the option. While

using the model, an investor should use a forward-looking volatility estimate. Historical

Volatility is determined by looking back in time and using the historical standard deviation for

the future estimate. Implied Volatility is more reflective of the current market conditions and

shows an estimate of future underlying asset volatility that would produce the current marketvalue of the option(s). In the sections that follow, we will examine and compare various

volatilities of Caterpillar and John Deere stocks along with the S&P500 index to see how

different choices of volatility impact option pricing estimates.

Historical Volatility - HV

Historical Volatility (HV) uses the historical data to measure volatility of the financial

instrument. There are numerous ways to calculate Historical Volatility. The most common one,

and one of the more accurate ways is by computing the standard deviation of the logarithmicprice relatives.15

PR will stand for the price relatives (i.e. return for a period of time), then the formulas for the

mean and variance would be:

Mean

(4)

PR=1/Tt=1T(lnPRt)

where:

PR= mean price relativeT= number of annual time periods

PRt= price change in the underlying asset

Variance

(5)

VAR(PR)=1T1t=1T(lnPRtPR)2

where:

VAR(PR) = varianceT= number of annual time periods

PRt= price change in the underlying asset

PR= average price relative


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Standard deviation

(6)

(PR)=1T1t=1T(lnPRtPR)216

Historical Volatility is significantly impacted by two factors: the time period of observations and

the price relative intervals. Depending on the choice of these two parameters, the results can vary

widely. A practitioner has to take them into account as indicators of the range of possible

volatilities. Historical and Future Volatilities are sometimes referred to as Realized Volatility.17

Historical Volatility is evaluated and compared between Caterpillar and Deere. The goal is to

calculate the Historical Volatility for both companies and compare them quantitatively and

graphically. Data for each companies' stock prices was obtained from Google

Finance,http://finance.google.com/financefor the past 5-years from 12/6/02 to 11/30/07.Historical Volatility values are calculated using the equations shown above.

Two sets of volatility estimates were calculated for each of the companies. One estimate for the

entire 5-year span and one estimate for the most recent 1-year span:

Caterpillar Volatility Estimates:

12/6/02 to 11/30/07: (5-year span)

Standard Deviation: 3.79%

Annualized Standard Deviation (Volatility Estimate):27.34%

1/5/07 to 11/30/07: (1-year span)


Annualized Standard Deviation (Volatility Estimate): 27.60%

Deere Volatility Estimates:

12/6/02 to 11/30/07: (5-year span)



1/5/07 to 11/30/07: (1-year span)


http://finance.google.com/financehttp://finance.google.com/financehttp://finance.google.com/financehttp://finance.google.com/finance


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Cannot fetch Flickr photo (id: 2091721660). The photo either does not exist, or is private Figure 2: Graph of Stock Comparisons 5-Years (Normalized)

Caterpillar and Deere

Cannot fetch Flickr photo (id: 2090969383). The photo either does not exist, or is private Figure 3: Graph of Stock Comparisons 1-Year (Normalized)

Caterpillar and Deere

Interpreting the Results:

By comparing the volatility estimations for Caterpillar, one can conclude that the volatility really

did not change between the 5-year span and the 1-year span. On the other hand, Deere's volatility

estimates were higher for the most recent 1-year span compared to the 5-year span. This means

that within the past 1-year, Deere's volatility increased.

By comparing the volatility estimates between Caterpillar and Deere's 5-year span, 27.34% and

27.40% respectively; one can conclude the volatility of both stocks remained fairly consistent.

This does not necessary mean that the stocks behaved similarly, it means that in either case, the

stock's probability of upward or downward movement are similar. On the other hand, by

comparing the 1-year span numbers: Caterpillar, 27.60% and Deere, 31.50%; one can conclude

that Deere had a higher probably of upward or downward movement in stock prices compared to

Caterpillar.

HV20 HV50 HV100 DATE CURIV Days/Percentile Close

STOCKHV2

0

HV5

0

HV10

0DATE

CURI

V

Days/Percentil

eClose

ONNOVEMBER 23, 2007

CAT 23 27 2707/11/2023

29.28 600/85%ile 68.63

DE 52 40 3607/12/2023

38.59 600/94%ile156.63


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S&P500(SPX)

24 24 1907/13/2023

22.86 600/97%ile144.07

ON

DECEMBER5, 2007

CAT 26 26 2607/11/2030

25.97 600/56%ile 71.88

DE 51 41 3707/11/2030

39.39 600/94%ile171.75

S&P500

(SPX) 26 20 21

07/11/203

0 22.70 600/96%ile

148.1

1

Table 1: Table of Volatilities as of November 23, 2007 and December 5, 200718

Where:

HV20: 20-day Historical Volatility - i.e. actual volatility

HV50: 50-day Historical Volatility

HV100: 100-day Historical Volatility

DATE: date of the last OPTION datacur_iv: the Implied Volatility of these options on DATE

Days: the number of days back for which Implied Volatility has been calculated

Percentile: measurement of the cur_iv, as compared to the past Days

Close: latest closing price of the underlying on November 23, 2007

Table 1 shows that Caterpillar's Implied Volatility is close to historical levels. A change from

29.28% to 25.97% in Implied Volatility values shows that it is not uncommon for CAT's IV to

change like this. Calculated by our team, Historical Volatility values are very close to Historical

Volatilities that are provided by Optionstrategist.com. The values of Historical Volatilitieschange depending on the time period, time interval, and exercise price of the options.

Since an observation has been made that Deere had a higher probability of movement compared

to Caterpillar in the past year, we should be able to make a similar observation on the price chart.

(See Figure 4)


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Two assumptions could be made to justify the higher volatility for Deere:

1. Overall, the slope of Deere's trend line is steeper, which yields a higher standarddeviation, therefore a higher volatility estimate.

2. Two significant spikes were seen in two relatively short time periods that are highlightedby the red circles below.

Cannot fetch Flickr photo (id: 2091120823). The photo either does not exist, or is private Figure 4: Graph of Stock Comparisons

1 year stock comparison of Caterpillar and Deere stock prices

An attempt to assess the accuracy of Historical Volatility: (Linear Regression)

Can one predict future stock prices by utilizing different regression techniques?

In order to assess the accuracy of Historical Volatility, a method has been developed to compare

the projection of Caterpillar and Deere's stocks via a regression line to the actual stock prices. A

total of two assessments will be conducted. 1.) Generating a linear regression line from year 1 toyear 4, and compare the year 5 projection to the actual year 5 prices shown in the red box (Figure

5) and 2.) Generating a linear regression line from year 3 to year 4, and compare the year 5

projection to the actual year 5 prices in the red box (Figure 6).

Cannot fetch Flickr photo (id: 2092009174). The photo either does not exist, or is private Figure 5: Graph of Caterpillar stock prices (5 years)

4-year linear regression model

Interpreting the above graph and regression line:

To further explain the above graph, you can see a regression line (blue) was calculated to reflectthe first 4 years of Caterpillar's stock prices (12/6/02 to last week of 06). The linear regression

line was then extended for another year for comparison of year 5 stock prices (shown in red box).

Cannot fetch Flickr photo (id: 2092009178). The photo either does not exist, or is private Figure 6: Graph of Caterpillar stock prices (2 years)

1-year linear regression model

Interpreting the above graph and regression line:

To further explain the above graph, you can see a regression line (blue) was calculated to reflect

the year 4 Caterpillar stock prices (2006). The linear regression line was then extended foranother year for comparison of year 5 stock prices (shown in red box).

Linear Regression Results

To start with, by looking at both linear regression charts above, you can clearly see the

regression chart with the smaller historical data did not predict the future stock prices as well as

the graph with the larger historical data regression line. In order to further analyze the accuracy


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of the regression estimates, the Mean Squared Error (MSE) was calculated for the projected

periods. To calculate the MSE, an investor first needs to take the square of the difference of the

projected price and the actual price over the sample period. Then, take the sum of all the squared

values and find the average.

(7)

MSE=t=1NEt2N

where:

N= the number of samples

Et= difference of the projected line and actual values

The MSE will ultimately indicate the accuracy of the regression model. A higher MSE means

that the prediction is poor, and a low MSE means that the prediction is more accurate.

The MSE for the 5-year sample is 51.86 and the MSE for the 2-year sample is 267.21. Since the

MSE for the 2-year sample is significantly higher than the 5-year sample, one can conclude that

the 5-year sample regression model is much more accurate compared to the 2-year model. This

result does not necessary mean the 5-year sample is truly accurate. It just means in this particular

case, the larger database yielded better regression predictions.

An attempt to assess accuracy of Historical Volatility: (Moving Average Method)

Linear regression models are not the only analysis tool to use in order to predict future stock

prices by using historical prices. Another tool is to use the moving average method. The moving

average method uses an average of recent prices to predict future prices. The below graph shows

how the moving average tool was used for Caterpillar for the last year of stock prices (2007).

Cannot fetch Flickr photo (id: 2092009182). The photo either does not exist, or is private Figure 7: Graph of Caterpillar stock prices (1 year)

1 and 2 month moving average models

To further clarify how the moving average is calculated for the chart above, the projection for the

first period is calculated by averaging the last 4 samples for the one month moving average

method and averaging the last 8 samples for the two month moving average method. Compared

to the linear regression models, you can clearly see the prediction with the moving average

method is much more accurate compared to the linear regression models. The comparison of

accuracy will be done once again by calculating the mean squared error.


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The MSE for the one-month moving average model is 10.92 while the two-month moving

average model is 17.36. Quantitatively, the accuracy for the moving average models are far more

accurate compared to the linear regression methods.

Historical Volatility charts provide useful information by giving a trader an idea about thehistorical levels of the volatility and risk in the underlying security and narrowing the range of

the parameter for the option pricing models. Reviewing the charts that are produced by various

tools online and by our simple model shows that Historical Volatility is not an exact number, but

rather a range that each investor should modify to their specific assumptions about the

fundamentals of the underlying asset and the market direction.

Charting the data is one of the best methods to quickly review this data. "Charts are not

mysterious, they are tools that allow investors to keep track of more opportunities & see quickly

when it's time to change strategies. They do not predict the future, but they are valuable in

determining the probably of success whether deciding to buy, to sell, or hold."19

When working with Historical Volatility measurements, the consideration should be made as to

using more data points vs. using more current data. The balance between the two is essential for

an accurate estimate of Historical Volatility.20

Also, Historical Volatility is used as an indicator of risk in the sense that stocks will require a

higher risk tolerance if the Historical Volatility is high.21In other words, stocks that have been

risky in the past have a greater chance of being risky in the future. Therefore, if an investor has

high risk tolerance, they would like to see a high level of volatility. An investor with a low risktolerance is liable to watch Historic Volatility closely in order to see which stocks have the

lowest rate.

As mentioned above, Historical Volatility works as a parameter in the Black-Scholes model

under the assumption that it is constant in time. However, in reality, volatility of the underlying

asset is constantly changing. Applying the value obtained from historical observations in the

model would not produce accurate results. Most practitioners are more concerned with

the estimatedorfuture volatility that would allow them to more accurately price the option

instead of having a range of Historical Volatilities. This is where Implied Volatility comes in andis used as additional information in the pricing model.

Implied Volatility - IV

According to Wikipedia, [I]mplied volatility of an option contract is the volatility implied bythe market price of the option based on an option pricing model. In other words, it is the


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volatility that, given a particular pricing model, yields a theoretical value for the option equal to

the current market price. The Implied Volatility is sometimes referred to as a measure of therelative value of the option.

It is the value of the volatility that makes the "fair" value of the option equal to its current price.Under the assumption that everybody in the market uses the same theoretical pricing model, such

as Black-Scholes, the discrepancy between the calculated value of the option and the market

value of the option is due to the difference in opinion about the inputs of the model .22

Using the previously discussed Black-Scholes model, the Implied Volatility can be determined

by knowing the following parameters for options with different strike price and maturity:

Current Stock Price Option Strike Price

Option Price Option Time to Maturity The Risk-Free Interest Rate

Once the parameters are substituted into the Black-Scholes model formula, the standard

deviation can be determined, which is the same as the Implied Volatility for an option. For the

majority of the stocks, several options with different expirations trade at the same time. Some

models take into account volatilities produced by each option to create weighted implied

standard deviations. Assumptions used in a model to average the volatilities impact the final

result. Most models give the highest weight to options that are the closest to "at-the-money"

positions. At-the-money options produce the least biased volatility estimates.

Numerous websites and programs provide Implied Volatility data to investors for a service fee or

for free. Optionstrategist.com is one of the sources that provide this information for free.

Looking again at thedata gathered from optionstrategist, the table shows that Implied Volatility

for both Caterpillar and Deere was different than any of the Historical Volatilities, but stayed

close to the range of historical values.

The phenomenon of the "Volatility Smile" is created and used to determine when the strike price

is in-the-money or out-of-the-money. The Volatility Smile occurs when a group of European

options with the same expiration date are graphed with Implied Volatility on the Y-Axis and

Strike Price on the X-Axis as shown below. It suggests that Black-Scholes under prices deep out-

of-the-money options as well as deep in-the-money options.
http://finance622volatility.wikidot.com/volatility-project#Table1http://finance622volatility.wikidot.com/volatility-project#Table1http://finance622volatility.wikidot.com/volatility-project#Table1http://finance622volatility.wikidot.com/volatility-project#Table1


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Figure 8: The volatility smile

It would appear that the lowest possible Implied Volatility levels are options at-the-money. Inpractice, options are frequently sold by quoting the Implied Volatility versus the price.

Investopedia also suggests that, "A volatility smile is used by investors to price options in the

foreign currency market and the equity option market."23Theoretical values of the options

predicted with the Black-Scholes model are not 100% accurate because of the Volatility Smilephenomenon. An investor would use a "neutral, bull, or bear trading strategy" to make

money24taking into account an analysis of the fundamentals of the underlying asset.

Implied Volatility has been found to be an excessively unstable predictor of Realized Volatility.

For example, on the 3-month Eurodollar, Implied Volatility has been falling since 1985. At thesame time, interest rates and inflation have been declining while the credible relationship

between the Fed and foreign markets improve.25

Also, there is a direct correlation between significant events in economic history and Implied

Volatility. The largest changes in Implied Volatility occurred on the same days as the 1987

stock-market crash, the Persian Gulf War fears, and the debt crisis in Russia. The following table

shows the Top 20 changes in Implied Volatility and the economic news event that happened on

that day.


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Table 2: News Events that Coincide with Large Changes in Three-Month Eurodollar IV

26

This leads us to believe that those who work closely with Implied Volatility must be constantly

monitoring the global environment in order to stay ahead. If an investor can catch a big story

related to the economy, they may be able to react accordingly based on what will soon happen

with Implied Volatility.

As options become more and more popular in the market, understanding them has become even

more important. Historical Volatility has become a big focus in the understanding of options.

"The main manifestation of rising volatility is in increased prices for traded options. The prices

of call and put options traded on the London International Financial Futures Exchange (Liffe) are

derived from a complex formula, but one element is the Historical Volatility of the underlying

security."27

For those that often use derivatives in their work or for personal investments, it is essential to

understand Historical Volatility in order to make a profit. In contrast, Historical Volatility is
http://finance622volatility.wikidot.com/local--files/volatility-project/NewsIV.bmp


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often overlooked by long-term investors. They often see it as a nuisance and a waste of time

because it is not the true factor in price trends. However, Nick Louth of Financial Times feels

that long-term investors should not dismiss Historical Volatility. Improved understanding of the

types of volatility and methods of applying it to managing their portfolios should allow investor

make better decisions and improve profitability.

Volatility Indexes

This section looks at implied volatility through the volatility indexes of the S&P 500, NASDAQ,

and Dow Jones Industrial Average (DJIA). This gives us an opportunity to review how volatility

is measured in todays market and how it can impact a stock portfolio. We reviewed the threeindexes for the above markets: CBOE Volatility Index (VIX), the CBOE NASDAQ-100

Volatility Index (VXN), and CBOE DJIA Volatility Index (VXD). Next, we will investigate two

studies that discuss strategies on how to use volatility options to reduce a stocks portfolio riskwithout impacting the portfolios average return.

CBOEChicago Board Options Exchange

The Chicago Board Options Exchange (CBOEreferred to as the "See-bo") was founded in1973 and focuses on the option contracts for individual equities, indexes, and interest rates. It is

the worlds largest options market and is the market leader in developing new functionalproducts and technological innovations, specifically with electronic trading.28The VIX, VXN,

and VXD are products of the CBOE.

As defined, the CBOE "volatility indexes are key measures of market expectations of near-term

volatility conveyed by stock index option prices. The indexes measure the market's expectation

of 30-day volatility implicit in the prices of near-term index options. The indexes are quoted in

percentage points, just like the standard deviation of a rate of return. The indexes are leading

barometers of investor sentiment and market volatility relating to key stock indexes.29

The relationship between the volatility index and the stock index option price can be easily

observed through highly negative correlation, as can be seen in Figure 9. The correlation is the

measurement of the relationship between two variables. A coefficient of +1 means that thevariables move perfectly together. A coefficient of -1 means the variables move in the opposite

direction. The VIX is compared with the S&P 500 Index Options (SPX), the VXN is compared

with DJX - the options based on The Dow Jones Industrial Average (DJIA), and VXD is

compared with NASDAQ-100 Index Options (NDX). In the first half of 2007, the volatility

indexes all had negative correlations to the daily returns of the related stock indexes:

'VIXandSPX-0.86
http://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspx


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VXDandDJX-0.85

VXNandNDX-0.78'30

In addition, the correlations for years 2004 through 2006 have been graphed and show that the

correlation has been relatively consistent between the volatility index and stock index option:

Figure 9: Negative Correlations of the VIX and S&P 500 Indexes31

Figure 9 shows that when the stock index moves one way, the volatility index moves in the

opposite direction. This can be very helpful when attempting to reduce risk within a stockportfolio. This will be discussed in more detail later on.

VIX - CBOE Volatility Index

Figure 10: VIX Index32

The CBOE Volatility Index (VIX) shows the markets expectation of 30-day volatility. It isconstructed using the Implied Volatilities of the S&P 500 index options and is meant to be

forward looking and is calculated from both calls and puts. The VIX is a widely used measureof market risk and is often referred to as the "investor fear gauge" .33
http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/vxn/http://www.cboe.com/micro/vxn/http://www.cboe.com/micro/vxn/http://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/vxn/http://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/vxd/


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The VIX was introduced in 1993 by the CBOE and was a weighted measure of the ImpliedVolatility of eight S&P 100 at-the-money put and call options.34The calculation for the VIXwas based on the Black-Scholes pricing model and, from the beginning, was considered to be

'the world's premier barometer of investor sentiment and market volatility'. It is widely followed

and has been cited in hundreds of news articles in the Wall Street Journal, Barron's, and otherleading financial publications.35

In 2003, a more robust methodology for the VIX was introduced. The fundamental features of

the VIX remain the same and it continues to provide a minute-by-minute snapshot of expectedstock market volatility over the next 30 calendar days.36In the opinion of many, this allows fora more accurate view of investors expectations on future market volatility. VIX values greaterthan 30 are generally associated with a large amount of volatility as a result of investor fear or

uncertainty, while values below 20 generally correspond to less stressful, even complacent, times

in the markets.37

According the CBOE, the most significant change is a new method of calculation. The newVIX estimates expected volatility from the prices of stock index options in a wide range of strike

prices, not just at-the-money strikes as in the original VIX. Also, the new VIX is not calculated

from the Black-Scholes option pricing model; the calculation is independent of any model. The

current VIX uses a newly developed formula to derive expected volatility by averaging the

weighted prices of out-of-the-money puts and calls. This simple and powerful derivation is based

on theoretical results that have spurred the growth of a new market where risk managers and

hedge funds can trade volatility, and market makers can hedge volatility trades with listed

options."38

The formula used to determine the current VIX calculation is:

Where


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39

The following link provides a detailed explanation on how the VIX is calculated:VIX

Calculation

The second significant change in the current calculation of VIX is that it now uses the options of

the S&P 500 instead of just the S&P 100. According to the CBOE, The S&P 500 is the primaryU.S. stock market benchmark and the reference point for the performance of many stock funds.

In addition, the S&P 500 underlies the most active stock index derivatives, and it is the domestic

index tracked by volatility and variance swaps.40

The updated VIX is considered to be more accurate than before and is now measured in the same

way that financial theorists, risk managers, and volatility traders have come to measure it. In

addition, the current VIX is considered more robust because it pools the information fromoption prices over the whole volatility skew, not just from at-the-money options."41

As discussed earlier, the relationship between the CBOE Indexes and the related stock indexes

are negatively correlated. This holds true for the VIX and S&P 500 Index as well.
http://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspx


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Figure 11: Negative Correlation of the Original VIX, the New VIX and the S&P 500

Indexes-4 Months42

The VIX is called the 'investor fear gauge' for a reason. Given the strong negative correlation

with the SPX, there are many points in history where drastic increases with one, correspond with

a significant decline in the other and vice versa. Specifically, this can be seen during both Gulf

Wars. These were times of extremely high volatility in the market where the VIX increased

drastically as the SPX declined just as drastically.


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43Figure 12: Negative Correlation of the Original VIX, the New VIX and the S&P 500

Indexes13-Year Span with Significant Events

VXN - CBOE NASDAQ-100 Volatility Index

Figure 14: VXN Index44

The CBOE NASDAQ-100 Volatility Index (VXN) is "the measure of the implied volatility for

the NASDAQ 100".45The VXN is calculated using the same formula and methodology that is

used to calculate the current VIX. As defined by the CBOE, "It measures the marketsexpectation of 30-day volatility implicit in the prices of near-term NASDAQ-100 options. VXN


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is quoted in percentage points, just like the standard deviation of a rate of return."46Similar to the

VIX, the VXN is the investor fear gauge for the NASDAQ-100 Index (NDX). It gauges the

investor sentiment and market volatility related to the NDX.47

Figure 15: NDX and VXN Indexes48

VXD - CBOE DJIA Volatility Index

Figure 16: VXD Index49

The CBOE DJIA Volatility Index (VXD) was introduced in 2005 and "tracks the volatility of the

Dow Jones Industrial Average (DIJA) by measuring Implied Volatility of the near-term DJX

options". It is designed to reflect investors' consensus view of expected volatility over the next

30 days in the DJIA, so like the other CBOE Indexes, it can be used as "a benchmark of investor

sentiment".50


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Figure 17: CBOE DJIA Volatility Index (VXD)51

Figure 18: DJX and VXD Indexes52

VIX, VXN , VXD Option Strategies


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Options for the VIX, VXN, and VXD are types of non-equity options that use the same specific

CBOE Volatility Index as the underlying asset. This gives individual investors the ability to trade

market volatility in order to try to reduce the risk of their overall portfolio. Trading CBOE

options can be a useful tool for investors wanting to hedge their portfolios against sudden market

declines, as well as to speculate on future moves in volatility.53

In 2007, the Fund Evaluation Group (FEG) completed a new study entitled "Evaluation of

BuyWrite and Volatility Indexes - Using the CBOE DJIA BuyWrite Index (BXD) and the CBOE

DJIA Volatility Index (VXD) for Asset Allocation and Diversification Purposes." The paper

covers a 9-year period from October 1997 to November 2006 and presents one strategy for

reducing risk in a portfolio using VXD options. The study concluded that "a small (10%)

allocation to the CBOE DJIA Volatility Index (VXD) could have reduced the volatility of an all-

stock portfolio by about 26%, without materially affecting returns". Likewise, as the graph below

illustrates, an almost optimal risk-return investment strategy would be an allocation of

approximately 20% to the VXD. This strategy would optimize the overall portfolio return, but at

the lowest possible risk. Another advantage found was how the VXD option reacted to

increasing and declining markets. "This showed that VXD increased more during market

declines (VXD reacted more to stock market declines than to stock market advances), indicating

that VXD has potential as a diversification tool." Likewise, "the inclusion of a small (5%)

allocation to the VXD Index Option boosted risk-adjusted returns for a stock-oriented portfolio,

and lowered the risk-adjusted returns for a fixed-income-oriented portfolio."54
http://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdf


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Figure 19: Efficient Frontier

55

A recent article from Barron's, Modern Portfolio Protection by Lawrence G. McMillan, discusses

a similar strategy regarding the use of the VIX option. McMillan discusses using VIX options as

an insurance policy. He explains, "The most popular form of "macro" protection has been buyingout-of-the-money SPX puts, but a new class of derivatives, based on the CBOE Volatility Index,

or VIX, has grown in popularity." "Buying volatility futures, or call options on volatility,protects against sharp increases in volatility, which typically occur when the stock market drops.

VIX calls are a better hedge for a broad-based equity portfolio than SPX puts, because they

provide dynamic protection." Also, according to McMillan, the cost of using a VIX option is

lower than that of a SPX put. He explains, "Also, owing to the extreme volatility of the VIX, you

need only protect about 10% of the value of a stock portfolio, thereby keeping the overall cost of

this insurance lower than that of protection using SPX puts."56

As we have seen from both studies, the market volatility indexes can be used to not only measure

risk, but also to reduce the overall risk of a stock portfolio. A relatively small percentage of ones'

overall stock portfolio can be invested in options in order to help reduce the overall risk of the

portfolio.

Conclusion


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Volatility represents the amount of uncertainty or risk that lays in the underlying security. It is

one of the most common measures of risk. This report looked at the practice of using Historical

Volatility data and Implied Volatility data to measure volatility in order to predict performance.

By analyzing the different methodologies of measuring volatility, we set out to determine which

method more accurately forecasts the actual volatility and in which conditions it works best.

The Black-Scholes Option Pricing Model is an easy way to value options, and is widely used and

accepted by large companies and major investors. However, it has the limitation of only

providing a value for the option at the expiration of the option (European Style). For an

American option, the Fischer Black Pseudo-American model can be used, but it is better applied

to calls as it is less reliable for puts.

Historical Volatility looks back in time and uses the historical standard deviation for future

estimates. It is significantly impacted by the time period of the observations as well as the price

relative intervals. So, the results can vary widely depending upon the choice of these two

parameters and ultimately upon the knowledge and skill of the investor making these choices.

Implied Volatility is more reflective of the current market conditions than Historical Volatility,

and provides an estimate of the future underlying asset volatility that would produce the current

market value of the option. Again, the knowledge and skill of the investor plays into the

determination of Implied Volatility because discrepancies between the calculated value of the

option and the actual market value of the option are due to the difference in opinions about the

inputs of the model.

Our analysis of the volatility of Caterpillar and Deere showed that Implied Volatility was

different than any of the Historical Volatility values, but did stay close to the range of historical

values. Therefore, for the time period of our study, we can say that the Historical and Implied

Volatilities were about the same.

Other works have found that for the three-month Eurodollar, Implied Volatility has been an

excessively unstable predictor of Realized Volatility. However, those that realize the relationship

between current events and Implied Volatility can arbitrage if they can be the first to learn of

major current events and react accordingly with respect to the Implied Volatility calculatedbefore and after breaking news.

Pricing options in the foreign currency market and the equity option market can be done with the

use of the Volatility Smile. The Volatility Smile is used to determining when the strike price is

in-the-money or out-of-the-money. However, it can under price deep out-of-the-money options


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as well as deep in-the-money options. Therefore, investors should use the appropriate neutral,

bull, or bear trading strategy and analyze the fundamentals of the underlying asset.

The previous strategies are applied generally for individual stocks, bonds, and portfolios.

However, the volatility options can be used to reduce a stock portfolios risk without loweringthe portfolios average return. Since volatility indices are key measures of market expectations ofnear-term volatility conveyed by stock index option prices, the VIX, VXN, and VXD are leading

barometers of investor sentiment and market volatility relating to key stock indexes. The

correlation between the volatility index and the stock index is highly negative. Therefore, when a

stock index moves one way, the volatility index moves in the opposite direction. This gives

investors the ability to trade market volatility in order to try to reduce the risk of their overall

portfolio. Also, VIX, VXN, and VXD options can be useful tools for investors wanting to hedge

their portfolios against sudden market declines, as well as to speculate on future moves in

volatility.

Ultimately, the knowledge, skill, and risk aversion of the investor will factor into the decisions

and trading style that will be undertaken by the investor. While Historical Volatility has been

shown to work well in the long run, Implied Volatility can be utilized better in the short term,

especially when an investor has the latest news and current events to consider. The best approach

seems to be a blend of both Historical and Implied Volatility with a good understanding of the

underlying assets.

Bibliography

Footnotes

1. Gleckman, Howard. "The Best Way of Valuing Options." Business Week Online (30 July2002): N.PAG. Business Source Elite. EBSCO. Cullom-Davis Library, Peoria, IL. 4 December2007..2. Tett, Gillian, The Australian, "The Father of derivatives tests his formula," May 22 2007, Nov25, 2007.http://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=e

host-live3. Unknown Author. "Stock Price Dynamics." OS Financial Trading System. 1999. 12 Dec. 2007.4. Unknown Author. "Stock Price Dynamics." OS Financial Trading System. 1999. 12 Dec. 2007.5. Risk Glossary, Encyclopedia & Resource Locator. "Black-Scholes (1973) Option PricingFormula". Riskglossary.com. 1996. 12 Dec.
http://ezproxy.bradley.edu:2451/login.aspx?direct=true&db=bsh&AN=7390055&site=ehost-livehttp://ezproxy.bradley.edu:2451/login.aspx?direct=true&db=bsh&AN=7390055&site=ehost-livehttp://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=ehost-livehttp://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=ehost-livehttp://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=ehost-livehttp://www.ftsmodules.com/public/texts/optiontutor/chap6.3.htmhttp://www.ftsmodules.com/public/texts/optiontutor/chap6.3.htmhttp://www.ftsmodules.com/public/texts/optiontutor/chap6.3.htmhttp://www.ftsmodules.com/public/texts/optiontutor/chap6.3.htmhttp://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=ehost-livehttp://search.ebscohost.com/login.aspx?direct=true&db=nfh&AN=200705221021580723&site=ehost-livehttp://ezproxy.bradley.edu:2451/login.aspx?direct=true&db=bsh&AN=7390055&site=ehost-livehttp://ezproxy.bradley.edu:2451/login.aspx?direct=true&db=bsh&AN=7390055&site=ehost-live


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2007http://www.riskglossary.com/link/black_scholes_1973.htmNov 23, 20076. Mark Grinblatt and Sheridan Tittman: Markets and Corporate Strategy2nd. McGraw-Hill,2004. 284-2857. Mark Grinblatt and Sheridan Tittman: Markets and Corporate Strategy2nd. McGraw-Hill,

2004. 2788. Wikipedia, free encyclopedia. "Black-Scholes." Wikipedia.com. 1997. 25 Nov.2007http://en.wikipedia.org/wiki/Black-Scholes9. Big Companies Go Black-Scholes RouteCFO.com, May 3, 2006 Wednesday, ACCOUNTING, 377 words, Marie Leone10.http://www.hoadley.net/options/bs.htmNov 23, 200711.http://www.fintools.com/doc/options/optionsBlackScholes_Models.html12. Cai, Lawrence. "Option trading - Akalawoo." Oct 2007. 1 Dec. 2007 .13. Wikipedia, free encyclopedia. "Volatility (finance)." Wikipedia.com. 1997. 23 Nov. 2007

14. Natenberg, Sheldon. Option Volatility & Pricing: Advanced Trading Strategies andTechniques. Irwin Profesional Publishing, 1994. 6915. Kolb, Robert W. Kolb. Futures, Options & Swaps. Third edition. Blackwell business, 2000.39916. Kolb, Robert W. Kolb. Futures, Options & Swaps. Third edition. Blackwell business, 2000.40117. Natenberg, Sheldon. Option Volatility & Pricing: Advanced Trading Strategies andTechniques. Irwin Profesional Publishing, 1994. 7018. The Option Strategist. McMillan Analysis Group. 2007. 26 Nov. 200719. Kahn, Michael N. Technical Analysis, Plain & Simple. FT P Financial Times, 2006. 17020. Kolb, Robert W. Futures, Options, & Swaps. Third edition. 1999. 40121. Investopedia. Historical volatility. 1997. 5 Dec. 2007.22. Natenberg, Sheldon. Option Volatility & Pricing: Advanced Trading Strategies andTechniques. Irwin Professional Publishing, 1994. 7323. Investopedia.com. Volatility Smile. 5 Dec.2007.24. In-The-Money.com, 5 Dec. 2007.25. Neely, Christopher, Review, "Using Implied Volatility to Measure Uncertainty AboutInterest Rates," May/Jun2005, Vol. 87 Issue 3, p407-425, Dec 5, 2007.26. Neely, Christopher, Review, "Using Implied Volatility to Measure Uncertainty AboutInterest Rates," May/Jun2005, Vol. 87 Issue 3, p407-425, Dec 5, 2007.27. Louth, Nick, Financial Times, "Treat volatility as an asset and not a liability PRACTICALINVESTOR. We can expect plenty of volatility in the coming year, says Nick Louth. But don't
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treat it as a nuisance; it can bring in a profit if used the right way" Jan 11, 2003, pg 7, Dec5, 2007.28. Investopedia.http://www.investopedia.com/terms/c/cboe.asp; Dec 8, 2007

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37. Investopedia.http://www.investopedia.com/terms/c/cboe.asp; Dec 8, 200738. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200739. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200740. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200741. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200742. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200743. Chicago Board of Exchange.http://www.cboe.com/micro/vix/vixwhite.pdf; Dec 1, 200744. Chicago Board of Exchange.http://www.cboe.com/default.aspx; Dec 1, 200745. Investopedia.http://www.investopedia.com/terms/c/cboe.asp; Dec 8, 200746. Chicago Board of Exchange.http://www.cboe.com/micro/vxn/; Dec 1, 200747. Chicago Board of Exchange.http://www.cboe.com/micro/vxn/; Dec 1, 200748. Chicago Board of Exchange.http://www.cboe.com/micro/vxn/; Dec 1, 200749. Chicago Board of Exchange.http://www.cboe.com/default.aspx; Dec 1, 200750. Chicago Board of Exchange.http://www.cboe.com/micro/vxd/; Dec 1, 200751. Chicago Board of Exchange.http://www.cboe.com/micro/vxd/; Dec 1, 200752. Chicago Board of Exchange.http://www.cboe.com/micro/vxd/; Dec 1, 200753. Investopedia.http://www.investopedia.com/terms/v/vixoption.asp; Dec 8, 200754. Chicago Board of Exchange.http://www.cboe.com/micro/vxd/; Dec 1, 200755. Chicago Board of Exchange.http://www.cboe.com/micro/vxd/; Dec 1, 200756. McMillan, Lawrence C. "The Striking Price." Barron's Online. 09 July 2007. McMillanAnalysis. 08 Dec. 2007 .

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