### The VIX Index: step by step

The VIX Index has been introduced by the CBOE in January 1993 and since then it has become the most popular and well known volatility index in the world. The VIX is an index extracted from S&P500 options and its calculation has been changed in September 2003 in order to obtain observations not linked or dependant from any model idiosyncrasy. The present study, inspired by the paper written and compiled by the CBOE, will show how to calculate the VIX Index in a step–by–step fashion. There will be ample explanations about how the index works and we will break down the formula in order to provide a better understanding of the function and weight of every component. Let’s start with the general formula for the VIX Index:

$\dpi{120}&space;\bg_white&space;\large&space;\sigma&space;^2&space;=&space;\frac{2}{T}\sum_\imath&space;\frac{\Delta&space;K_\imath}{K_\imath&space;^2}&space;e^&space;R^T&space;Q&space;(K_\imath&space;)-&space;\frac{1}{T}&space;\left&space;[&space;\frac{F}{K_0}-1&space;\right&space;]^2$                                     (1)

Where

$\dpi{120}&space;\bg_white&space;\large&space;\sigma$ = $\dpi{120}&space;\bg_white&space;\frac{VIX}{100}\Rightarrow&space;VIX&space;=&space;\sigma&space;x&space;100$

T = Time to expiration

F = Forward Index price (the ATM strike price)

K0=  First strike price below the Forward price F

Ki= Strike of the ith out–of–the–money option. A call option strike will be considered if Ki>K0 while a put option strike will be used if Ki=K0

ΔKi = it is the midpoint between the strike prices. Specifically, the strikes used will be on both the sides of Ki. In other words, ΔK for the lowest strike is just the difference between the lowest strike and the penultimate strike of the option chain while ΔK for the highest strike is equivalent to the difference between the highest strike and the strike immediately below the top one. The formula is the following $\dpi{120}&space;\bg_white&space;\Delta&space;K_\imath&space;=&space;\frac{K_\imath_+_1&space;-&space;K_\imath_-_1}{2}$

R = Risk free interest rate (the interest rate used is usually the Treasury Bond yield that more closely matches the option expiration date)

Q (Ki) = The midpoint of the spread between bid price and ask price for each option with a strike Ki

This is the formula that it is currently employed to calculate the most famous risk management index in the world. Before we provide any real example it is necessary to clarify some points. First of all, the VIX index is calculated using 2 expirations: the front month and the second front month contracts. In our study we will be using S&P500 option prices recorded on the 10th of October 2013 and therefore the front month options will be those expiring on the 18th of October 2013 (which will provide us with information about the volatility in the near term) while the second front month options will expire on the 15th of November 2013 and they will provide us with information about the volatility in the medium term. It is important to point out that near term options must have at least 7 days to expiration and when this criteria is no longer met the model automatically rolls to the next available contract. Let’s give an example, our front month options expire on the 18th of October while the medium term options will expire on the 15th of November, hence, on Friday the 12th the VIX will mechanically roll. At this point, the front month options used will be the ones expiring on the 15th of November while the second front month options employed in the calculation will become those expiring in December. This measure has to be adopted in order to avoid mispricing issues that commonly happen when options are about to expire. Another important detail to mention is that T (time to expiration) is calculated in minutes using calendar days, consequently, the time to expiration will be given by

$\dpi{120}&space;\bg_white&space;T&space;=&space;\frac&space;{M_c_u_r_r_e_n_t_&space;\hspace&space;{1&space;mm}_d_a_y&space;+&space;M&space;_s_e_t_t_l_e_m_e_n_t&space;\hspace&space;{1&space;mm}_d_a_y&space;+&space;M_o_t_h_e_r&space;\hspace&space;{1&space;mm}_d_a_y_s&space;}{Minutes\hspace&space;{1&space;mm}in&space;\hspace&space;{1&space;mm}a\hspace&space;{1&space;mm}&space;year}$                                  (2)

M current day= minutes remaining until midnight of the same day

M settlement day= minutes remaining from midnight until 08:30 am on SPX settlement day

M other days= total minutes in the days between current day and settlement day

If we assume that the data have been recorded exactly at 08:30 am on the 10th of October and that options will expire at 08:30 am of the 18th of October:

$\dpi{120}&space;\bg_white&space;T_1&space;=\frac&space;{930+510+1,920}{525,600}&space;=&space;0.0246575$

$\dpi{120}&space;\bg_white&space;T_2&space;=\frac&space;{930+510+3,540}{525,600}&space;=&space;0.1013699$

Needless to say that T1 and T2 refer to near term and medium term options respectively. The yield for 1 month Treasury Bills as of the 9th of October 2013 was 0.27% and therefore, given that our October options expire in 9 days while the November options will expire in 37 days, we will use this figure for both near and medium terms.

We now need to determine the F price of the index and in order to do so we take the strike price at which puts and calls have the smallest difference in absolute terms. The below reported table displays call and put prices with their absolute differential:

The red back–grounded figures are the strike prices at which calls and puts have the lowest difference, however, the two expiration dates have 2 different strike prices: ATM for October options is at 1,650 while ATM for November options is 1,655. In order to simplify calculations, and given the fact that the difference is extremely small, we will take 1,650 as F price because it is the front month contract. We can calculate F using the following formula:

$\dpi{120}&space;\bg_white&space;F&space;=&space;Strike&space;\hspace&space;{1&space;mm}Price&space;+&space;e^R^T&space;x&space;(&space;Call&space;\hspace&space;{1&space;mm}Price&space;-&space;Put&space;\hspace&space;{1&space;mm}&space;Price)$                                 (3)

Obviously, there will be two forward index values, F1 and F2 , the first for near term and the second one for medium term options respectively:

$\dpi{120}&space;\bg_white&space;F_1:&space;1,650+&space;e^(^0^.^0^0^2^7&space;\hspace&space;{1mm}^x&space;\hspace&space;{1mm}^0^.^0^2^4^6^5^7^5^)x&space;(23.75&space;-&space;22.7)&space;=&space;1,651.05$

$\dpi{120}&space;\bg_white&space;F_2:&space;1,650+&space;e^(^0^.^0^0^2^7&space;\hspace&space;{1mm}^x&space;\hspace&space;{1mm}^0^.^1^0^1^3^6^9^9^)x&space;(39.75&space;-&space;37.3)&space;=&space;1,652.45$

Now we have both  F1 and F2 , so we can identify K0 which is the strike price immediately below the forward prices. In our case, the closest strike right below F1 and F2 is 1,650. Consequentially, K0,1= 1,650 and K0,2 = 1,650.

The next step is to select all put options whose strike is lower than K0 and all call options whose strike is higher than K0. The selection excludes every option with a bid equal to 0 and it terminates when there are 2 consecutive strike prices that equal zero:

The above reported table explains very well what stated before. In the month of October the lowest puts are at 790 and 795 but the two red back–grounded options cannot be accepted in our calculation because there are two consecutive 0 in their bid prices and therefore they are our stopping point. In other words, no option with a lower than 800 strike price will be considered into the calculation of the VIX. The same principle applies to calls and in fact 2,095 and 2,100 are the highest call option strike prices that will be considered. The same procedure will be applied to November options. It is important to point out that the option chains will rarely have the same amount of strikes available because according to volatility fluctuations the number of strikes that will be priced by market makers will vary. It goes without saying that high volatility explosions will obligate market makers to price even Far–Away–From–The–Money options on both sides because the demand for these instruments will rise. Consequentially, the number of strikes that will be used for the purpose of calculating the VIX will vary according to volatility fluctuations and swings in S&P500 futures. The following table lists the options that will be used for calculating the volatility index:

The ATM strike is highlighted in blue. The put options in the near term contract that will be used start from strike 800 until strike 1,645 while the call options range from strike 1,655 until strike 2,100. The medium term strikes, instead, goes from 985 until 1,645 for put options and from strike 1,655 until strike 2,100 for calls. We now calculate the specific weight that every single strike will have in the calculation by using the following formula and we will take the 800 put as an example:

$\dpi{120}&space;\bg_white&space;\frac{\Delta&space;K_8_0_0&space;\hspace&space;{1&space;mm}Put}&space;{K^2_8_0_0&space;\hspace&space;{1&space;mm}Put}&space;e&space;^R^T^1&space;Q(800&space;Put)$                        (4)

Please bear in mind that Q is the midpoint between bid and ask while ΔK is the difference between the last option’s strike and the closest next strike, hence, in our case ΔK is (805 – 800) = 5. Let’s proceed with the calculation:

$\dpi{120}&space;\bg_white&space;\frac{5}{(800)^2}\hspace&space;{2&space;mm}2.71828&space;^(^0^.^0^0^2^7&space;\hspace&space;{1&space;mm}^x&space;\hspace&space;{1&space;mm}&space;^0^.^0^2^4^6^5^7^5^)&space;(0.05)=&space;0.000000390651$

The next table summarizes the contribution of each option strike to the overall computation of the VIX Index:

Now, the weights need to be added up and multiplied by 2/T1 for the near term and by 2/T2for the medium term and by performing this calculation we would have 0.074319519775 for the near term and 0.041553088 for the medium term.

The equation (1) that we presented at the beginning of this study is almost completed, in fact, the final part is the only one yet to be estimated. Let’s proceed:

$\dpi{120}&space;\bg_white&space;\frac{1}{T_1}&space;\left&space;[&space;\frac{F_1}{K_0}-1&space;\right&space;]^2&space;=&space;\frac&space;{1}{0.0246575}&space;\left&space;[\frac&space;{1,651.05}{1,650}-1&space;\right&space;]^2&space;=&space;0.000016423$

$\dpi{120}&space;\bg_white&space;\frac{1}{T_2}&space;\left&space;[&space;\frac{F_2}{K_0}-1&space;\right&space;]^2&space;=&space;\frac&space;{1}{0.1013699}&space;\left&space;[\frac&space;{1,652.45}{1,650}-1&space;\right&space;]^2&space;=&space;0.000021750$

We can now complete the calculation by subtracting the two members of equation (1):

Near Term σ21 : 0.074319519775 – 0.000016423 = 0.074303096

Medium Term σ22 : 0.041553088 – 0.000021750 = 0.041531338

The final VIX computation is given by the following formula:

$\dpi{120}&space;\bg_white&space;VIX&space;=&space;100&space;\hspace&space;{1&space;mm}&space;x&space;\hspace&space;{1&space;mm}\sqrt\left&space;\left&space;\{&space;T_1&space;\sigma_1^2&space;\left&space;[&space;\frac{N_T_2-N_3_0}{N_T_2&space;-&space;N_T_1}\right]&space;-&space;T_2\sigma&space;_2^2&space;\left&space;[&space;\frac{N_3_0&space;-&space;N_T_1}{N_T_2&space;-&space;N_T_1}&space;\right&space;]\right&space;\}&space;x&space;\frac&space;{N_3_6_5}{N_3_0}$

The second term of the equation is nothing but the computation of the square root of the 30 day average of σ21 and σ22which are subsequently multiplied by the first term 100.  The weights of σ21 and σ22 are less than 1 or almost equal to 1 when the near term options have less 30 days and the medium term options have more than 30 days to expiration. However, it is worth noting that when the VIX rolls both near and medium term options have more than 30 days to expiration. Let’s now conclude the VIX estimation:

NT1= minutes left before the settlement of near – term options

NT2= minutes left before the settlement of medium – term options

N30= number of minutes in 30 days

N365=  number of minutes in a year composed by 365 days

If we plug in the numbers we have the following outcome:

$\dpi{120}&space;\bg_white&space;\sqrt\left&space;\{&space;0.0246575&space;\hspace&space;{1&space;mm}x&space;\hspace&space;{1&space;mm}0.074303096&space;\hspace&space;{1&space;mm}&space;x&space;\left&space;[&space;\frac{53,280&space;-&space;43,200}{53,280&space;-&space;12,960}\right]&space;+&space;0.1013699&space;\hspace&space;{1&space;mm}&space;x&space;\hspace&space;{1&space;mm}&space;0.041531338&space;\hspace&space;{1&space;mm}x&space;\hspace&space;{1&space;mm}\left&space;[&space;\frac{43,200&space;-&space;12,960}{53,280&space;-&space;12,960}&space;\right&space;]\right&space;\}&space;\hspace&space;{1&space;mm}&space;x&space;\hspace&space;{1&space;mm}&space;\frac&space;{525,600}{43,200}$

This leads to the very last step: VIX = 100 x 0.209736087 = 20.9%

Now, the number we came out with is fairly accurate but it is limited to the moment in which the option prices have been “frozen”. Consequentially, the aforementioned procedure and calculation, in order to be precise, has to be automated and repeated every instant because any changes in option premiums will inevitable affect the final assessment of the VIX.

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### E-Mini S&P500 Futures Volatility Forecast (05/10/2011)

E-Mini S&P500 futures opened at 1,158 on Monday, achieved 1,170 on Tuesday, plummeted to 1,149 on Wednesday, and rose to 1,157 on Thursday but on Friday the price dropped to 1,122 in the last minutes of the trading session.

The current volatility is 2% (31.7% annualised) and the plot is still showing a volatility curve which is trading at very high levels but now moving laterally with a shy downward sloping tendency.

The conditional variance is indeed very high but it is likely that the mean reverting process will start to become more and more “heavy” meaning that the volatility curve will probably tend to mean revert towards the next hours even if it will be unlikely to see a full and complete return to the long term equilibrium point.

The HyperVolatility team is bullish E-Mini S&P500 futures because the softening of the oscillation rate should support the price action which could potentially achieve the 1,180 – 1,200 area by Friday.

Nevertheless, unexpected bad news could drag futures prices back down and this time, should this happen, a retesting of the 1,000 support would be almost inevitable.

### E-Mini S&P500 Futures Volatility Forecast (27/09/2011)

E-Mini S&P500 futures opened at 1,197 on Monday, touched 1,195 on Tuesday, dropped to 1,156 on Wednesday, plunged to 1,124 on Thursday and closed at 1,130 on Friday.

The actual volatility is 2% (31.7% annualised) and the TGARCH plot is manifestly showing an upward sloping curve which is just the natural consequence of the sharp drop futures prices experienced the last week. However, the curve is not extremely steep, although pretty high, implying that the upside potential of the conditional variance could be limited.

The European sovereign debt crises is still the main issue causing sleepless nights to investors and traders but the intention expressed by EU policymakers to bail out European banks, extend the peripheral bond buying programme and increase the EFSF will probably cool down investors’ nervousness.

The HyperVolatility team is bullish E-Mini S&P500 futures because the volatility should keep mean reverting over the next trading hours whilst the price action is likely to retest the 1,250 – 1,260 area by Friday.

### E-Mini S&P500 Futures Volatility Forecast (19/09/2011)

E-Mini S&P500 futures opened at 1,163 on Monday, rose to 1,171 on Tuesday, jumped to 1,187 on Wednesday, achieved 1,211 on Thursday and closed at 1,214 on Friday.

The actual volatility is 1.4% (22.2% annualised), which is still a fairly high value if we consider an average fluctuations rate of 0.7% – 0.8% (11.1% -12.6% in annual terms), but the slope of the curve is now clearly downward sloping. Hence, the upcoming days should see an ulterior flattening of the conditional variance which, ceteris paribus, should complete the mean reverting process and settle around the abovementioned average values.

The current week does not present much macro-economic news coming from the States, apart from the FOMC announcement, and as a consequence most of investors’ eyes will focus on the European sovereign debt crises.

The HyperVolatility team is moderately bullish E-Mini S&P500 futures because, should the volatility keep in its downward trend, the price could touch the 1,230 – 1,240 by Friday.

The market remains highly unstable and any bad news coming from European politicians is going to provoke a short term sell-off which would completely twist our analysis.

### E-Mini S&P500 Futures Volatility Forecast (06/09/2011)

E-Mini S&P500 futures opened at 1,208 on Monday, tested the 1,204 level on Tuesday, achieved 1,219 on Wednesday, plunged to 1,201 on Thursday and closed at 1,169 on Friday.

The actual volatility is 1.52% (24.1% annualised) and the TGARCH plot is evidently displaying an upward sloping curve which seems suggesting that an ulterior augment in the oscillation rate should be expected over the next trading days. Obviously, last Friday’s market drop considerably influenced the fluctuations rate but it is worth pointing out that the mean reverting speed started to decrease well before Friday. Should the market keep plummeting we could see readings around the 2.5% (39.6% in annual terms) by Friday again.

The HyperVolatility team is bearish E-Mini S&P500 futures because the expected increase in the conditional variance is going to drag the price action back down into the 1,110 area by Friday.

It is worth pointing out that the situation could completely change should Obama or Bernanke announce a new fiscal/monetary stimulus package because many investors who got burnt after the August sell-off would probably get back to buy equities or increase their exposure to risky markets.

### E-Mini S&P500 Futures Volatility Forecast (30/08/2011)

E-Mini S&P500 futures opened at 1,124 on Monday, rose to 1,158 on Tuesday, achieved 1,171 on Wednesday, plummeted to 1,157 on Thursday and closed at 1,175 on Friday.

The current volatility is 2.2% (34.9% in annual terms) and the TGARCH plot displays a sideways movement of the volatility curve which clearly implies the fact that many traders and investors have been waiting for Bernanke’s speech. The conditional variance is still extremely high and the mean reverting process should manifest itself over the next trading hours although some short term volatility increases are quite likely to occur.

However, it is important to point out that the overall interpretation of the chart highlights that a softening of the market fluctuations rate is quite likely to happen.

The HyperVolatility team is bullish E-Mini S&P500 futures because the decrease in market volatility should favour a recovery of the price which could eventually retest the 1,230 points by Friday.

It is worth noting that the majority of investors and traders will focus on the macroeconomics news such as manufacturing index, NFP and initial jobless claims and therefore a great deal of attention will be needed during their announcement.

### E-Mini S&P500 Futures Volatility Forecast (22/08/2011)

The market headed north in the first half of the week, meeting our last week’s expectations, but the disappointing figures related to the US manufacturing industry and the medieval approach to economics that some European politicians have, certainly helped to destroy investors’ confidence even further.

E-Mini S&P500 futures opened at 1,196 on Monday, dropped to 1,192 on Tuesday, plummeted to 1,189 on Wednesday, plunged to 1,144 on Thursday and closed to 1,123 on Friday.

The current volatility is 2.6% (41.2% in annual terms) and the TGARCH plot is showing an insistently upward sloping curve which seems suggesting that the upcoming days will see an ulterior increase in the oscillation rate although the present readings are well higher than the equilibrium point and still dangerously close to what we saw in the 2008- 2009 crash.

The HyperVolatility team remains bearish E-Mini S&P500 futures because the conditional variance is still very high and the VIX Index is not showing any sign of rest. There will be some short term uptrend but the overall week should not see a robust recovery of the price because it will take some times for the market to attract some buyers.

We expect futures prices to plunge over the next trading days and eventually retest the 1,000 level by Friday. No long position should be entered during this week even if the market could potentially show some movement on the upside.

This week all eyes will be on Europe because quite a few macroeconomics data are going to be released but on Friday a great deal of attention will be needed since investors will switch the focus on Bernanke’s speech.

### E-Mini S&P500 Futures Volatility Forecast (14/08/2011)

The HyperVolatility team was right once again and the profit targets we suggested the last week have been successfully hit by futures prices making our analysis, once again, accurate and profitable. In particular, E-Mini S&P500 futures opened at 1,111 on Monday, rose to 1,171 on Tuesday, dropped to 1,123 on Wednesday, rose to 1,168 on Thursday and closed at 1,177 on Friday.

The actual volatility is 3% (47.6% annualised) and the TGARCH plot is evidently showing a downward sloping curve whose mean reverting process seems to be on its way. The fluctuations rate should decrease over the next trading hours because the conditional variance will try to settle around the 1% threshold (15.8% in annual terms) even if a short term, although not violent, increases of volatility are very likely to occur along the way down.

It is worth noting that this week most of the macroeconomics news coming from the US has been encouraging: better than expected initial and continuing jobless claims, better than expected core sales and lower than expected Federal budget deficit.

The HyperVolatility is bullish E-Mini S&P500 futures because the plummeting volatility curve is a clear signal that the down movement is now over and that many investors and traders are now buying back the market.

Moreover, the positive macroeconomics news and the decrease in the fluctuations of the VIX are all factors that are pointing towards a recovery of the price action. E-Mini S&P500 futures should eventually touch 1,250 – 1,260 points by Friday, credit rating agencies permitting.

### E-Mini S&P500 Futures Volatility Forecast (08/08/2011)

The American economy has been hit by very bad macroeconomics figures and the US debt ceiling problems accentuated traders ‘concerns dragging the index down. Specifically, E-Mini S&P500 futures opened at 1,279 on Monday, dropped to 1,247 on Tuesday, rose to 1,254 on Wednesday, plummeted to 1,199 on Thursday and closed at 1,197 on Friday.

The volatility is now 1.78% (28.2% annualised) and the TGARCH plot is undoubtedly upward sloping implying that the big drop is far from being over. Particularly, the volatility does not show any retracement and its explosion was as rapid as “clean” (with no short term fluctuations of the curve) and consequently it is very probable that the conditional variance will keep augmenting over the next hours.

The US debt concerns, the S&P downgrading of US sovereign debt, the bad macroeconomics data about the manufacturing industry, the concerns about a potential default of big European countries have completely knocked down investors’ confidence and the panic is the only thing left in the market.

Furthermore, the slightly better than expected news coming from the NFP did not really help market sentiment even if on Tuesday the FOMC will try to regain a bit of credibility.

The HyperVolatility team is extremely bearish on E-Mini S&P500 futures because the oscillation rate should increase even more over the next hours dragging the price back down in the 1,050 – 1,100 area by Friday.

Beware of market up movements because, with this volatility, they are more likely to be bull traps rather than a bargain opportunity.

### E-Mini S&P500 Futures Volatility Forecast (02/08/2011)

E-Mini S&P500 futures opened at 1,333 on Monday, dropped to 1,326 on Tuesday, plummeted to 1,299 on Wednesday, settled at 1,297 on Thursday and closed at 1,292 on Friday.

The actual volatility is 1% (15.8% annualised) and the TGARCH plot is evidently displaying a volatility curve which has dropped significantly over the last two weeks but that sharply mean reverted and jumped back up again during the last 5 trading days. Furthermore, the conditional variance seems now “intentioned” to trade within the actual ranges even if the fair equilibrium point for this market is around the 0.65% – 0.73% threshold (10.3% – 11.5% annualised).

Many markets are now close to break through their 2 years highs and the 1,350 – 1,360 threshold is the E-Mini S&P500 futures obstacles that bulls will have to overcome if they want to see the market keeps rallying higher. Specifically, the price action attempted to surpass this level quite a few times in the past but it never successfully managed to remain above it for a time sufficient enough to boost investors’ confidence.

The HyperVolatility team is bullish E-Mini S&P500 futures because the volatility plot shows a fairly robust curve which should keep the weekly average price up. In particular, we expect the 1,330 – 1,335 area to be retested by Friday even if some short term explosions of the conditional variance could easily bring back down futures prices (particularly in the first half of the week).