The Volatility Smile

Document Sample
The Volatility Smile Powered By Docstoc
					                VILLANOVA UNIVERSITY




   VIX and the Volatility
          Smile
The Impact of Volatility on Stock and Option
                  Prices



              MAT 5900: Quantitative Finance




                      Stephen Perno
                    Kaitlin Cherundolo




                       May 8, 2009
                                               Table of Contents
Table of Contents ................................................................................................................ 2
Table of Figures .................................................................................................................. 3
Introduction ......................................................................................................................... 4
VIX: The Fear Index ........................................................................................................... 4
What is the VIX? ................................................................................................................ 4
History of the VIX .............................................................................................................. 4
Calculating the VIX ............................................................................................................ 5
Interpreting the Numbers .................................................................................................. 11
Reputation as the ―Fear Index‖ ......................................................................................... 12
Excel Graph Data: Correlative and Predictive Fit ........................................................... 12
   Extension to Other Volatility Indexes........................................................................... 16
The Volatility Smile .......................................................................................................... 16
Impact of Time to Expiry on Volatility ............................................................................ 22
Correlation Between % Short and Bond Rating ............................................................... 24
Measuring the Problem ..................................................................................................... 25
   Dollar Error ................................................................................................................... 25
   Minimax Dollar Error ................................................................................................... 25
   Minimax Percentage Error ............................................................................................ 26
   Pricing ........................................................................................................................... 26
   Stock Market Crash of 1987 ......................................................................................... 26
Implied Binomial Trees .................................................................................................... 26
   Binomial Trees According to Black-Scholes ................................................................ 26
   Market Behavior Tree ................................................................................................... 27
   Uses of the Implied Binomial Tree Model ................................................................... 28
Conclusion ........................................................................................................................ 30
Bibliography ..................................................................................................................... 32
Addendum ......................................................................................................................... 33
   Appendix A: VIX vs. S&P 500 Calculations and Graph .............................................. 33
   Appendix B: VIX vs. Dow Calculations and Graph ..................................................... 43
   Appendix C: VIX vs. S&P 500 Prediction Calculations and Graph............................. 50
   Appendix D: Volatility Smile Calculations (Maple) .................................................... 58
   Appendix E: Maple Class Example (Empty Worksheet) ............................................. 70
   Appendix F: Maple Class Example (Exxon Mobil) ..................................................... 71
                                           Table of Figures
Figure 1: At-The-Money Strike Price Table ....................................................................... 7
Figure 2: Mid-Quote Calculation Table .............................................................................. 9
Figure 3: VIX vs. S&P 500—Correlation......................................................................... 12
Figure 4: VIX vs. S&P 500 (SPX) During the LTCM Crisis ........................................... 13
Figure 5: Negative Correlation of VIX with S&P 500 (SPX) .......................................... 13
Figure 6: 2008-2009 VIX vs. S&P 500 ............................................................................ 14
Figure 7: VIX Value vs. S&P 500 Value: Prediction ....................................................... 15
Figure 8: Volatility Smile Graph ...................................................................................... 16
Figure 9: Microsoft Option Data....................................................................................... 17
Figure 10: Microsoft Volatility Smile............................................................................... 17
Figure 11: Citigroup Option Data ..................................................................................... 18
Figure 12: Citigroup Volatility Smile ............................................................................... 18
Figure 13: JP Morgan Option Data ................................................................................... 18
Figure 14: JP Morgan Volatility Smile ............................................................................. 20
Figure 15: General Motors Option Data ........................................................................... 19
Figure 16: General Motors Volatility Smile ..................................................................... 20
Figure 17: Amazon Option Data ....................................................................................... 20
Figure 18: Amazon Volatility Smile ................................................................................. 21
Figure 19: Exxon Mobil Option Data ............................................................................... 21
Figure 20: Exxon Mobil Volatility Smile ......................................................................... 22
Figure 21: JP Morgan Volatility Smile-- Impact of Time to Expiry ................................ 22
Figure 22: VOD Implied Volatility................................................................................... 23
Figure 23: Percent Short/Bond Rating Correlation ........................................................... 24
Figure 24: Black-Scholes Binomial Tree .......................................................................... 26
Figure 25: Implied Binomial Tree Example ..................................................................... 27
Figure 26: Implied Risk-Neutral Stock Price Distribution at 5 Years .............................. 28
Figure 27: Lognormal Probability Distribution ................................................................ 28
Figure 28: (Implied-Lognormal) Probability Distribution ................................................ 29
Figure 29: Implied Local Volatility .................................................................................. 30
Introduction
Volatility has played a major role in shaping the stock market behavior amongst
particular markets like the S&P 500, the Nasdaq, and others. Calculating the implied
volatility used to be done solely using the Black-Scholes method. The VIX, a ticker
symbol for the Chicago Boards of Option Exchange Volatility Index, and the volatility
smile, a characteristic of implied volatility plots in which out-of-the-money options
exhibit higher volatilities than at-the-money options, they both go against the Black
Scholes model to create the formula or method to measure the quantity or estimate the
behavior. This paper will intend on investigating those methodologies, by calculating the
VIX value step by step and studying the factors that affect the implied volatility in the
market today and how to account for that through modeling, and embarking on new
territory on volatility smiles and their relationship to other financial concepts like the
percent short and bond rating of a company.

VIX: The Fear Index
What is the VIX?
VIX is the ticker symbol for the Chicago Board Options Exchange Volatility Index. It is
a popular measure of the implied volatility of S&P 500 index options. It also measures
market expectations of near term volatility (30 days) conveyed by stock index option
prices. More specifically, the VIX can be a measure of the expected movement in the
S&P 500 index over a 30-day period, on an annualized basis. It is speculated that a high
value corresponds to a more volatile market and therefore more costly options. These
more costly options come from investors that charge a higher premium to insure against
someone investing in a big change in the index price to make a maximal profit with
minimal effort.

History of the VIX
The VIX ticker, which was incepted in 1990, used to measure the regular volatility of
market indexes, specifically the S&P 100. It began as a value based on the S&P 100
Index option prices and it extracted implied volatilities from option-pricing models like
Black & Scholes and others.

In 1993, the VIX index was introduced to the general public on a grand scale and
popularized by Professor Robert E. Whaley of Duke University who wrote countless
papers on the VIX and the potential of the VIX as a volatility index.

Not much changed in the VIX until 10 years later, when in 2003, the underlying index
changed from the S&P 100 to the S&P 500. The original VIX that measured the S&P
100 Index option prices became the VXO ticker symbol. Also, in 2003, more robust
methodology to compute the VIX value improved estimates such that the expected
volatility from option prices in a wide range of strike prices could be computed, not just
at at-the-money strikes as in the original VIX.
After 2003, the VIX and its popularity took off. On March 26, 2004, the first-ever
trading in VIX futures on the CBOE Futures Exchange (CFE) took place. Now people
could estimate the value of the VIX and put a money stake in their estimations. VIX
options were then launched in February 2006 where people could bet on the movement of
the VIX.

Some of the statistical data1 on the VIX from its inception in 1990 are shocking. On
10/24/2008, the VIX reached an intraday high of 89.53. This very, very high number
reflects the recent economical turmoil that has pervaded the S&P 500 and other market
indexes. The lowest recorded value of the VIX between 1990 and 2008 occurred on
12/24/1993 when it fell to 9.48. This marked a time when the S&P 500 was more stable.
Between 1990 and October 2008, the average value of the VIX was 19.04.


Calculating the VIX2
The old VIX was calculated using the Black Scholes pricing model. The new VIX uses a
formula, developed from discussions on volatility and variance swaps which became
available in the early 2000s. The formula is used to derive the expected volatility by
averaging the weighted prices of non-zero out-of-the-money puts and calls. This
particular formula is independent of any model and gives more flexibility and generality
to the VIX value. The updated formula for the VIX value is as follows:
                                                                    2
                            2  K                 1F    
                             2i e RT Q( K i )     1
                            2
                                 K                T K
                            T i  i                   0 

where σ = VIX/100 and therefore VIX = 100*σ, and T is the maturity of the options. The
options can either be near-term options or next-term options. The difference between
them is that ―[n]ear-term‖ options must have at least one week to expiration; a
requirement intended to minimize pricing anomalies that might occur close to expiration.
When the near-term options have less than a week to expiration, VIX ―rolls‖ to the
second and third […] contract months.‖3 Going back to the formula, F is the forward
index level from the index prices or the spot (current) price of the index in question. Ki is
the strike price of i-th out-of-the-money option; it is a call if Ki > F and a put if Ki < F.
ΔKi is the interval between the strike prices calculated by halving the difference of the
two strikes surrounding Ki as shown here:
                                               K i 1  K i 1
                                      K i 
                                                      2
K0, in particular, is the first strike below the spot price F. R is the risk-free interest rate
and Q(Ki) is the mid-quote price for each out-of-the-money option with strike Ki, whether
it is for a call or a put.

1
  "VIX." Wikipedia, the free encyclopedia. 26 Feb. 2009. <http://en.wikipedia.org/wiki/VIX>.
2
  ―THE CBOE VOLATILITY INDEX – VIX.‖ CBOE – Home. Chicago Board of Options Exchange. 22
Mar. 2009 <http://www.cboe.com/micro/vix/vixwhite.pdf>.
3
  Ibid.
There are two components to every option (call, put, etc.), the bid and ask prices. To
retrieve a certain overall price for an option, the mid-quote price, and an average of the
bid and ask prices must be done. The mid quote price of an option depends on the
location of the strike with respect to K0. Out-of-the-money puts come from strikes that
are less than K0 while out-of-the-money calls come from strikes that are greater than K0.
The mid-quote price for the puts is the put price, calculated from the bid price and the ask
price and the mid-quote price for the calls is the call price, also calculated from the bid
and ask price. For the strike K0 the mid quote price is the average of the call and put
prices both calculated from the bid and ask prices.

For example, consider a strike K0 of $920. Since K0 is the strike closest to the spot price
F, one must consider the bid and ask prices for both the call option and for the put option.
With that in mind, compute the mid-quote price for the call where the bid price is $35.20
and the ask price is $39.10.


                                              35.20  39.10
                          mid  quotecall                   37.15
                                                    2

Now, compute the mid-quote price for the put option where the bid price is $35.20 and
the ask price $38.10.

                                              35.20  38.10
                          mid  quoteput                    36.65
                                                    2

Finally, to get the mid-quote price for K0, one must compute the average of the mid-quote
call and the mid-quote put prices.

                                              37.15  36.65
                          mid  quoteK0                     36.90
                                                    2

Further, consider any out-of-the-money option, say a put, with a strike price K = $915
where the bid price is $30.80 and the ask price is $36.30. The mid quote price for that
particular strike would only be

                                                30.80  36.30
                         mid  quoteputonly                   33.55
                                                      2

Now that the ingredients and preliminary explanations have been formulated, the
procedure to find the VIX value can begin. The VIX calculation will first need two
options with different maturities, one near-term and the other next-term. Starting with
options that have maturities 9 days and 37 days, assume that the options begin at
8:30AM Chicago time. The time of settlement for these options is assumed to be also
8:30AM Chicago time. In order to have accurate calculations, the time to maturity must
be converted to minutes per year. The new maturity T will then be

               T = {MCurrent Day + MOther days + MSettlement Day} / Minutes in a year
where…
             MCurrent Day = # of minutes from start until midnight of the current day
     MOther Days = # of minutes of days in between the current days and the settlement day
      MSettlement Day = # of minutes from midnight until settlement time on the last day

Given the initial time is 8:30AM, T1 and T2 for the respective options are:

T1 = {(15.5 hrs*60 mins) + (8 days*24 hrs*60 mins) + (8.5 hrs*60 mins)} / (365 days*24
                                     hrs*60 mins)
                   T1 = {930 + 11,520 + 510) / 525,600 = 0.0246575

      T2 = {(15.5 hrs*60 mins) + (36 days*24 hrs*60 mins) + (8.5 hrs*60 mins)} / (365
                                   days*24 hrs*60 mins)
                      T2 = {930 + 51,840 + 510) / 525,600 = 0.1013699

Assume R to be 0.38% for both options in this example. The riskless interest rate R is
retrieved by ―the bond-equivalent yield of the U.S. T-bill maturing closest to the [option]
expiration dates.‖4 So, the VIX calculation may use different riskless interest rates for
near- and next-term options.

The selected options for the calculation of the spot price F, are out-of-the-money calls
and puts centered around an at-the-money strike, K0. The at-the-money strike price can
be found by finding the strike price where the difference between the call and put prices
is the smallest. From the table below, the at-the-money strike price is 920.

Figure 1: At-The-Money Strike Price Table5




4
    ―THE CBOE VOLATILITY INDEX – VIX.‖
5
    Ibid.
  NOTE: The dots in the various blocks means that values have been omitted to save space. Please see
                     http://www.cboe.com/micro/vix/vixwhite.pdf for full tables.

Just to check that the strike price that was chosen was the right choice, one can calculate
the spot prices for both the near- and next-term options. Listed below they are:

                       F = Strike Price + eRT × (Call Price – Put Price)

                 F1 = 920 + e(0.0038 × 0.0246575) × (37.15 – 36.65) = 920.50005
                 F2 = 920 + e(0.0038 × 0.1013699) × (61.55 – 60.55) = 921.00039

K0 should be the strike price that is immediately below the spot prices so 920 is indeed
the strike price K0.

Next, choose out-of-the-money put options with strike prices less than K0. Start with the
put strike directly below K0 and move to consecutively lower strike prices. The
constraints throughout this process include omitting puts with a zero bid price and that
the collection of put options will cease when two puts with consecutive strike prices that
are zero are seen. Once those options are collected, follow the same procedure with the
call options with the same constraints. With the options that were collected, calculate the
mid-quote price as detailed above. For the strike price 920, K0, average the mid-quote
prices for both the call and the put option at strike to get the mid-quote price for 920.

Next, bring in the full formula and begin to compute values for individual terms in the
formula. Take the left hand side of our formula:

                                      2  K i RT         
                                         K 2 e Q( K i ) 
                                          
                                      T i  i             
                                                          

ΔKi, as explained above, is the interval between the strike prices. Middle values of this
interval can be calculated by taking the difference of the strikes surrounding Ki and
dividing it by 2. The upper end strike and the lower end strike intervals can be calculated
simply by subtracting the end strike with the adjacent strike. For example if the listing of
put strikes were the following: 350, 355, 360, 365, 370, and 375, the interval for one of
the middle values, like 365, would be

                                                 370  360 10
                                  K 365 Put                5
                                                     2      2

and for one of the end values like 350, the calculation for ΔK350 Put would be

                                     K 350 Put  355  350  5

The interval is the same for this particular example set but each case is shown so to find
the interval at different locations in the strike distribution.
Knowing about the strike interval one can now begin to calculate values for the left term
of the VIX formula. For the near-term put at strike 400, the left term of the formula
would come out to be the following:




Each contribution from all the out-of-the-money puts and calls is then summed and
multiplied by 2/T1 for near-term options and 2/T2 for next-term options. A table of some
of these particular values and the subsequent end value is shown below.

Figure 2: Mid-Quote List and Left Term Calculation Table6




     NOTE: The dots in the various blocks means that values have been omitted to save space. Please see
                        http://www.cboe.com/micro/vix/vixwhite.pdf for full tables.

So the near-term total contribution of the strikes is 0.4727799 and the next-term total
contribution is 0.3668297.

For the right term of the VIX formula, directly plug in the variables to get the right term
values for both near- and next-term options.



6
    Ibid.
With both the left hand term and the right hand term of the formula calculated, the
variance σ2 for both types of options can finally be calculated.




In order to get the solitary volatility σ, take the square root of the variance. But before
that happens, compute the 30-day weighted average of the near-term and the next-term
options so to get one value to take the square root of. There are some things that need to
be considered when thinking about weighted averages. Firstly, if the near-term maturity
of one option is less than 30 days and the next-term maturity for the other option is
greater than 30 days, then the VIX calculation reflects an interpolation wherein each
individual option weight is less than or equal to 1 and the sum of the weights equal to 1.
Conversely, if both the near-term and the next-term options are greater than 30 days, then
the VIX calculation reflects an extrapolation wherein the sum of the option weights is
still equal to 1, but the near-term option weight is greater than 1, and the next-term option
weight is less than 0.

In this particular problem, the weighted average calculation calls for an interpolation.
The VIX value with the appropriate interpolation term under the radical is shown below




Where:

            NT1 = # of minutes to settlement of the near-term options (12,960)
            NT2 = # of minutes to settlement of the next-term options (53,280)
                   N30 = # of minutes in 30 days (30 × 1,440 = 43,200)
             N365 = # of minutes in a 365-day year (365 ×1,440 = 525,600)

The equation with the values plugged in look like the following:
The specific weights that come into play with the different variances intend on narrowing
the disparity between the two variances. With T2 closer to 30 days at 37 days, its
contribution to the interpolation will be less than the contribution from T1 which is 9
days. Once the interpolated variances are found between the near-term and the next-term
options, one can take the square root of that value and multiply by 100 to get a final VIX
value of 61.22.




Interpreting the Numbers
If the VIX value is 61.22, what exactly does that mean for the investor or speculators?
Quantitatively, a VIX value of 61.22 ―represents an expected annualized change of
[61.22%] over the next 30 days; thus one can infer that the index option markets expect
                                     61.22%               
the S&P 500 to move up or down                   17.67% over the next 30-day period.‖7
                                     12months             
As far as the index option prices, this means that the options were priced with about a
68% chance that the S&P 500‘s 30-day return will be above or below 17.67%.
Qualitatively, it means the depending on the number of the VIX, the more unstable the
S&P 500. Listed below is the subjective distribution of stability given particular values
of the VIX:

VIX ‘anxiety’ levels:8

       5-10 = extremely low anxiety = extreme complacency
       10-15 = very low anxiety = high complacency
       15-20 = low anxiety = moderate complacency
       20-25 = moderate anxiety = low complacency
       25-30 = moderately high anxiety
       30-35 = high anxiety
       35-40 = very high anxiety
       40-45 = extremely high anxiety
       45-50 = near panic
       50-55 = moderate panic
       55-60 = panic
7
  "Volatility smile." Wikipedia, the free encyclopedia. 15 Apr. 2009
<http://en.wikipedia.org/wiki/Volatility_smile>.
8
  Krupansky, Jack. "VIX - CBOE Volatility Index." Finaxyz. 30 Jan. 2006. 5 May 2009
<http://www.finaxyz.com/vix.htm>.
       60-65 = intense panic
       65+ = extreme panic

A good comfortable range to be in is between 18 and 27. That means that the economy is
stable and people‘s feelings about the market are mainly complacent.

Reputation as the “Fear Index”
Computing these values and finding their implications as far the movement of the stock
market in a 30 day time span is both enlightening for some investors but more often than
not, it strikes fear into investors‘ hearts. Investors believe that a higher value of VIX
translates into a greater degree of market uncertainty while a low value is consistent with
greater stability. When investors anticipate a large upside volatility, they are unwilling to
sell upside ―call‖ stock options unless they receive a high premium. The higher premium
is wanted because the writer of the option wants to charge more for an option that
potentially has great growth in a highly changing market. The same situation occurs for
put options but with opposite trends. For a large downside volatility, a writer for a put
option will want to raise his price because of the high risk for drastic movement.


Excel Graph Data: Correlative and Predictive Fit
How exactly does all this interpretive data fit with actual data of stock indexes? A graph
was tabulated showing the performance of the VIX, both old and new, imposed with the
S&P 500 index prices from 1990 through to 2003.

Figure 3: VIX vs. S&P 500—Correlation9




9
  ―THE CBOE VOLATILITY INDEX – VIX.‖ CBOE – Home. Chicago Board of Options Exchange. 22
Mar. 2009 <http://www.cboe.com/micro/vix/vixwhite.pdf>. (NOTE: Graphs were on an older version of
this page and have been eliminated in the update)
In it, multiple dates exhibit a negative correlation between the S&P 500 index price and
the VIX value. Some good examples include the beginning and the end of 1998,
September 11, 2001, and the dot-com crisis of 2002. The end of 1998 signified the crises
of Russian debt and the problems that arose with Long Term Capital Management. It
was during this time, among other times, that there was a high negative correlation
between the VIX values and the S&P 500 index prices. The table below shows the
disparity between the S&P 500 and the VIX between August 3, 1998 and the end of
November, 1998 when the Russian debt and the LTCM crisis was going on.

Figure 4: VIX vs. S&P 500 (SPX) During LTCM and Russian Debt Crisis10




How much does this correlation extend for the 13 year period? In this next plot below,
see that the slope of the VIX values have a negative slope when comparing the daily VIX
changes and the daily index changes.

Figure 5: Negative Correlation of VIX with S&P 500 (SPX)11




10
   ―THE CBOE VOLATILITY INDEX – VIX.‖ (NOTE: Graphs were on an older version of this page and
have been eliminated in the update)
11
   Ibid. (NOTE: Graphs were on an older version of this page and have been eliminated in the update)
The data seem very suggestive to the VIX having a negative correlation with stock prices.
Does the same relationship hold for today‘s stock behavior? Indeed it does. Below is an
Excel plot of the VIX vs. the S&P 500 for the time period of 2008-2009.

Figure 6: 2008-2009 VIX vs. S&P 500

                                                    VIX vs. S&P500 (2008-2009) Correlation

                 0.3




                 0.2




                 0.1
  Daily Change




                                                                                                                   VIX
                   0
                                                                                                                   S&P500
                 2/22/2008   4/12/2008   6/1/2008      7/21/2008   9/9/2008   10/29/2008   12/18/2008   2/6/2009


                 -0.1




                 -0.2




                 -0.3
                                                                   Date
On a number of occasions, specifically the time around October 10 and November 24, the
VIX movement and the S&P 500 movement are opposites of each other.

It has been thoroughly established that the VIX correlates negatively with the index
options, but does it offer predictions for market movement? If it is, then investors would
know when significant changes would be happen so that they can either make profit in
option investments or hedge against bigger risks of loss. However, the evidence that has
been explored does not show any predictive qualities. In fact, a lot of the previous
assessments show that the VIX value change is a response to Index changes rather than a
prediction of the Index changes. The Excel plot below shows this evidence—see the
points below marked with *. The VIX value spikes a day or two after the S&P 500 index
drops.

Figure 7: VIX Value vs. S&P 500 Value: Prediction




So it turns out that this ―fear index,‖ as investors call it, does not have any predictive
qualities. It does correlate well with Index prices but, as far as one can tell, it does not
provide any predictive evidence for market behavior.

Other critiques of the VIX value extend further than the predictive value. VIX measures
index option volatility over a 30-day interval. However, equity options have 2-6 month
maturities. There is no way to weigh that length of maturity into a 30-day VIX value
without having a gross amount of error. Also, volatility is usually high in technology
stocks and low in utility stocks. So the VIX may be too simplistic to estimate for all
types of stocks. Granted the VIX is supposed to measure the volatility in the S&P 500
market index in general, but the influence of the volatility on particular industries can
skew the VIX value.

Extension to Other Volatility Indexes
Not only can this analysis be done for the VIX, there are other volatility indexes that have
come about the in the recent that serve to measure the movement of various markets.
Other volatility indexes include the Nasdaq-100 Volatility Index, the DJIA Volatility
Index, the Russell 2000 Volatility Index, the S&P 500 3-Month Volatility Index, and
volatility indexes on commodities and foreign currencies like crude oil, gold, and
EuroCurrency.


                              The Volatility Smile
The Volatility Smile is the phenomenon where, when the Black-Scholes implied
volatility is calculated using option prices based on the current market, the volatility of
in-the-money options and out-of-the-money options tends to be higher than at-the-money
options. When strike price is plotted against implied volatility, the graph looks something
like:

Figure 8: Volatility Smile Graph12




The Volatility Smile results from the probability of extreme moves. The greater the
difference of the strike price (K) from the initial stock price (S0) the higher the implied
volatility of the option.

Traders use the Black-Scholes pricing method to find the price of a put or call option.
When the options are priced using this method, they are priced with a consistent implied
volatility. That is, for options with the same underlying stock and time to expiry, but
different K, Black-Scholes assumes a constant σ. However, using data from actual stock
prices, it can be found that the volatility of the options is not consistent.


12
  "Volatility smile." Wikipedia, the free encyclopedia. 15 Apr. 2009
<http://en.wikipedia.org/wiki/Volatility_smile>.
Using option data from April 16, 200913, the Black-Scholes function in Maple was used
to calculate the implied volatility of options with varied strike prices for six different
companies: Microsoft, Citigroup, JP Morgan, General Motors, Amazon and Exxon
Mobil. In these representations, it is important to note that plot of the volatility against
the strike price for the company does not always reflect a ‗smile‘ curve like the one in
Figure 8. There are two possible causes for this: 1) the option prices on
finance.yahoo.com are out of date and 2) not all companies trade in a strike price range
wide enough to reflect the entirety of the smile.

Figure 9: Microsoft Option Data14

Microsoft                           16-Apr-09 Strike Price Call Option Price       Volatility
Stock Price                           $19.76         $10.00              $9.80      0.635053
Date of Expiry                      16-Oct-09        $13.00              $7.01      0.536304
Riskless Intrest Rate                  -0.029        $18.00              $3.15       0.43624
                                                     $21.00              $1.83      0.438989
                                                     $24.00              $0.83      0.401399
                                                     $30.00              $0.12      0.363736


Figure 10: Microsoft Volatility Smile




13
     Data from finance.yahoo.com.
14
     Data from finance.yahoo.com
Note that the Microsoft stock price at the time the graph was generated was $19.76. This
curve reflects the left hand side of the smile.

Figure 11: Citigroup Option Data15

Citigroup                   16-Apr-09 Strike Price Call Option Price Volatility
Stock Price                     $4.01         $4.00              $0.98  1.1584
Date of Expiry              18-Sep-09         $5.00              $0.75 1.20201
Riskless Intrest Rate           -0.335        $6.00              $0.58 1.22284
                                              $7.00              $0.50 1.28384
                                              $8.00              $0.38 1.26572
                                              $9.00              $0.32 1.28768
                                             $10.00              $0.23 1.24566
                                             $15.00              $0.13 1.34701

Figure 12: Citigroup Volatility Smile




The price of Citigroup stock at the time the graph was generated was $4.01. Note that this
graph is unique (out of the ones studied) because it reflects the right side of the smile as
opposed to the left. It is believed this happened because of how low Citigroup‘s stock
is—that there are not any options with a strike price below the underlying.

Figure 13: JP Morgan Option Data16




15
     Data from finance.yahoo.com
16
     Ibid.
JP Morgan                    16-Apr-09 Strike Price Call Option Price Volatility
Stock Price                    $33.24          $7.50             $25.90 1.31616
Date of Expiry               18-Sep-09        $17.00             $16.80 0.83719
Riskless Intrest Rate            -0.014       $23.00             $12.40 0.85417
                                              $27.00              $9.90 0.83692
                                              $30.00              $7.60 0.73562
                                              $35.00              $4.89 0.66598
                                              $40.00              $2.99 0.62278
                                              $45.00              $1.66 0.58113
                                              $50.00              $0.80 0.53802
                                              $55.00              $0.62 0.57696
                                              $60.00              $0.33 0.56012
                                              $65.00              $0.19 0.55616

Figure 14: JP Morgan Volatility Smile




JP Morgan has more call options available for trading, resulting in a wider range of strike
prices. Because of the more pronounced smile, it is believed that the option data for JP
Morgan was also more up-to-date when the graph was generated.

Figure 15: General Motors Option Data17




17
     Data from finance.yahoo.com.
General Motors              16-Apr-09 Strike Price Call Option Price Volatility
Stock Price                     $1.94         $1.00              $0.92 1.45429
Date of Expiry              18-Sep-09         $2.00              $0.54 1.54094
Riskless Intrest Rate           -0.755        $3.00              $0.33 1.51958
                                              $4.00              $0.21 1.49524
                                              $5.00              $0.16 1.53495

Figure 16: General Motors Volatility Smile




General Motors has a very low underlying stock price of $1.94 and only five different
strike prices available for trade. However, the implied volatilities of these options are
more consistent than those of other companies. Something that is important to note,
however, is that the volatility is consistently very high—approximately 1.5.

Figure 17: Amazon Option Data18




18
     Data from finance.yahoo.com.
Amazon                        16-Apr-09 Strike Price Call Option Price Volatility
Stock Price                     $77.25         $40.00             $37.35 0.55971
Date of Expiry                16-Oct-09        $45.00             $33.05 0.59459
Riskless Intrest Rate            -0.017        $50.00             $28.80 0.58462
                                               $55.00             $25.50 0.62443
                                               $60.00             $21.45 0.58119
                                               $70.00             $16.32 0.61589
                                               $80.00             $11.40 0.59057
                                               $85.00              $9.15 0.56818
                                               $90.00              $7.15 0.54467
                                               $95.00              $5.90 0.54457
                                              $100.00              $4.40 0.52042
                                              $120.00              $1.49 0.48745

Figure 18: Amazon Volatility Smile




The plot of Amazon‘s implied volatilities exhibits some of the characteristics of the
volatility smiles, but it seems that the option prices for 40 ≤ K ≤ 70 is outdated because
the graph is inconsistent.

Figure 19: Exxon Mobil Option Data19




19
     Exxon Mobil calculation suggested by Richard Hurst. 22 April 2009. Data from finance.yahoo.com.
Exxon Mobil              22-Apr-09 Strike Price Call Option Price Volatility
Stock Price                $64.75         $50.00             $14.85 0.68934
Date of Expiry           16-May-09        $55.00             $10.10 0.56726
Riskless Intrest Rate       -0.077        $60.00              $5.40 0.40197
                                          $65.00              $1.99 0.33733
                                          $70.00              $0.47 0.32083
                                          $75.00              $0.10 0.33593
                                          $80.00              $0.05 0.39957

Figure 20: Exxon Mobil Volatility Smile




The Exxon Mobil volatility smile is an almost perfect example of the theoretical smile
shown in Figure 8. This is most likely attributed to the high volume of trading on Exxon
Mobil options as well as the moderate impact of the economy on the oil industry.

Impact of Time to Expiry on Volatility
In addition to strike price, time to expiry affects the volatility smile. In Figure 21, the red
line represents the volatility of JP Morgan call options that expire on May 15, 2009, the
blue line represents the volatilities of options that expire on June 19, 2009, and the green
line represents the volatilities of the options that expire on September 18, 2009. The
volatility smile with the greatest volatility and the steepest slope is the one that is the
closest to expiry.

Figure 21: JP Morgan Volatility Smile-- Impact of Time to Expiry
In the example of the impact of time to expiry below, as in the first example, the
downward slope of the implied volatilities is greater with a shorter time to expiry. In this
example, the range of strike prices is wide enough to show the full volatility smile, so you
can see that the options with the least time to expiry also have a steeper upwards slope.

Figure 22: VOD Implied Volatility20




20
  Dermen, Emanuel. "Laughter in the Dark-- The Problem of the Volatility Smile." 26 May 2003. 17 Apr.
2009, 4.
Correlation Between % Short and Bond Rating
There appears to be a correlation between the bond rating for a company, and the % short
value, which gives a perception percentage that people believe the company‘s stock price
will drop. Good companies will have good bond ratings and low percent shorts and vise
versa.

Figure 23: Percent Short/Bond Rating Correlation21,22

Company               Ticker Symbol    % Short     Bond Rating Average Default Rate   Definition
                                                                                      Minimum
                                                                                      Investment
Amazon                AMZN                   8.00% Baa2                       0.15%   Grade
Citigroup             C                NA          A3                         0.02%   High Quality
General Motors        GM                    17.10% Ca                        24.73%   Very Poor Quality
JP Morgan             JPM                    1.90% Aa3                        0.01%   Very High Quality
                                                                                      Highest Rating
Microsoft             MSFT                   1.30% Aaa                        0.00%   Available
                                                                                      Highest Rating
Exxon Mobil           XOM                    0.70% Aaa                        0.00%   Available

Its connection to the volatility smile is not exactly clear. The hypothesis was that, the
more shallow the smile, the better the company performed both in % short value and in
its bond rating. It turns out that for the limited number of companies with a somewhat
limited amount of strikes to evaluate per company, it seems as though the company with

21
     Percent Short Information from finance.yahoo.com
22
     Bond Rating information from moodys.com
the steeper smile ends up having a better performance report on the other quantities. The
reason there could be 2 reasons: 1) The bond ratings take a long time to evaluate, so
some long term ratings may either be out of date (though it is a good rating, Exxon
Mobil‘s long term rating is Aaa as of June 6, 1995) and 2) Though the smile may be
steep, the volatility that it is approaching may not be as high as some companies with less
pronounced smiles or odd plots overall.

As far as is known for now, there can be no definitive correlation between the volatility
smile, the % short value, and the bond rating. More data needs to be collected –
hopefully more accurate data – to make a better assessment of this hypothesized
relationship.

Measuring the Problem
Mark Rubenstein proposed two methods of measuring the difference between the Black-
Scholes price and the market price: dollar error and minimax dollar error. Dollar error is
used to calculate minimax dollar error. Rubenstein then used the minimax dollar error to
calculate the effectiveness of Black-Scholes.

Dollar Error23
To calculate dollar error, first select two options with the same underlying stock price, the
same time to expiry, and different strike prices. For a given volatility, calculate the
Black-Scholes price of each option.

                                 Dollar error=|market – BS price|

Record the maximum difference of the dollar errors from either option 1 or option 2.
Repeat this process, altering assumed volatility each time so that the volatilities
eventually range from 0 to ∞ by an interval of your choosing.

Minimax Dollar Error24
Using the set of maximum dollar errors given by calculating dollar error over the range 0
to ∞ for option 1 and option 2, the minimax dollar error (MDE) is the minimum of the set
of maximums. The MDE for the two options will be such that:

                          Implied σ[op1] < σ[MDE] < implied σ[op2]
                                             and
                             dollar error[op1] = dollar error[op2]




23
   Rubenstein, Mark. "Implied Binomial Trees." Haas School of Business, University of California
Berkeley - MBA, MFE, PhD, and Undergraduate Programs. 15 Apr. 2009
24
   Ibid.
Minimax Percentage Error25
The minimax percentage errors are the percentage errors that occur at the volatility that
―equalizes the absolute values of the ratio of the [minimax] dollar error divided by the
corresponding option price.‖26 Symbolically, this means if DE is the dollar error, that at a
particular volatility σ:

                                       DE[op1]      DE[op2]
                                                  
                                       priceop1   priceop2

Pricing27
In order to put a price on the MDE, Rubenstein proposed a scaled version of the MDE in
which:

                       Scaled MDE=(MDE*100)/concurrent underlying

This formula accounts for the difference in the original underlying stock prices. It would
seem that the higher the stock price, the greater the MDE.

Stock Market Crash of 198728
In 1986, the worst case scenario compared calls that were 9% in-the-money with calls
that were 9% out-of-the-money. The percentage error was less than 1% and the scaled
MDE was $0.04. For the market index that was approximately $225, the unscaled error
was $0.10. So, if the Black-Scholes formula was correct, this meant that the average error
was $0.10. As the stock market fell in 1987, the MDE percent error approximately
doubled and by 1992, it had increased from 1% to 6.5%. Because of the drastic increase
in the error after the crash of 1987, it became necessary to find a way to model the
difference between the market prices and the Black-Scholes prices.

Implied Binomial Trees
Binomial Trees According to Black-Scholes
Black-Scholes assumes a uniform volatility of options with the same underlying stock
price, the same time to expiry and different strike prices. The binomial tree that uses the
Black-Scholes pricing model looks like the picture in Figure 21, all of the nodes are of an
equal probability and equally spaced.

Figure 24: Black-Scholes Binomial Tree29



25
   Rubenstein, Mark..
26
   Ibid.
27
   Ibid.
28
   Ibid.
29
   Dermen, Emanuel, and Iraj Kani. "The Volatility Smile and Its Implied Tree." Jan. 1994. 15 Apr. 2009,
     6.
Market Behavior Tree
However, as seen in the volatility smiles from different companies, the volatility of
options with the same underlying and time to expiry but different strike prices is
different. Thus, multiple sources have proposed an implied binomial tree that is unique
according to the market prices. The Black-Scholes tree is now skewed toward lower
index prices. As puts travels more and more out-of-the-money, the implied volatility
increases, hence augmenting the change of the index price at this range of the index.
That is why the spacing in the implied tree is skewed to lower index prices. There is a
larger chance that prices will drop at a larger rate which corresponds to market behavior.
Implied trees can also be used to ―calculate both the distribution and volatility of the
index at future times and market levels, as implied by option prices.‖30

Figure 25: Implied Binomial Tree Example31




30
  Dermen, Emanuel, and Iraj Kani..
31
  Dermen, Emanuel. "Laughter in the Dark-- The Problem of the Volatility Smile." 26 May 2003. 17 Apr.
2009, 3.
Uses of the Implied Binomial Tree Model32
Once you have the model to construct an implied tree, of which takes an innumerable
amount of time to comprehend and reproduce, you can look at distributions of future
stock prices. All of the following distributions are from an implied 5-yr tree with 500
levels to the following smile:
             For all T, at-the-money (K=100) implied σ=10%
             σ increases/decreases by 1% point for every corresponding 10% drop/rise
                in the strike price
             Assume r=3% compounded continuously

Figure 26: Implied Risk-Neutral Stock Price Distribution at 5 Years33




The mean stock price for the above distribution is $116.18 with a standard deviation of
21.80%. This distribution reflects the implied tree distribution where the implied
volatility is skewed toward the lower index prices like those less than $100.

Figure 27: Lognormal Probability Distribution34




32
   Dermen, Emanuel, and Iraj Kani. "The Volatility Smile and Its Implied Tree." Jan. 1994. 15 Apr. 2009,
14.
33
   Ibid.
34
   Dermen, Emanuel, and Iraj Kani. "The Volatility Smile and Its Implied Tree." Jan. 1994. 15 Apr. 2009,
14.
This distribution has the same mean stock price and standard deviation as the implied
probability distribution, but this particular plot reflects what should be happening in the
Black Schole model, where the random walk of index prices takes place.

Figure 28: (Implied-Lognormal) Probability Distribution35




35
  Dermen, Emanuel, and Iraj Kani. "The Volatility Smile and Its Implied Tree." Jan. 1994. 15 Apr. 2009,
14.
This is the difference between the two previous distributions. The most important set of
data to consider is the first set of positive bars show how the implied tree skews its
volatility values to lower index prices.

Figure 29: Implied Local Volatility36




The three-dimensional image above plots the index vs. volatility vs. time. In the Black
Scholes pricing model, this would be a flat plane. Using the implied market behavior tree,
it is a surface where the implied volatility is higher at a lower index and at a shorter time
to expiry. Not only can these distributions be plotted for smiles that are independent of
expiration time, but one can compute distributions where expiration time is significant,
such as European options as well as more exotic options at different expiration times.

                                  Conclusion
The VIX, its history, a walkthrough of a detailed example of how to calculate the value of
the VIX, and the interpretation of the data gathered over the years on the VIX has been
presented here. The VIX is certainly negatively correlated to the S&P 500 market index
but it does not serve to predict the movement of the market. It must be remembered that
the VIX value was intended to measure the implied volatility of the S&P 500 index. Any
subjective speculations about its affect on the market or how it reacts to the markets
cannot be fully confirmed with this current analysis. Also, as more is learned about the
volatility smile phenomenon, scientists and economists continually work with modeling
the volatility smile and to effectively price options that accurately reflect market
behavior. The two main areas of study are the implied binomial tree, as shown above, and

36
     Ibid.
stochastic models using differential equations. All in all, the VIX and the volatility smile
are both ways for economists to study the behavior of the market and discover how they
can potentially predict, if at all possible, the movement of the market. Only time will tell
the wondrous discoveries that will be made about the seemingly random movements of
volatility and the stock market.
                                  Bibliography
Bringo, Damiano. "Volatility-Smile Modeling with Density-Mixture Stochastic
       Differential Equations." 18 Dec. 2002. 15 Apr. 2009.

Dermen, Emanuel, and Iraj Kani. "The Volatility Smile and Its Implied Tree." Jan.
      1994. 15 Apr. 2009 <http://www.ederman.com/new/docs/gs
      volatility_smile.pdf>.

Dermen, Emanuel. "Laughter in the Dark-- The Problem of the Volatility Smile." 26
      May 2003. 17 Apr. 2009.

Krupansky, Jack. "VIX - CBOE Volatility Index." Finaxyz. 30 Jan. 2006. 5 May 2009
      <http://www.finaxyz.com/vix.htm>.

Moodys.com. 21 Apr. 2009 <http://www.moodys.com>.

Poon, Ser-Huang. "Volatility Smile and Skew." 28 Sept. 2008. 17 Apr. 2009.
       Rubenstein, Mark. "Implied Binomial Trees." Haas School of Business,
       University of California Berkeley - MBA, MFE, PhD, and Undergraduate
       Programs. 15 Apr. 2009
       <http://www.haas.berkeley.edu/groups/finance/WP/rpf232.pdf>.

―THE CBOE VOLATILITY INDEX – VIX.‖ CBOE – Home. Chicago Board of
      Options Exchange. 22 Mar. 2009
      <http://www.cboe.com/micro/vix/vixwhite.pdf>.

"VIX." Wikipedia, the free encyclopedia. 22 Mar. 2009
       <http://en.wikipedia.org/wiki/VIX>.

"Volatility Skew." Optionistics. 15 Apr. 2009
       <http://www.optionistics.com/i/volatility_skew>.

"Volatility smile." Wikipedia, the free encyclopedia. 15 Apr. 2009
       <http://en.wikipedia.org/wiki/Volatility_smile>.
                                 Addendum
Appendix A: VIX vs. S&P 500 Calculations and Graph
             Adj                                  Adj
Date         Close   VIX             Date         Close     S&P500
                             -                                        -
 2/26/2009     44.66 0.0002239        2/26/2009    752.83   0.015905668
                             -                                        -
 2/25/2009     44.67 0.0181904        2/25/2009     764.9   0.010715038
                             -
 2/24/2009     45.49 0.1456038        2/24/2009    773.14   0.039320053
                                                                      -
 2/23/2009     52.62 0.0651722        2/23/2009    743.33   0.035315356
                                                                      -
 2/20/2009      49.3 0.0460758        2/20/2009    770.05   0.011478573
                             -                                        -
 2/19/2009     47.08 0.0288904        2/19/2009    778.94   0.012096922
                             -                                        -
 2/18/2009     48.46 0.0041186        2/18/2009    788.42   0.000950817
                                                                      -
 2/17/2009     48.66 0.1252865        2/17/2009    789.17   0.046629446
                                                                      -
 2/13/2009     42.93 0.0399198        2/13/2009    826.84   0.010048038
 2/12/2009     41.25 -0.076512        2/12/2009    835.19   0.001737641
                             -
 2/11/2009     44.53 0.0469384        2/11/2009    833.74   0.007923456
 2/10/2009     46.67 0.0671274        2/10/2009    827.16   -0.05036862
  2/9/2009     43.64 0.0062062         2/9/2009    869.89   0.001484047
                             -
  2/6/2009     43.37 0.0082664         2/6/2009     868.6   0.026540681
                             -
  2/5/2009     43.73 0.0027404         2/5/2009    845.85   0.016233194
                                                                      -
  2/4/2009     43.85 0.0181802         2/4/2009    832.23   0.007517662
                             -
  2/3/2009     43.06 0.0555573         2/3/2009    838.51    0.01570993
                                                                      -
  2/2/2009     45.52 0.0150512         2/2/2009    825.44   0.000532907
 1/30/2009     44.84 0.0505424        1/30/2009    825.88   -0.02305281
                                                                      -
 1/29/2009     42.63 0.0722151        1/29/2009    845.14   0.033681051
                             -
 1/28/2009     39.66 0.0632612        1/28/2009    874.09   0.033006834
                             -
 1/27/2009     42.25 0.0782751        1/27/2009    845.71   0.010866312
 1/26/2009     45.69         -        1/26/2009    836.57   0.005537856
                     0.0339964
 1/23/2009   47.27   -0.000423    1/23/2009   831.95   0.005363236
                                                                 -
 1/22/2009   47.29 0.0185685      1/22/2009    827.5   0.015278458
                           -
 1/21/2009   46.42 0.1991616      1/21/2009   840.24   0.042572033
                                                                 -
 1/20/2009   56.65 0.2058621      1/20/2009   805.22   0.054261984
                           -
 1/16/2009   46.11 0.1007958      1/16/2009   850.12   0.007533126
 1/15/2009      51 0.0371523      1/15/2009   843.74   0.001328305
                                                                 -
 1/14/2009   49.14 0.1272138      1/14/2009   842.62   0.034032484
                           -
 1/13/2009   43.27 0.0576975      1/13/2009   871.79   0.001756552
                                                                 -
 1/12/2009   45.84 0.0681518      1/12/2009   870.26   0.022822626
                                                                 -
  1/9/2009   42.82 0.0060904       1/9/2009   890.35   0.021533208
                           -
  1/8/2009   42.56 0.0193142       1/8/2009   909.73   0.003391364
                                                                 -
  1/7/2009   43.39 0.1180135       1/7/2009   906.65   0.030469134
                           -
  1/6/2009   38.56 0.0133954       1/6/2009    934.7   0.007786738
                           -                                     -
  1/5/2009   39.08 0.0028108       1/5/2009   927.45   0.004679315
                           -
  1/2/2009   39.19 0.0204578       1/2/2009    931.8   0.031118829
                           -
12/31/2008      40 0.0399416     12/31/2008   903.25   0.014059065
                           -
12/30/2008   41.63 0.0530933     12/30/2008   890.64   0.024113983
                                                                 -
12/29/2008    43.9 0.0119158     12/29/2008   869.42   0.003880112
                           -
12/26/2008   43.38 0.0189525     12/26/2008    872.8   0.005341924
                           -
12/24/2008   44.21 0.0181558     12/24/2008   868.15   0.005764437
                                                                 -
12/23/2008   45.02 0.0102702     12/23/2008   863.16   0.009764948
                           -                                     -
12/22/2008   44.56 0.0082691     12/22/2008   871.63   0.018471577
                           -
12/19/2008   44.93 0.0522499     12/19/2008   887.88    0.00293262
                           -                                     -
12/18/2008   47.34 0.0514623     12/18/2008   885.28   0.021389875
                                                               -
12/17/2008   49.84 -0.049516   12/17/2008   904.42   0.009639159
                           -
12/16/2008   52.37 0.0804979   12/16/2008   913.18   0.050084832
                                                               -
12/15/2008   56.76  0.044676   12/15/2008   868.57   0.012766861
                           -
12/12/2008   54.28 0.0272595   12/12/2008   879.73   0.007003884
                                                               -
12/11/2008   55.78 0.0008968   12/11/2008   873.59   0.028938804
                           -
12/10/2008   55.73 0.0554923   12/10/2008   899.24   0.011823999
                                                               -
 12/9/2008   58.91 0.0071551    12/9/2008   888.67   0.023388912
                           -
 12/8/2008   58.49 0.0243214    12/8/2008    909.7   0.037668878
 12/5/2008   59.93 -0.060065    12/5/2008   876.07   0.035849048
                                                               -
 12/4/2008   63.64 0.0469691    12/4/2008   845.22   0.029746476
                           -
 12/3/2008   60.72 0.0365441    12/3/2008   870.74   0.025508056
                           -
 12/2/2008   62.98 0.0841625    12/2/2008   848.81   0.039163694
                                                               -
 12/1/2008   68.51 0.2145685    12/1/2008   816.21   0.093536559
11/28/2008   55.28 0.0065336   11/28/2008   896.24   0.009596917
                           -
11/26/2008   54.92 0.1033556   11/26/2008   887.68   0.034718427
11/25/2008    60.9 -0.060528   11/25/2008   857.39   0.006529394
                           -
11/24/2008    64.7 0.1161674   11/24/2008   851.81    0.06271427
                           -
11/21/2008   72.67 0.1067906   11/21/2008   800.03   0.061327968
                                                               -
11/20/2008   80.86 0.0851468   11/20/2008   752.44   0.069481827
                                                               -
11/19/2008   74.26 0.0933729   11/19/2008   806.58   0.063105523
                           -
11/18/2008   67.64 0.0220785   11/18/2008   859.12   0.009790296
                                                               -
11/17/2008   69.15 0.0419373   11/17/2008   850.75   0.026149375
11/14/2008   66.31 0.1028335   11/14/2008   873.29   -0.04259349
                           -
11/13/2008   59.83 0.1050931   11/13/2008   911.29   0.066922601
                                                               -
11/12/2008   66.46 0.0785392   11/12/2008    852.3   0.053288838
11/11/2008   61.44 0.0240499   11/11/2008   898.95   -0.02228719
                                                                -
11/10/2008   59.98 0.0668754   11/10/2008    919.21   0.012733931
                           -
 11/7/2008    56.1 0.1267347    11/7/2008    930.99   0.028446198
                                                                -
 11/6/2008   63.68 0.1545695    11/6/2008    904.88   0.051571193
                                                                -
 11/5/2008   54.56 0.1337409    11/5/2008    952.77   0.054115279
                           -
 11/4/2008   47.73 0.1174804    11/4/2008   1005.75   0.040014466
                           -                                    -
 11/3/2008   53.68 0.1094691    11/3/2008     966.3   0.002532236
                           -
10/31/2008   59.89 0.0490366   10/31/2008    968.75   0.015248574
                           -
10/30/2008    62.9 0.1063775   10/30/2008    954.09   0.025476651
                                                                -
10/29/2008   69.96 0.0438282   10/29/2008    930.09   0.011140926
                           -
10/28/2008   66.96 0.1786809   10/28/2008    940.51   0.102457328
                                                                -
10/27/2008   80.06 0.0116843   10/27/2008    848.92   0.032279747
                                                                -
10/24/2008   79.13 0.1545299   10/24/2008    876.77   0.035120816
                           -
10/23/2008    67.8 0.0269205   10/23/2008    908.11   0.012554947
                                                                -
10/22/2008   69.65 0.2711175   10/22/2008    896.78   0.062953125
                                                                -
10/21/2008   53.11 0.0026395   10/21/2008    955.05   0.031283955
                           -
10/20/2008   52.97 0.2834727   10/20/2008     985.4    0.04658284
                                                                -
10/17/2008   70.33 0.0394425   10/17/2008    940.55   0.006232201
                           -
10/16/2008   67.61 0.0239672   10/16/2008    946.43    0.04162886
                                                                -
10/15/2008   69.25 0.2280291   10/15/2008    907.84   0.094695145
                                                                -
10/14/2008   55.13 0.0025427   10/14/2008    998.01   0.005336384
                           -
10/13/2008   54.99 0.2406294   10/13/2008   1003.35   0.109571959
                                                                -
10/10/2008   69.95 0.0901484   10/10/2008    899.22   0.011828963
                                                                -
 10/9/2008   63.92 0.1053258    10/9/2008    909.92   0.079224042
                                                                -
 10/8/2008   57.53 0.0692661    10/8/2008    984.94   0.011397429
                                                                -
10/7/2008   53.68 0.0308357     10/7/2008    996.23   0.059107758
                                                                -
10/6/2008   52.05  0.142436     10/6/2008   1056.89   0.039279301
                          -                                     -
10/3/2008   45.14 0.0026549     10/3/2008   1099.23   0.013598522
                                                                -
10/2/2008   45.26 0.1283055     10/2/2008   1114.28   0.041124925
                                                                -
10/1/2008   39.81 0.0106062     10/1/2008   1161.06   0.003164505
                          -
9/30/2008   39.39 0.1706604     9/30/2008   1164.74   0.051368308
                                                                -
9/29/2008   46.72 0.2962806     9/29/2008   1106.42   0.092189616
9/26/2008   34.74 0.0568537     9/26/2008   1213.27    0.00337675
                          -
9/25/2008   32.82 0.0697239     9/25/2008   1209.18   0.019465761
                          -                                     -
9/24/2008   35.19 0.0149488     9/24/2008   1185.87   0.001979707
                                                                -
9/23/2008   35.72 0.0537718     9/23/2008   1188.22   0.015756115
                                                                -
9/22/2008   33.85  0.054018     9/22/2008   1207.09   0.038986811
                          -
9/19/2008   32.07 0.0316123     9/19/2008   1255.08   0.039467421
                          -
9/18/2008    33.1 0.0900782     9/18/2008   1206.51   0.042428811
                                                                -
9/17/2008   36.22 0.1784637     9/17/2008   1156.39   0.048288065
9/16/2008    30.3 -0.045169     9/16/2008    1213.6   0.017371504
                                                                -
9/15/2008    31.7 0.2113833     9/15/2008    1192.7   0.048282983
9/12/2008   25.66 0.0507601     9/12/2008    1251.7   0.002119365
                          -
9/11/2008   24.39 0.0053159     9/11/2008   1249.05    0.01371193
                          -
9/10/2008   24.52 0.0380122     9/10/2008   1232.04   0.006130568
                                                                -
 9/9/2008   25.47  0.117783      9/9/2008   1224.51   0.034734463
                          -
 9/8/2008   22.64 0.0183813      9/8/2008   1267.79   0.020302677
                          -
 9/5/2008   23.06 0.0412035      9/5/2008   1242.31   0.004420895
                                                                -
 9/4/2008   24.03 0.1145112      9/4/2008   1236.83   0.030378838
                                                                -
 9/3/2008   21.43   -0.025796    9/3/2008   1274.98   0.002037171
                                                               -
 9/2/2008   21.99 0.0628725     9/2/2008   1277.58   0.004100911
8/29/2008   20.65 0.0608971    8/29/2008   1282.83   -0.01381863
                          -
8/28/2008   19.43 0.0168414    8/28/2008   1300.68   0.014731092
                          -
8/27/2008   19.76 0.0362773    8/27/2008   1281.66   0.007950942
                          -
8/26/2008   20.49 0.0231559    8/26/2008   1271.51    0.00367956
                                                               -
8/25/2008   20.97 0.1087042    8/25/2008   1266.84   0.019820581
                          -
8/22/2008   18.81 0.0523029    8/22/2008    1292.2   0.011268953
                          -
8/21/2008   19.82 0.0298233    8/21/2008   1277.72    0.00249191
                          -
8/20/2008   20.42 0.0412529    8/20/2008   1274.54    0.00617813
8/19/2008   21.28 0.0141981    8/19/2008   1266.69   -0.00935853
8/18/2008   20.98 0.069061     8/18/2008    1278.6   -0.01521296
                          -
8/15/2008   19.58 0.0380808    8/15/2008    1298.2   0.004067729
                          -
8/14/2008   20.34 0.0577864    8/14/2008   1292.93   0.005506536
                                                               -
8/13/2008   21.55 0.0177907    8/13/2008   1285.83   0.002919914
                                                               -
8/12/2008   21.17 0.0508707    8/12/2008   1289.59   0.012123883
                          -
8/11/2008   20.12 0.0264851    8/11/2008   1305.32    0.00691874
                          -
 8/8/2008   20.66 0.0234404     8/8/2008   1296.32   0.023611867
 8/7/2008   21.15 0.0444733     8/7/2008   1266.07     -0.0180965
                          -
 8/6/2008   20.23 0.0440003     8/6/2008   1289.19   0.003348785
                          -
 8/5/2008   21.14 0.1054078     8/5/2008   1284.88   0.028314091
                                                                -
 8/4/2008   23.49 0.0399532     8/4/2008   1249.01   0.009006485
                          -                                     -
 8/1/2008   22.57 0.0162605     8/1/2008   1260.31   0.005594055
7/31/2008   22.94 0.0784093    7/31/2008   1267.38     -0.0132309
                          -
7/30/2008   21.21 0.0379324    7/30/2008   1284.26   0.016534493
                          -
7/29/2008   22.03 0.0951864    7/29/2008    1263.2   0.023087465
                                                               -
7/28/2008   24.23   0.056018   7/28/2008   1234.37   0.018771643
                          -
7/25/2008   22.91 0.0228705    7/25/2008   1257.76   0.004158871
                                                               -
7/24/2008   23.44 0.0952675    7/24/2008   1252.54   0.023396064
7/23/2008   21.31 0.0061191    7/23/2008   1282.19   0.004055976
                          -
7/22/2008   21.18 0.0846084    7/22/2008     1277    0.013401856
                          -                                    -
7/21/2008   23.05 0.0424692    7/21/2008     1260    0.000539537
                          -
7/18/2008   24.05 0.0391407    7/18/2008   1260.68   0.000285601
                          -
7/17/2008   25.01 0.0035921    7/17/2008   1260.32   0.011941012
                          -
7/16/2008    25.1 0.1284388    7/16/2008   1245.36   0.024754645
7/15/2008   28.54 0.0021045    7/15/2008   1214.91     -0.0109611
                                                                -
7/14/2008   28.48 0.0353798    7/14/2008    1228.3   0.009068905
                                                                -
7/11/2008   27.49 0.0716207    7/11/2008   1239.49   0.011151876
7/10/2008   25.59 0.0141679    7/10/2008   1253.39   0.006965378
                                                                -
 7/9/2008   25.23  0.086039     7/9/2008   1244.69   0.023039548
                          -
 7/8/2008   23.15 0.1076042     7/8/2008    1273.7   0.016936205
                                                               -
 7/7/2008   25.78 0.0391587     7/7/2008   1252.31   0.008420818
                          -
 7/3/2008   24.79 0.0445745     7/3/2008    1262.9   0.001093321
                                                               -
 7/2/2008   25.92 0.0916518     7/2/2008   1261.52   0.018371334
                          -
 7/1/2008   23.65 0.0126052     7/1/2008   1284.91   0.003828599
6/30/2008   23.95 0.0215244    6/30/2008     1280    0.001266427
                          -                                    -
6/27/2008   23.44 0.0206889    6/27/2008   1278.38   0.003724341
                                                               -
6/26/2008   23.93 0.1239659    6/26/2008   1283.15   0.029805056
                          -
6/25/2008   21.14 0.0587864    6/25/2008   1321.97   0.005826452
                          -
6/24/2008   22.42 0.0097648    6/24/2008   1314.29   -0.00281884
                          -
6/23/2008   22.64 0.0101078    6/23/2008     1318    5.31122E-05
                                                               -
6/20/2008   22.87   0.058059   6/20/2008   1317.93   0.018717003
6/19/2008   21.58          -   6/19/2008   1342.83   0.003745379
                    0.0301255
                                                                -
6/18/2008   22.24 0.0511986     6/18/2008   1337.81   0.009759296
                                                                -
6/17/2008   21.13 0.0085552     6/17/2008   1350.93   0.006794391
                          -
6/16/2008   20.95 0.0128055     6/16/2008   1360.14   8.08773E-05
6/13/2008   21.22 -0.094796     6/13/2008   1360.03   0.014934164
                          -
6/12/2008   23.33 0.0333013     6/12/2008   1339.87   0.003274329
                                                                -
6/11/2008   24.12 0.0397515     6/11/2008   1335.49   0.017038717
                                                                -
6/10/2008   23.18 0.0025918     6/10/2008   1358.44   0.002440998
                          -
 6/9/2008   23.12 0.0188523      6/9/2008   1361.76   0.000793406
                                                                -
 6/6/2008   23.56 0.2347772      6/6/2008   1360.68   0.031376343
                          -
 6/5/2008   18.63 0.1101798      6/5/2008   1404.05   0.019308465
                                                                -
 6/4/2008    20.8 0.0272921      6/4/2008    1377.2   0.000326697
                                                                -
 6/3/2008   20.24 0.0204649      6/3/2008   1377.65   0.005804628
                                                                -
 6/2/2008   19.83 0.1063135      6/2/2008   1385.67   0.010559851
5/30/2008   17.83 -0.017237     5/30/2008   1400.38   0.001515022
5/29/2008   18.14 -0.049997     5/29/2008   1398.26   0.005320725
                          -
5/28/2008   19.07 0.0294519     5/28/2008   1390.84   0.003955066
5/27/2008   19.64 0.004593      5/27/2008   1385.35   0.006822949
                                                                -
5/23/2008   19.55 0.0798296     5/23/2008   1375.93   0.013298491
                          -
5/22/2008   18.05 0.0294781     5/22/2008   1394.35   0.002613949
                                                                -
5/21/2008   18.59 0.0558619     5/21/2008   1390.71   0.016183741
5/20/2008   17.58 0.0329605     5/20/2008    1413.4   -0.00931687
5/19/2008   17.01 0.0322609     5/19/2008   1426.63   0.000897622
5/16/2008   16.47 0.0103754     5/16/2008   1425.35   0.001249597
                          -
5/15/2008    16.3 0.0801371     5/15/2008   1423.57   0.010528903
                          -
5/14/2008   17.66 0.0179578     5/14/2008   1408.66   0.003997587
                                                                -
5/13/2008   17.98 0.0106235     5/13/2008   1403.04   0.000384805
5/12/2008   17.79         -     5/12/2008   1403.58   0.010960545
                  0.0871519
                                                              -
 5/9/2008   19.41 0.0005153    5/9/2008   1388.28   0.006748148
                          -
 5/8/2008    19.4 0.0168673    5/8/2008   1397.68   0.003662758
                                                              -
 5/7/2008   19.73 0.0801694    5/7/2008   1392.57   0.018279807
 5/6/2008   18.21 -0.037191    5/6/2008   1418.26   0.007622792
                                                              -
 5/5/2008    18.9 0.0388398    5/5/2008   1407.49   0.004543867
                          -
 5/2/2008   18.18 0.0377811    5/2/2008    1413.9   0.003230334
                          -
 5/1/2008   18.88 0.0963689    5/1/2008   1409.34   0.016995468
                                                              -
4/30/2008   20.79 0.0268113   4/30/2008   1385.59   0.003853736
                                                              -
4/29/2008   20.24 0.0300925   4/29/2008   1390.94   0.003896235
                                                              -
4/28/2008   19.64 0.0025491   4/28/2008   1396.37   0.001052176
                          -
4/25/2008   19.59 0.0237086   4/25/2008   1397.84   0.006473722
                          -
4/24/2008   20.06 0.0099207   4/24/2008   1388.82   0.006421692
                          -
4/23/2008   20.26 0.0296642   4/23/2008   1379.93   0.002895639
                                                              -
4/22/2008   20.87 0.0178878   4/22/2008   1375.94   0.008849199
                                                              -
4/21/2008    20.5 0.0182136   4/21/2008   1388.17   0.001554796
4/18/2008   20.13 -0.011852   4/18/2008   1390.33   0.017976528
4/17/2008   20.37 -0.007824   4/17/2008   1365.56   0.000622649
                          -
4/16/2008   20.53 0.1039957   4/16/2008   1364.71   0.022437717
                          -
4/15/2008   22.78 0.0446426   4/15/2008   1334.43   0.004589249
                                                              -
4/14/2008   23.82 0.0152287   4/14/2008   1328.32   0.003389515
                                                              -
4/11/2008   23.46 0.0651639   4/11/2008   1332.83   0.020584529
                          -
4/10/2008   21.98 0.0370661   4/10/2008   1360.55    0.00446403
                                                              -
 4/9/2008   22.81 0.0199254    4/9/2008   1354.49   0.008124955
                          -                                   -
 4/8/2008   22.36 0.0026798    4/8/2008   1365.54   0.005113083
 4/7/2008   22.42         -    4/7/2008   1372.54    0.00156037
                  0.0013372
                          -
 4/4/2008   22.45 0.0332926    4/4/2008    1370.4   0.000795705
 4/3/2008   23.21 -0.009434    4/3/2008   1369.31    0.00130077
                                                              -
 4/2/2008   23.43 0.0325338    4/2/2008   1367.53   0.001935925
                          -
 4/1/2008   22.68 0.1214994    4/1/2008   1370.18   0.035267016
                          -
3/31/2008   25.61 0.0038971   3/31/2008    1322.7    0.00567115
                          -                                   -
3/28/2008   25.71 0.0065904   3/28/2008   1315.22   0.007981928
                          -                                   -
3/27/2008   25.88 0.0076983   3/27/2008   1325.76   0.011526662
                                                              -
3/26/2008   26.08 0.0138998   3/26/2008   1341.13   0.008804416
                          -
3/25/2008   25.72 0.0003887   3/25/2008   1352.99   0.002301259
                          -
3/24/2008   25.73 0.0340052   3/24/2008   1349.88   0.015205246
3/20/2008   26.62 -0.114187   3/20/2008   1329.51   0.023662313
                                                              -
3/19/2008   29.84  0.145863   3/19/2008   1298.42   0.024587038
                          -
3/18/2008   25.79 0.2232211   3/18/2008   1330.74   0.041534885
                                                              -
3/17/2008   32.24 0.0340727   3/17/2008    1276.6   0.008999024
                                                              -
3/14/2008   31.16 0.1326149   3/14/2008   1288.14   0.021002301
3/13/2008   27.29 0.0025683   3/13/2008   1315.48   0.005113853
                                                              -
3/12/2008   27.22 0.0321043   3/12/2008   1308.77   0.009036275
                          -
3/11/2008   26.36 0.1084665   3/11/2008   1320.65    0.03645711
                                                              -
3/10/2008   29.38 0.0664919   3/10/2008   1273.37   0.015584286
                          -                                   -
 3/7/2008   27.49 0.0021802    3/7/2008   1293.37   0.008445951
                                                              -
 3/6/2008   27.55 0.1132561    3/6/2008   1304.34   0.022259869
 3/5/2008    24.6 -0.036716    3/5/2008    1333.7   0.005224692
                          -                                   -
 3/4/2008   25.52 0.0293457    3/4/2008   1326.75   0.003453611
                          -
 3/3/2008   26.28 0.0098448    3/3/2008   1331.34    0.00053344
                                                              -
2/29/2008   26.54 0.1203768   2/29/2008   1330.63   0.027463359
                                                                                                                     -
 2/28/2008                       23.53 0.0363519                             2/28/2008       1367.68       0.008982118
 2/27/2008                       22.69                                       2/27/2008       1380.02


                                                   VIX vs. S&P500 (2008-2009) Correlation

                0.3




                0.2




                0.1
 Daily Change




                                                                                                                         VIX
                  0
                                                                                                                         S&P500
                2/22/2008   4/12/2008   6/1/2008      7/21/2008   9/9/2008   10/29/2008   12/18/2008   2/6/2009



                -0.1




                -0.2




                -0.3
                                                                  Date




Appendix B: VIX vs. Dow Calculations and Graph
                             Adj
Date                         Close        VIX                            Date       Adj Close DOW
 2/26/2009                      44.66     -0.00022                        2/26/2009 7182.08 -0.01229
 2/25/2009                      44.67     -0.01819                        2/25/2009 7270.89 -0.01095
 2/24/2009                      45.49       -0.1456                       2/24/2009 7350.94 0.032654
 2/23/2009                      52.62     0.065172                        2/23/2009 7114.78 -0.03466
 2/20/2009                       49.3     0.046076                        2/20/2009 7365.67 -0.01352
 2/19/2009                      47.08     -0.02889                        2/19/2009 7465.95 -0.01194
 2/18/2009                      48.46     -0.00412                        2/18/2009 7555.63 0.000401
 2/17/2009                      48.66     0.125286                        2/17/2009   7552.6 -0.03867
 2/13/2009                      42.93      0.03992                        2/13/2009 7850.41 -0.01044
 2/12/2009                      41.25     -0.07651                        2/12/2009 7932.76 -0.00085
 2/11/2009                      44.53     -0.04694                        2/11/2009 7939.53     0.0064
 2/10/2009                      46.67     0.067127                        2/10/2009 7888.88 -0.04729
  2/9/2009                      43.64     0.006206                         2/9/2009 8270.87 -0.00117
  2/6/2009                      43.37     -0.00827                         2/6/2009 8280.59 0.02662
  2/5/2009   43.73   -0.00274      2/5/2009   8063.07   0.013285
  2/4/2009   43.85    0.01818      2/4/2009   7956.66   -0.01518
  2/3/2009   43.06   -0.05556      2/3/2009   8078.36   0.017675
  2/2/2009   45.52   0.015051      2/2/2009   7936.83   -0.00804
 1/30/2009   44.84   0.050542     1/30/2009   8000.86   -0.01835
 1/29/2009   42.63   0.072215     1/29/2009   8149.01   -0.02741
 1/28/2009   39.66   -0.06326     1/28/2009   8375.45   0.024257
 1/27/2009   42.25   -0.07828     1/27/2009   8174.73   0.007207
 1/26/2009   45.69      -0.034    1/26/2009   8116.03   0.004751
 1/23/2009   47.27   -0.00042     1/23/2009   8077.56   -0.00559
 1/22/2009   47.29   0.018568     1/22/2009    8122.8   -0.01288
 1/21/2009   46.42   -0.19916     1/21/2009    8228.1   0.034498
 1/20/2009   56.65   0.205862     1/20/2009   7949.09   -0.04093
 1/16/2009   46.11     -0.1008    1/16/2009   8281.22   0.008334
 1/15/2009      51   0.037152     1/15/2009   8212.49   0.001505
 1/14/2009   49.14   0.127214     1/14/2009   8200.14   -0.02984
 1/13/2009   43.27     -0.0577    1/13/2009   8448.56      -0.003
 1/12/2009   45.84   0.068152     1/12/2009   8473.97   -0.01467
  1/9/2009   42.82    0.00609      1/9/2009   8599.18   -0.01652
  1/8/2009   42.56   -0.01931      1/8/2009   8742.46   -0.00311
  1/7/2009   43.39   0.118014      1/7/2009    8769.7     -0.0276
  1/6/2009   38.56     -0.0134     1/6/2009    9015.1   0.006925
  1/5/2009   39.08   -0.00281      1/5/2009   8952.89     -0.0091
  1/2/2009   39.19   -0.02046      1/2/2009   9034.69   0.029006
12/31/2008      40   -0.03994    12/31/2008   8776.39   0.012382
12/30/2008   41.63   -0.05309    12/30/2008   8668.39   0.021509
12/29/2008    43.9   0.011916    12/29/2008   8483.93   -0.00372
12/26/2008   43.38   -0.01895    12/26/2008   8515.55   0.005543
12/24/2008   44.21   -0.01816    12/24/2008   8468.48   0.005802
12/23/2008   45.02    0.01027    12/23/2008   8419.49   -0.01183
12/22/2008   44.56   -0.00827    12/22/2008   8519.69   -0.00695
12/19/2008   44.93   -0.05225    12/19/2008   8579.11   -0.00301
12/18/2008   47.34   -0.05146    12/18/2008   8604.99   -0.02517
12/17/2008   49.84   -0.04952    12/17/2008   8824.34   -0.01125
12/16/2008   52.37     -0.0805   12/16/2008   8924.14   0.041131
12/15/2008   56.76   0.044676    12/15/2008   8564.53   -0.00758
12/12/2008   54.28   -0.02726    12/12/2008   8629.68   0.007513
12/11/2008   55.78   0.000897    12/11/2008   8565.09   -0.02266
12/10/2008   55.73   -0.05549    12/10/2008   8761.42   0.008032
 12/9/2008   58.91   0.007155     12/9/2008   8691.33   -0.02756
 12/8/2008   58.49   -0.02432     12/8/2008   8934.18   0.034012
 12/5/2008   59.93   -0.06006     12/5/2008   8635.42   0.030473
 12/4/2008   63.64   0.046969     12/4/2008   8376.24     -0.0254
 12/3/2008   60.72   -0.03654    12/3/2008 8591.69    0.020294
 12/2/2008   62.98   -0.08416    12/2/2008 8419.09    0.032595
 12/1/2008   68.51   0.214569    12/1/2008 8149.09    -0.08014
11/28/2008   55.28   0.006534   11/28/2008 8829.04    0.011669
11/26/2008   54.92   -0.10336   11/26/2008 8726.61    0.028729
11/25/2008    60.9   -0.06053   11/25/2008 8479.47    0.004264
11/24/2008    64.7   -0.11617   11/24/2008 8443.39    0.048157
11/21/2008   72.67   -0.10679   11/21/2008 8046.42    0.063376
11/20/2008   80.86   0.085147   11/20/2008 7552.29    -0.05725
11/19/2008   74.26   0.093373   11/19/2008 7997.28    -0.05207
11/18/2008   67.64   -0.02208   11/18/2008 8424.75    0.018106
11/17/2008   69.15   0.041937   11/17/2008 8273.58    -0.02668
11/14/2008   66.31   0.102834   11/14/2008 8497.31       -0.039
11/13/2008   59.83   -0.10509   11/13/2008 8835.25    0.064585
11/12/2008   66.46   0.078539   11/12/2008 8282.66    -0.04846
11/11/2008   61.44    0.02405   11/11/2008 8693.96    -0.02011
11/10/2008   59.98   0.066875   11/10/2008 8870.54    -0.00823
 11/7/2008    56.1   -0.12673    11/7/2008 8943.81    0.028123
 11/6/2008   63.68    0.15457    11/6/2008 8695.79    -0.04974
 11/5/2008   54.56   0.133741    11/5/2008 9139.27    -0.05181
 11/4/2008   47.73   -0.11748    11/4/2008 9625.28    0.032249
 11/3/2008   53.68   -0.10947    11/3/2008 9319.83    -0.00183
10/31/2008   59.89   -0.04904   10/31/2008 9336.93    0.016875
10/30/2008    62.9   -0.10638   10/30/2008 9180.69    0.020883
10/29/2008   69.96   0.043828   10/29/2008 8990.96    -0.00821
10/28/2008   66.96   -0.17868   10/28/2008 9065.12    0.103259
10/27/2008   80.06   0.011684   10/27/2008 8175.77    -0.02455
10/24/2008   79.13    0.15453   10/24/2008 8378.95    -0.03659
10/23/2008    67.8   -0.02692   10/23/2008 8691.25    0.019993
10/22/2008   69.65   0.271117   10/22/2008 8519.21    -0.05991
10/21/2008   53.11    0.00264   10/21/2008 9045.21    -0.02405
10/20/2008   52.97   -0.28347   10/20/2008 9265.43    0.045622
10/17/2008   70.33   0.039443   10/17/2008 8852.22    -0.01425
10/16/2008   67.61   -0.02397   10/16/2008 8979.26    0.045727
10/15/2008   69.25   0.228029   10/15/2008 8577.91    -0.08201
10/14/2008   55.13   0.002543   10/14/2008 9310.99      -0.0082
10/13/2008   54.99   -0.24063   10/13/2008 9387.61    0.105083
10/10/2008   69.95   0.090148   10/10/2008 8451.19    -0.01503
 10/9/2008   63.92   0.105326    10/9/2008 8579.19    -0.07616
 10/8/2008   57.53   0.069266    10/8/2008   9258.1   -0.02021
 10/7/2008   53.68   0.030836    10/7/2008 9447.11    -0.05242
 10/6/2008   52.05   0.142436    10/6/2008   9955.5   -0.03648
 10/3/2008   45.14   -0.00265    10/3/2008 10325.38   -0.01514
10/2/2008   45.26   0.128306    10/2/2008   10482.85   -0.03268
10/1/2008   39.81   0.010606    10/1/2008   10831.07   -0.00181
9/30/2008   39.39   -0.17066    9/30/2008   10850.66   0.045748
9/29/2008   46.72   0.296281    9/29/2008   10365.45   -0.07235
9/26/2008   34.74   0.056854    9/26/2008   11143.13   0.010924
9/25/2008   32.82   -0.06972    9/25/2008   11022.06   0.018025
9/24/2008   35.19   -0.01495    9/24/2008   10825.17   -0.00268
9/23/2008   35.72   0.053772    9/23/2008   10854.17   -0.01477
9/22/2008   33.85   0.054018    9/22/2008   11015.69   -0.03328
9/19/2008   32.07   -0.03161    9/19/2008   11388.44   0.032915
9/18/2008    33.1   -0.09008    9/18/2008   11019.69   0.037919
9/17/2008   36.22   0.178464    9/17/2008   10609.66   -0.04148
9/16/2008    30.3   -0.04517    9/16/2008   11059.02   0.012878
9/15/2008    31.7   0.211383    9/15/2008   10917.51   -0.04517
9/12/2008   25.66    0.05076    9/12/2008   11421.99   -0.00103
9/11/2008   24.39   -0.00532    9/11/2008   11433.71   0.014518
9/10/2008   24.52   -0.03801    9/10/2008   11268.92   0.003395
 9/9/2008   25.47   0.117783     9/9/2008   11230.73   -0.02463
 9/8/2008   22.64   -0.01838     9/8/2008   11510.74   0.025497
 9/5/2008   23.06     -0.0412    9/5/2008   11220.96   0.002921
 9/4/2008   24.03   0.114511     9/4/2008   11188.23   -0.03034
 9/3/2008   21.43     -0.0258    9/3/2008   11532.88   0.001385
 9/2/2008   21.99   0.062872     9/2/2008   11516.92   -0.00231
8/29/2008   20.65   0.060897    8/29/2008   11543.55   -0.01476
8/28/2008   19.43   -0.01684    8/28/2008   11715.18    0.01832
8/27/2008   19.76   -0.03628    8/27/2008   11502.51   0.007824
8/26/2008   20.49   -0.02316    8/26/2008   11412.87   0.002335
8/25/2008   20.97   0.108704    8/25/2008   11386.25   -0.02101
8/22/2008   18.81     -0.0523   8/22/2008   11628.06   0.017161
8/21/2008   19.82   -0.02982    8/21/2008   11430.21   0.001119
8/20/2008   20.42   -0.04125    8/20/2008   11417.43   0.006051
8/19/2008   21.28   0.014198    8/19/2008   11348.55   -0.01146
8/18/2008   20.98   0.069061    8/18/2008   11479.39     -0.0156
8/15/2008   19.58   -0.03808    8/15/2008    11659.9   0.003778
8/14/2008   20.34   -0.05779    8/14/2008   11615.93   0.007168
8/13/2008   21.55   0.017791    8/13/2008   11532.96   -0.00945
8/12/2008   21.17   0.050871    8/12/2008   11642.47   -0.01194
8/11/2008   20.12   -0.02649    8/11/2008   11782.35   0.004085
 8/8/2008   20.66   -0.02344     8/8/2008   11734.32   0.026151
 8/7/2008   21.15   0.044473     8/7/2008   11431.43   -0.01946
 8/6/2008   20.23      -0.044    8/6/2008   11656.07   0.003463
 8/5/2008   21.14   -0.10541     8/5/2008   11615.77   0.028965
 8/4/2008   23.49   0.039953     8/4/2008   11284.15   -0.00373
 8/1/2008   22.57   -0.01626     8/1/2008   11326.32   -0.00455
7/31/2008   22.94   0.078409    7/31/2008   11378.02   -0.01791
7/30/2008   21.21   -0.03793    7/30/2008   11583.69   0.016199
7/29/2008   22.03   -0.09519    7/29/2008   11397.56   0.023658
7/28/2008   24.23   0.056018    7/28/2008   11131.08     -0.0213
7/25/2008   22.91   -0.02287    7/25/2008   11370.69   0.001885
7/24/2008   23.44   0.095268    7/24/2008   11349.28   -0.02464
7/23/2008   21.31   0.006119    7/23/2008   11632.38   0.002572
7/22/2008   21.18   -0.08461    7/22/2008    11602.5   0.011718
7/21/2008   23.05   -0.04247    7/21/2008   11467.34   -0.00255
7/18/2008   24.05   -0.03914    7/18/2008   11496.57   0.004351
7/17/2008   25.01   -0.00359    7/17/2008   11446.66   0.018283
7/16/2008    25.1   -0.12844    7/16/2008   11239.28   0.024931
7/15/2008   28.54   0.002105    7/15/2008   10962.54   -0.00842
7/14/2008   28.48    0.03538    7/14/2008   11055.19   -0.00409
7/11/2008   27.49   0.071621    7/11/2008   11100.54   -0.01151
7/10/2008   25.59   0.014168    7/10/2008   11229.02   0.007292
 7/9/2008   25.23   0.086039     7/9/2008   11147.44   -0.02102
 7/8/2008   23.15     -0.1076    7/8/2008   11384.21   0.013464
 7/7/2008   25.78   0.039159     7/7/2008   11231.96   -0.00502
 7/3/2008   24.79   -0.04457     7/3/2008   11288.53    0.00649
 7/2/2008   25.92   0.091652     7/2/2008   11215.51   -0.01476
 7/1/2008   23.65   -0.01261     7/1/2008   11382.26   0.002837
6/30/2008   23.95   0.021524    6/30/2008   11350.01   0.000308
6/27/2008   23.44   -0.02069    6/27/2008   11346.51   -0.00938
6/26/2008   23.93   0.123966    6/26/2008   11453.42   -0.03081
6/25/2008   21.14   -0.05879    6/25/2008   11811.83   0.000373
6/24/2008   22.42   -0.00976    6/24/2008   11807.43   -0.00295
6/23/2008   22.64   -0.01011    6/23/2008   11842.36    -2.8E-05
6/20/2008   22.87   0.058059    6/20/2008   11842.69   -0.01844
6/19/2008   21.58   -0.03013    6/19/2008   12063.09   0.002825
6/18/2008   22.24   0.051199    6/18/2008   12029.06   -0.01085
6/17/2008   21.13   0.008555    6/17/2008    12160.3   -0.00891
6/16/2008   20.95   -0.01281    6/16/2008   12269.08   -0.00311
6/13/2008   21.22     -0.0948   6/13/2008   12307.35   0.013561
6/12/2008   23.33     -0.0333   6/12/2008   12141.58   0.004773
6/11/2008   24.12   0.039752    6/11/2008   12083.77     -0.0169
6/10/2008   23.18   0.002592    6/10/2008   12289.76   0.000768
 6/9/2008   23.12   -0.01885     6/9/2008   12280.32   0.005758
 6/6/2008   23.56   0.234777     6/6/2008   12209.81   -0.03181
 6/5/2008   18.63   -0.11018     6/5/2008   12604.45   0.017121
 6/4/2008    20.8   0.027292     6/4/2008   12390.48      -0.001
 6/3/2008   20.24   0.020465     6/3/2008   12402.85   -0.00811
 6/2/2008   19.83   0.106314     6/2/2008   12503.82     -0.0107
5/30/2008   17.83   -0.01724    5/30/2008   12638.32   -0.00062
5/29/2008   18.14       -0.05   5/29/2008   12646.22   0.004135
5/28/2008   19.07   -0.02945    5/28/2008   12594.03   0.003634
5/27/2008   19.64   0.004593    5/27/2008   12548.35   0.005491
5/23/2008   19.55    0.07983    5/23/2008   12479.63   -0.01163
5/22/2008   18.05   -0.02948    5/22/2008   12625.62   0.001937
5/21/2008   18.59   0.055862    5/21/2008   12601.19   -0.01789
5/20/2008   17.58    0.03296    5/20/2008   12828.68   -0.01543
5/19/2008   17.01   0.032261    5/19/2008   13028.16    0.00318
5/16/2008   16.47   0.010375    5/16/2008    12986.8   -0.00045
5/15/2008    16.3   -0.08014    5/15/2008   12992.66   0.007283
5/14/2008   17.66   -0.01796    5/14/2008   12898.38   0.005146
5/13/2008   17.98   0.010624    5/13/2008   12832.18   -0.00341
5/12/2008   17.79   -0.08715    5/12/2008   12876.05   0.010161
 5/9/2008   19.41   0.000515     5/9/2008   12745.88   -0.00944
 5/8/2008    19.4   -0.01687     5/8/2008   12866.78   0.004083
 5/7/2008   19.73   0.080169     5/7/2008   12814.35   -0.01598
 5/6/2008   18.21   -0.03719     5/6/2008   13020.83   0.003947
 5/5/2008    18.9    0.03884     5/5/2008   12969.54   -0.00681
 5/2/2008   18.18   -0.03778     5/2/2008    13058.2   0.003698
 5/1/2008   18.88   -0.09637     5/1/2008      13010   0.014702
4/30/2008   20.79   0.026811    4/30/2008   12820.13   -0.00092
4/29/2008   20.24   0.030093    4/29/2008   12831.94     -0.0031
4/28/2008   19.64   0.002549    4/28/2008   12871.75   -0.00156
4/25/2008   19.59   -0.02371    4/25/2008   12891.86   0.003334
4/24/2008   20.06   -0.00992    4/24/2008   12848.95   0.006694
4/23/2008   20.26   -0.02966    4/23/2008   12763.22   0.003374
4/22/2008   20.87   0.017888    4/22/2008   12720.23     -0.0082
4/21/2008    20.5   0.018214    4/21/2008   12825.02     -0.0019
4/18/2008   20.13   -0.01185    4/18/2008   12849.36   0.017972
4/17/2008   20.37   -0.00782    4/17/2008   12620.49   9.67E-05
4/16/2008   20.53      -0.104   4/16/2008   12619.27    0.02056
4/15/2008   22.78   -0.04464    4/15/2008   12362.47   0.004899
4/14/2008   23.82   0.015229    4/14/2008   12302.06     -0.0019
4/11/2008   23.46   0.065164    4/11/2008   12325.42     -0.0206
4/10/2008   21.98   -0.03707    4/10/2008   12581.98   0.004359
 4/9/2008   22.81   0.019925     4/9/2008   12527.26   -0.00392
 4/8/2008   22.36   -0.00268     4/8/2008   12576.44   -0.00286
 4/7/2008   22.42   -0.00134     4/7/2008   12612.43   0.000239
 4/4/2008   22.45   -0.03329     4/4/2008   12609.42   -0.00132
 4/3/2008   23.21   -0.00943     4/3/2008   12626.03   0.001356
 4/2/2008   23.43   0.032534     4/2/2008   12608.92     -0.0036
 4/1/2008   22.68     -0.1215    4/1/2008   12654.36   0.031424
3/31/2008   25.61     -0.0039   3/31/2008   12262.89   0.003798
3/28/2008   25.71   -0.00659    3/28/2008    12216.4   -0.00702
3/27/2008   25.88     -0.0077   3/27/2008   12302.46   -0.00974
3/26/2008   26.08      0.0139   3/26/2008   12422.86   -0.00879
3/25/2008   25.72   -0.00039    3/25/2008    12532.6   -0.00128
3/24/2008   25.73   -0.03401    3/24/2008   12548.64    0.01504
3/20/2008   26.62   -0.11419    3/20/2008   12361.32   0.021395
3/19/2008   29.84   0.145863    3/19/2008   12099.66   -0.02393
3/18/2008   25.79   -0.22322    3/18/2008   12392.66   0.034513
3/17/2008   32.24   0.034073    3/17/2008   11972.25   0.001769
3/14/2008   31.16   0.132615    3/14/2008   11951.09   -0.01616
3/13/2008   27.29   0.002568    3/13/2008   12145.74   0.002927
3/12/2008   27.22   0.032104    3/12/2008   12110.24   -0.00384
3/11/2008   26.36   -0.10847    3/11/2008   12156.81   0.034875
3/10/2008   29.38   0.066492    3/10/2008   11740.15   -0.01299
 3/7/2008   27.49   -0.00218     3/7/2008   11893.69   -0.01226
 3/6/2008   27.55   0.113256     3/6/2008   12040.39   -0.01767
 3/5/2008    24.6   -0.03672     3/5/2008   12254.99   0.003367
 3/4/2008   25.52   -0.02935     3/4/2008    12213.8   -0.00369
 3/3/2008   26.28   -0.00984     3/3/2008    12258.9   -0.00061
2/29/2008   26.54   0.120377    2/29/2008   12266.39   -0.02542
2/28/2008   23.53   0.036352    2/28/2008   12582.18   -0.00887
2/27/2008   22.69               2/27/2008   12694.28
                                                   VIX vs. DOW (2008-2009) Correlation

                0.3




                0.2




                0.1
 Daily Change




                                                                                                                    VIX
                  0
                                                                                                                    DOW
                2/22/2008   4/12/2008   6/1/2008     7/21/2008   9/9/2008     10/29/2008    12/18/2008   2/6/2009



                -0.1




                -0.2




                -0.3
                                                                 Date




Appendix C: VIX vs. S&P 500 Prediction Calculations and Graph

                                                                            S&P
Date                        VIX         Log Change                          500            Log Change
                                                  -
 1/2/2009                     39.19     0.020457842                           931.8        0.031118829
                                                  -
12/1/2008                         40    0.323531725                          903.25        0.007791136
                                                  -                                                  -
11/3/2008                     55.28     0.080098367                          896.24        0.077798346
                                                                                                     -
10/1/2008                     59.89     0.418997569                          968.75        0.184246584
                                                                                                     -
 9/2/2008                     39.39     0.645796658                         1164.74        0.096570689
                                                  -
 8/1/2008                     20.65     0.105166792                         1282.83        0.012116797
                                                  -
 7/1/2008                     22.94     0.043086212                         1267.38          -0.0099083
 6/2/2008                     23.95     0.295085892                            1280        -0.08988355
 5/1/2008                     17.83     -0.15358967                         1400.38        0.010617587
                                                  -
 4/1/2008                     20.79     0.208510798                         1385.59         0.04645094
                                                  -                                                  -
 3/3/2008                     25.61     0.035670129                          1322.7        0.005977412
                                                                                                     -
 2/1/2008                     26.54     0.012893618                         1330.63        0.035379708
 1/2/2008                      26.2     0.152244102                         1378.55        -0.06311391
                              -
12/3/2007    22.5   0.016310699   1468.36   -0.00866593
                                                      -
11/1/2007   22.87   0.210434968   1481.14   0.045042789
10/1/2007   18.53   0.029019282   1549.38   0.014713558
                              -
 9/4/2007     18    0.261509198   1526.75   0.035168284
                              -
 8/1/2007   23.38   0.005970167   1473.99   0.012781559
                                                      -
 7/2/2007   23.52   0.370989742   1455.27   0.032504501
                                                      -
 6/1/2007   16.23   0.218073248   1503.35   0.017976931
                              -
 5/1/2007   13.05   0.085861291   1530.62   0.032030724
                              -
 4/2/2007   14.22   0.029108084   1482.37   0.042379836
 3/1/2007   14.64   -0.05190786   1420.86   0.009930484
                                                      -
 2/1/2007   15.42   0.391938332   1406.82   0.022088306
                              -
 1/3/2007   10.42   0.103823827   1438.24   0.013961173
12/1/2006   11.56   0.057871063    1418.3   0.012536836
                              -
11/1/2006   10.91   0.017265308   1400.63   0.016332505
                              -
10/2/2006    11.1   0.076293484   1377.94   0.031021837
                              -
 9/1/2006   11.98   0.027173348   1335.85   0.024269376
 8/1/2006   12.31   -0.19429936   1303.82   0.021051125
 7/3/2006   14.95   0.133626954   1276.66   0.005072924
                              -
 6/1/2006   13.08   0.228633044    1270.2   8.66043E-05
                                                      -
 5/1/2006   16.44   0.349574732   1270.09   0.031404914
 4/3/2006   11.59    0.01740688   1310.61   0.012082374
                              -
 3/1/2006   11.39   0.080110241   1294.87   0.011034734
 2/1/2006   12.34   -0.04824977   1280.66   0.000452994
 1/3/2006   12.95   0.070372753   1280.08   0.025147961
12/1/2005   12.07   0.000828844   1248.29   -0.00095285
                              -
11/1/2005   12.06   0.239264973   1249.48   0.034581238
                                                      -
10/3/2005   15.32   0.250941503   1207.01   0.017899994
                              -
 9/1/2005   11.92   0.055479152   1228.81   0.006924907
                                                      -
 8/1/2005    12.6   0.085281273   1220.33   0.011285468
                              -
 7/1/2005   11.57   0.039818899   1234.18   0.035336452
                              -                       -
 6/1/2005   12.04   0.098777433   1191.33   0.000142687
                              -
 5/2/2005   13.29   0.141494337    1191.5   0.029512223
                                                       -
 4/1/2005   15.31   0.088021328    1156.85   0.020313519
                                                       -
 3/1/2005   14.02   0.148933689    1180.59   0.019302752
                              -
 2/1/2005   12.08   0.059455259     1203.6   0.018726935
                              -                        -
 1/3/2005   12.82   0.036005421    1181.27   0.025615748
12/1/2004   13.29   0.003769322    1211.92   0.031942491
                              -
11/1/2004   13.24   0.206080371    1173.82   0.037868779
10/1/2004   16.27   0.198555881     1130.2   0.013916956
                              -
 9/1/2004   13.34   0.136431979    1114.58   0.009320337
                              -
 8/2/2004   15.29   0.001960144    1104.24   0.002284721
                                                       -
 7/1/2004   15.32   0.066106329    1101.72   0.034892238
                              -
 6/1/2004   14.34   0.077787189    1140.84   0.017829189
                              -
 5/3/2004    15.5   0.103487795    1120.68   0.012011024
                                                       -
 4/1/2004   17.19   0.026526754     1107.3   0.016933393
                                                       -
 3/1/2004   16.74   0.140210071    1126.21   0.016494221
 2/2/2004   14.55     -0.1336173   1144.94   0.012135101
                               -
 1/2/2004   16.63   0.096239065    1131.13   0.017128882
12/1/2003   18.31   0.115056009    1111.92   0.049518899
11/3/2003   16.32   0.013572078     1058.2   0.007103225
                               -
10/1/2003    16.1   0.344426322    1050.71   0.053504268
                                                       -
 9/2/2003   22.72   0.198472409     995.97   0.012016233
                              -
 8/1/2003   18.63   0.045128329    1008.01   0.017715344
                              -
 7/1/2003   19.49   0.001538067     990.31   0.016093506
 6/2/2003   19.52   0.002564762      974.5   0.011258626
                              -
 5/1/2003   19.47   0.085597949     963.59   0.049645665
                              -
 4/1/2003   21.21   0.317982144     916.92    0.07792735
                              -
 3/3/2003   29.15   0.016332449     848.18   0.008322874
                              -                        -
 2/3/2003   29.63   0.050668732     841.15   0.017149844
                                                       -
 1/2/2003   31.17    0.08535032      855.7   0.027797494
                                                       -
12/2/2002   28.62   0.039919769     879.82   0.062229277
                               -
11/1/2002    27.5   0.124307162     936.31   0.055500585
10/1/2002   31.14     -0.2426061    885.76   0.082914421
                                                      -
 9/3/2002   39.69   0.195560737    815.28   0.116561168
 8/1/2002   32.64   0.018865566    916.07   0.004869544
                                                      -
 7/1/2002   32.03    0.23192379    911.62   0.082299872
                                                      -
 6/3/2002    25.4   0.240017401    989.82   0.075214343
                                                      -
 5/1/2002   19.98   -0.09221138   1067.14   0.009122942
                                                      -
 4/1/2002   21.91   0.230472947   1076.92   0.063384683
                              -
 3/1/2002    17.4   0.215760038   1147.39   0.036080076
                                                      -
 2/1/2002   21.59    0.02343125   1106.73   0.020984887
                              -                       -
 1/2/2002   21.09   0.120886586    1130.2   0.015696374
                              -
12/3/2001    23.8   0.045985113   1148.08   0.007545292
                              -
11/1/2001   24.92   0.297664188   1139.45    0.07248435
10/1/2001   33.56   0.049788875   1059.78   0.017937188
                                                      -
 9/4/2001   31.93   0.247875313   1040.94   0.085256615
                                                      -
 8/1/2001   24.92   0.142051882   1133.58   0.066255606
                                                      -
 7/2/2001   21.62   0.126026914   1211.23   0.010798221
                              -                       -
 6/1/2001   19.06   0.172126355   1224.38   0.025354152
                              -
 5/1/2001   22.64   0.118175577   1255.82   0.005077288
                              -
 4/2/2001   25.48   0.116910511   1249.46   0.074007011
                                                      -
 3/1/2001   28.64   0.010177312   1160.33   0.066358544
 2/1/2001   28.35   0.252675899   1239.94   -0.09683109
 1/2/2001   22.02   -0.19831469   1366.01   0.034050246
                              -
12/1/2000   26.85   0.099196304   1320.28   0.004045193
                                                      -
11/1/2000   29.65   0.226945034   1314.95   0.083456134
                                                      -
10/2/2000   23.63   0.138683388    1429.4   0.004961785
                                                      -
 9/1/2000   20.57   0.200076695   1436.51   0.054966292
                              -
 8/1/2000   16.84   0.208307194   1517.68   0.058928158
                                                      -
 7/3/2000   20.74   0.059600556   1430.83   0.016476253
                              -
 6/1/2000   19.54   0.190899468    1454.6   0.023651631
                              -                       -
 5/1/2000   23.65   0.102396296    1420.6   0.022158698
 4/3/2000    26.2   0.101550987   1452.43             -
                                             0.031279977
 3/1/2000   23.67   0.012755275    1498.58   0.092323812
                              -                        -
 2/1/2000   23.37   0.065420674    1366.42   0.020313063
                                                       -
 1/3/2000   24.95      0.0641378   1394.46   0.052244823
                               -
12/1/1999    23.4   0.032789823    1469.25      0.0562328
11/1/1999   24.18   0.085433556    1388.91   0.018882473
                               -
10/1/1999    22.2   0.135050508    1362.93   0.060661767
                                                       -
 9/1/1999   25.41   0.038512581    1282.71   0.028967267
                              -                        -
 8/2/1999   24.45   0.007740923    1320.41   0.006273778
                                                       -
 7/1/1999   24.64   0.155572144    1328.72   0.032570815
                              -
 6/1/1999   21.09   0.185556401    1372.71    0.05300824
                                                       -
 5/3/1999   25.39   0.012683484    1301.84   0.025287466
 4/1/1999   25.07   0.074936765    1335.18   0.037241816
                              -
 3/1/1999   23.26   0.181174439    1286.37   0.038060601
                                                       -
 2/1/1999   27.88   0.060243597    1238.33   0.032815091
 1/4/1999   26.25    0.07226352    1279.64   0.040190831
                              -
12/1/1998   24.42   0.063078611    1229.23   0.054843528
                              -
11/2/1998   26.01   0.075507553    1163.63   0.057444072
                              -
10/1/1998   28.05   0.378363178    1098.67   0.077233407
                              -
 9/1/1998   40.95   0.078181298    1017.01   0.060526299
 8/3/1998   44.28   0.579689455     957.28   -0.15758607
                                                       -
 7/1/1998    24.8   0.229717532    1120.67   0.011683381
                              -
 6/1/1998   19.71   0.078519478    1133.84   0.038680395
                                                       -
 5/1/1998   21.32   0.006588259    1090.82   0.019005643
                              -
 4/1/1998   21.18   0.134121398    1111.75   0.009035526
 3/2/1998   24.22   0.266708949    1101.75    0.04873843
                              -
 2/2/1998   18.55   0.146186823    1049.34   0.068078429
                              -
 1/2/1998   21.47   0.111813798     980.28   0.010098973
                              -
12/1/1997   24.01   0.133166895     970.43   0.015609171
                              -
11/3/1997   27.43   0.246278885      955.4   0.043621422
                                                       -
10/1/1997   35.09   0.426342693     914.62   0.035086042
                              -
 9/2/1997   22.91   0.077655951    947.28   0.051789019
                                                      -
 8/1/1997   24.76   0.142107178    899.47   0.059182865
                              -
 7/1/1997   21.48   0.002325042    954.31   0.075242742
 6/2/1997   21.53   0.115058001    885.14   0.042535055
                              -
 5/1/1997   19.19   0.044338473    848.28   0.056925444
                              -
 4/1/1997   20.06   0.098658145    801.34   0.056763565
 3/3/1997   22.14   0.048112887    757.12   -0.04354862
 2/3/1997    21.1   0.080398221    790.82   0.005910048
 1/2/1997   19.47   -0.07183082    786.16   0.059510648
                                                      -
12/2/1996   20.92   0.199290726    740.74   0.021739987
                              -
11/1/1996   17.14   0.055049359    757.02   0.070808965
10/1/1996   18.11   0.066196438    705.27   0.025766182
                              -
 9/3/1996   16.95   0.003533573    687.33   0.052785302
 8/1/1996   17.01   -0.13455967    651.99   0.018639176
                                                      -
 7/1/1996   19.46   0.352426165    639.95   0.046827521
                              -
 6/3/1996   13.68   0.161019268    670.63   0.002254153
 5/1/1996   16.07   0.015047306    669.12    0.02259616
                              -
 4/1/1996   15.83   0.176196287    654.17   0.013342046
 3/1/1996   18.88   0.102539639     645.5   0.007885385
 2/1/1996   17.04   0.307437752    640.43   0.006909816
 1/2/1996   12.53   0.000798403    636.02   0.032096689
12/1/1995   12.52   0.078047894    615.93   0.017293479
                              -
11/1/1995   11.58   0.177560674    605.37   0.040228869
                                                      -
10/2/1995   13.83   0.082093496     581.5   0.004991819
 9/1/1995   12.74   0.100661995    584.41   0.039314487
                               -                      -
 8/1/1995   11.52   0.157864015    561.88   0.000320302
 7/3/1995   13.49   0.170091242    562.06   0.031281632
                               -
 6/1/1995   11.38   0.190634884    544.75   0.021055362
 5/1/1995   13.77   0.158639072     533.4   0.035667976
                               -
 4/3/1995   11.75   0.129160151    514.71   0.027576544
 3/1/1995   13.37   0.129160151    500.71   0.026962467
                               -
 2/1/1995   11.75   0.017714508    487.39   0.035438711
                               -
 1/3/1995   11.96   0.098649081    470.42    0.02398764
12/1/1994    13.2      -0.189242   459.27   0.012224127
                                                      -
11/1/1994   15.95   0.091180786    453.69   0.040306091
10/3/1994   14.56   0.019418086   472.35   0.020619728
                                                     -
 9/1/1994   14.28   0.176456437   462.71   0.027245343
 8/1/1994   11.97   0.072759354   475.49   0.036909143
                              -
 7/1/1994   11.13   0.296404033   458.26   0.031004223
                                                     -
 6/1/1994   14.97   0.138793807   444.27   0.027156214
                              -
 5/2/1994   13.03   0.055237922    456.5   0.012320937
 4/4/1994   13.77   -0.39549057   450.91   0.011464639
                                                     -
 3/1/1994   20.45   0.318637122   445.77   0.046825875
                                                     -
 2/1/1994   14.87   0.335665568   467.14   0.030505659
                              -
 1/3/1994   10.63   0.092483989   481.61   0.031983824
                              -
12/1/1993   11.66   0.165601652   466.45   0.010040591
11/1/1993   13.76   0.182903121   461.79   -0.01299474
                              -
10/1/1993   11.46   0.125317119   467.83   0.019207289
                                                     -
 9/1/1993   12.99   0.091851963   458.93   0.010038133
 8/2/1993   11.85   0.010178205   463.56   0.033852458
                                                     -
 7/1/1993   11.73    0.04089304   448.13   0.005341299
                              -
 6/1/1993   11.26   0.179208368   450.53   0.000754952
 5/3/1993   13.47   0.081156914   450.19   0.022463264
                              -                      -
 4/1/1993   12.42   0.008817692   440.19   0.025745373
                              -
 3/1/1993   12.53   0.049056157   451.67   0.018524635
 2/1/1993   13.16   0.057873849   443.38   0.010429042
                              -
 1/4/1993   12.42   0.012004946   438.78   0.007021264
12/1/1992   12.57   -0.03440527   435.71   0.010057059
                              -
11/2/1992   13.01   0.216201757   431.35   0.029812921
10/1/1992   16.15   0.123060093   418.68   0.002104056
 9/1/1992   14.28   0.050261835    417.8   0.009064414
                                                     -
 8/3/1992   13.58   0.030656606   414.03   0.024290181
                              -
 7/1/1992   13.17   0.013574869   424.21   0.038618363
                              -                      -
 6/1/1992   13.35   0.037490609   408.14   0.017511285
                              -
 5/1/1992   13.86   0.113766643   415.35   0.000963507
                              -
 4/1/1992   15.53   0.041002274   414.95   0.027510774
                              -
 3/2/1992   16.18   0.030434485   403.69   -0.02207368
 2/3/1992   16.68             -    412.7   0.009543823
                    0.042259809
                                                      -
 1/2/1992    17.4   -0.10415289    408.78   0.020124912
                              -
12/2/1991   19.31   0.048025402    417.09   0.105789505
11/1/1991   20.26   0.269099631    375.22   -0.04489662
                              -
10/1/1991   15.48   0.023620632    392.45   0.011764691
                                                      -
 9/3/1991   15.85   0.091783284    387.86   0.019329331
                              -
 8/1/1991   14.46   0.048592555    395.43   0.019458251
                              -
 7/1/1991   15.18   0.252996514    387.81   0.043882292
 6/3/1991   19.55   0.204771163    371.16   -0.04907751
                              -
 5/1/1991   15.93   0.135412861    389.83   0.037878465
 4/1/1991   18.24   0.077487495    375.34   0.000319761
                              -
 3/4/1991   16.88   0.229285787    375.22   0.021959955
 2/1/1991   21.23   0.015187762    367.07   0.065114417
                              -
 1/2/1991   20.91   0.232378634    343.93   0.040679049
12/3/1990   26.38   0.174317285    330.22   0.024524557
                              -
11/1/1990   22.16   0.304240965    322.22   0.058206841
                                                      -
10/1/1990   30.04   0.031448069      304    0.006720786
                              -                       -
 9/4/1990   29.11   0.026776723    306.05   0.052540678
                                                      -
 8/1/1990    29.9   0.348111619    322.56   0.099062825
 7/2/1990   21.11   0.308906838    356.15   -0.00523686
                               -                      -
 6/1/1990    15.5   0.113904556    358.02   0.008926024
                               -
 5/1/1990   17.37   0.116695001    361.23   0.088000911
                               -                      -
 4/2/1990   19.52   0.010700739     330.8   0.027255168
                               -
 3/1/1990   19.73   0.108447485    339.94   0.023965543
                               -
 2/1/1990   21.99   0.142585325    331.89   0.008502706
 1/2/1990   25.36      #DIV/0!     329.08      #DIV/0!
                                                  VIX Value vs. S&P 500 Value (Prediction?)


                  1800                                                                                            70


                                                                                                                             S&P 500
                  1600
                                                                                                                  60         VIX


                  1400

                                                                                                                  50
                  1200




                                                                                                                       VIX
                                                                                                                  40
    Stock Value




                  1000


                   800
                                                                                                                  30


                   600
                                                                                                                  20

                   400

                                                                                                                  10
                   200


                     0                                                                                            0
                    8/30/89   5/26/92   2/20/95        11/16/97      8/12/00     5/9/03       2/2/06   10/29/08
                                                                  Date




Appendix D: Volatility Smile Calculations (Maple)
Volatility Smile
                                                                                                     Stephen Perno
                                                                                                Kaitlin Cherundolo
                                                                                                     April 22, 2009
Using the Black-Scholes function on Maple, we calculated the 'volatility smiles' of
Microsoft, Citigroup, JP Morgan, General Motors, and Amazon.

>
Microsoft
Option Data from April 16, 2009 with expiry October 16, 2009.

The first step is to solve for the riskless interest rate, 'r':
>




>



Then we use the BlackScholes function in Maple to find the implied volatilities for the
options at various strike prices. The following is the call option data at six different strike
prices:

>


>




>
>




Citigroup
Option Data from April 16, 2009 with expiry September 18, 2009.

The first step is to solve for the riskless interest rate, 'r':
>




>



Then we use the BlackScholes function in Maple to find the implied volatilities for the
options at various strike prices. The following is the call option data at eight different
strike prices:

>


>
>




>




JP Morgan
Option Data from April 16, 2009 with expiry September 18, 2009.

The first step is to solve for the riskless interest rate, 'r':
>




>



Then we use the BlackScholes function in Maple to find the implied volatilities for the
options at various strike prices. The following is the call option data at eleven different
strike prices:

>




>
>




>
Compute the same interest rate r and use the strikes and option prices to calculate the
implied volatility for expiry June 19, 2009.

>




>
>




>




>




>
Finally, compute the last interest rate r and use the strikes and option prices to calculate
the implied volatility for expiry September 18, 2009.

>




>
>




>




>
>




Combine the plots into one complete plot and notice how the volatility smile with the
greatest volatility and the steepest slope is the one closest to expiry.

>
>


Appendix E: Maple Class Example (Empty Worksheet)

    Volatility Smile Walkthrough Example for the Class
                    Stephen Perno and Kaitlin Cherundolo

>
First, choose a company to compute its volatility smile. Start the calculation by listing
the stock price S at closing or where it stands at the moment, the at-the-money Call price
C, the at-the-money Put price P, the time to maturity T, and the at-the-money strike price
K. The at-the-money numbers can be found in the options page of your company at the
border of the highlighted and non-highlighted prices. Try to keep T as accurate as
possible (denom = 12 for months, 52 for weeks, 252 for days)

Name of Company:
>
To find the riskless interest rate, r, we use the equation for r solved through the Put-Call
Parity.




>

Create two arrays choosing strike prices K that have high volumes (can be found on the
right side of the chart of option data) for the one array and a different array of the option
prices x for the chosen strike prices.

>
Create the for loop to compute the implied volatility for each pair of strike and option
prices.

>




Create yet another array this time pairing the particular implied volatilities with their
respective strike prices which can be plotted.

>



Plot the volatility smile being sure not to forget to set the ranges for both the volatility
and for the strike prices.

>


Appendix F: Maple Class Example (Exxon Mobil)

    Volatility Smile Walkthrough Example for the Class
                      Stephen Perno and Kaitlin Cherundolo

>
First, choose a company to compute its volatility smile. Start the calculation by listing
the stock price S at closing or where it stands at the moment, the at-the-money Call price
C, the at-the-money Put price P, the time to maturity T, and the at-the-money strike price
K. The at-the-money numbers can be found in the options page of your company at the
border of the highlighted and non-highlighted prices. Try to keep T as accurate as
possible (denom = 12 for months, 52 for weeks, 252 for days)

Name of Company:Exxon Mobil
>




To find the riskless interest rate, r, we use the equation for r solved through the Put-Call
Parity.




>



Create two arrays choosing strike prices K that have high volumes (can be found on the
right side of the chart of option data) for the one array and a different array of the option
prices x for the chosen strike prices.

>




Create the for loop to compute the implied volatility for each pair of strike and option
prices.

>
Create yet another array this time pairing the particular implied volatilities with their
respective strike prices which can be plotted.

>




Plot the volatility smile being sure not to forget to set the ranges for both the volatility
and for the strike prices.

>
>

				
DOCUMENT INFO