### 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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### VIX Index Volatility Forecast (05/10/2011)

The VIX index opened at 39% on Monday, dropped to 37.7 on Tuesday, climbed to 41 on Wednesday, plummeted to 38.8 on Thursday and closed to 42.9 on Friday.

The volatility is around the 7.8% – 8% level (26.6% – 27.7% monthly) and the TGARCH chart is now displaying a volatility curve which is primarily moving sideways even if it is still very close to the 4% (13.8% monthly) long term equilibrium point. The actual situation of the VIX is neither indicating panic nor greed but uncertainty and indecision because many investors are not really sure which way the price is going to head to. Additionally, most of the uncertainty has been caused by both European and American politicians who keep promising new and robust policies to tackle the crises but are not implementing them.

The HyperVolatility team is bearish the VIX index because the stochastic volatility of the S&P500 implied volatility index is not giving any signal of potential bursts. As a consequence, we believe that the high volatility levels at which the VIX is currently trading will not hold for long implying that we could have a closing around the 38% – 39% by Friday.

However, a further round of sell-offs would push the VIX even higher and given the panic surrounding financial markets readings around the 48% – 50% could be easily achieved.

### VIX Index Volatility Forecast (27/09/2011)

The VIX index opened at 32.7 on Monday, rose to 32.8 on Tuesday, jumped to 37.3 on Wednesday, achieved 41.3 on Thursday in a very volatile session and closed to 41.2 on Friday.

The actual volatility is 8% (27.7% monthly) and the TGARCH plot is showing an upward sloping curve which, in the very last part, does present some signs of retracement that, however, should be carefully interpreted given the delicate financial situation we are in.

The massive sell-off we saw the last week provoked an augment in volatility of volatility but the spike was not proportioned to the violence of the underlying movements. The VIX closed for 2 consecutive days above the 41% level and even during the big market drop occurred in May 2010 the aforementioned point was only barely touched implying that the probability for a mean reverting movement are now very much increased.

The HyperVolatility team is bearish the VIX index because the very last part of the chart should be considered as a short term explosion which is not going to affect VIX fluctuation over the medium term. Consequently, we could see the S&P500 implied volatility index to retrace and eventually close around the 30% level before Friday.

### VIX Index Volatility Forecast (19/09/2011)

The VIX Index opened at 38.59 on Monday, dropped to 36.91 on Tuesday, plunged to 34.6 on Wednesday, settled at 31.97 on Thursday and closed at 30.98 on Friday.

The current volatility is 7.5% (25.9% monthly) and the TGARCH chart is showing a volatility curve which is quite close to its long term equilibrium point and therefore about to complete its mean reverting process. The last week saw some ups and downs in volatility, which are evident in the very last part of curve, but a break through the 7% level (24.2% monthly) should signal an ulterior drop of the oscillation rate and a potential settlement of the latter around the 4% threshold (13.8% monthly).

The HyperVolatility team is moderately bearish the VIX Index because the overall movement is still pointing towards a total retracement of the volatility towards it long term equilibrium point. Consequently, the VIX should touch the 25% area by the next Friday but, given the actual market conditions, short term bursts of the volatility are very likely to occur.

Even in this case the FOMC statement is going to play a key role.

### VIX Index Volatility Forecast (06/09/2011)

The VIX Index opened at 32.2 on Monday, rose to 32.8 on Tuesday, dropped to 31.6 on Wednesday, touched 31.8 on Thursday and closed at 33.9 on Friday.

The current volatility is around 8% (27.7% monthly) and the TGARCH plot is still showing a downward sloping curve which is still trying to complete its mean reverting process and settle around the 4% threshold (13.8% monthly). However, like for the VXN Index, the speed at which the curve was collapsing decreased significantly meaning that the selling pressure that hit the market on Friday had a significant impact on the implied volatility and consequently on the volatility of volatility.

The HyperVolatility team is bullish the VIX Index because its mean reverting speed diminished drastically and a short term burst of the conditional variance is expected to occur within this week.

Should the panic prevail once again we would, almost certainly, see the S&P500 implied volatility index to retest the 40% – 41% although Thursday’s speech will prove decisive in the medium term.

### VIX Index Volatility Forecast (30/08/2011)

The VIX Index opened at 42.4 on Monday, dropped to 36.2 on Tuesday, settled at 35.9 on Wednesday, jumped back up on Thursday, when it touched 39.7 and closed at 35.5 on Friday.

The volatility is around 13% (45% monthly) and the TGARCH plot is displaying an overall downward sloping curve which should continue its mean reverting journey in the upcoming hours. The very last part of the chart shows a small retracement but, as mentioned for the VXN Index, the waiting for Bernanke’s speech on Friday made many investors a bit nervous and this caused the conditional variance to increase.

However, even if some short retracements of the volatility can always happen it is worth pointing out that the oscillations of the VIX Index are decreasing and, given what we can see in the chart, it is likely that the next trading days will see an ulterior flattening of the variance.

The HyperVolatility team is bearish the VIX Index because its rate of fluctuations should diminish and the mean reverting process should become more and more evident. Consequently, we expect the VIX to retest the 30% threshold by Friday.

### VIX Index Volatility Forecast (22/08/2011)

The volatility decreased in the first half of the week, as we managed to forecast, but the bad macroeconomics data accompanied by some nonsensical rants pushed the fear index through the roof. In fact, the VIX Index opened at 31.87% on Monday, rose to 32.85% on Tuesday, settled at 31.58% on Wednesday, jumped to 42.67% on Thursday and closed at 43.05% on Friday.

The actual volatility is 17% (58.8% monthly) and the TGARCH plot is showing an upward sloping curve implying that the panic in the market is far from being over and that the next trading days are likely to experience even wilder market oscillations. Additionally, the contemporary volatility readings are higher than the ones observed during the big drop that the market experienced in March 2004. Also, the VIX Index climbed so high only in the middle of the credit crunch: rough waters ahead!!

Panic and fears are still the predominant feelings amongst investors and the fact that the VIX closed 2days in a row above the 40% threshold is more than a word of warning (again, such a phenomenon did not occur since the first phase of the credit crunch).

The HyperVolatility team is bullish the VIX Index because many traders and investors will keep buying downside protection in the options market driving the index up. In particular, we are expecting the VIX to reach the 50% level by Friday.

On the other hand, good macroeconomics news could push the implied volatility index down towards the 30% level but it’s difficult that we will see readings below this level because most of the market participants will wait for Bernanke’s speech on Friday before making up their minds.

### VIX Index Volatility Forecast (14/08/2011)

The HyperVolatility team forecasted an increase in the VIX index which would have pushed its value around the 40% level and, also in this case, the previously mentioned target was not just achieved but surpassed. In fact, the VIX Index opened at 48% on Monday, plummeted to 35% on Tuesday, touched 42.9% on Wednesday, plunged to 39% on Thursday and closed at 36.3% on Friday.

The current volatility is 18% (62% monthly) and the TGARCH plot is displaying a significant explosion of the conditional variance, which is even higher than the one we had during May 2010, but that is probably going to mean revert soon towards the 4% threshold (13.8% monthly).

The volatility is now moving a bit sideways but it is normal, for the conditional variance, to experience some short term “interruptions” during its journey towards the long term balance level and such a phenomenon is observable in every mean reverting process which follows a big spike in the volatility.

The HyperVolatility team is bearish the VIX because the stochastic volatility plot of the S&P500 implied volatility index is now displaying an-out-of-order and unsustainably high volatility curve. Therefore, the VIX should keep plummeting and possibly touch 25% – 28% by Friday.

Furthermore, the rumours of a potential downgrading of France’s triple A have been immediately vanished by the credit rating agencies themselves and the bargain hunting type of trading which is now hitting the market will push more and more traders to buy equities: all these factors together should favour a recovery of the price and a softening of the conditional variance.

### VIX Index Volatility Forecast (08/08/2011)

The S&P500 implied volatility index is clearly reflecting, with its high price, the great fear that many investors and traders have about the US economy.     The VIX Index opened at 23.6% on Monday, rose to 24.8% on Tuesday, dropped to 23.3% on Wednesday, jumped to 31.6% on Thursday and closed at 32% on Friday.

The actual volatility is 13% (45% monthly) and the TGARCH plot is displaying an impressively high volatility curve whose slope is dangerously steep.

The volatility of the VIX Index has been increasing over the last 2 weeks and the great explosion in the conditional variance is a clear signal that many investors and traders are now panicking and fearing a double dip recession.

The volatility is very high and it will likely to remain high in the upcoming hours with a great potential for an ulterior explosion: there are no upside boundaries in this market.

The HyperVolatility team is bullish the VIX Index because its volatility is very elevated but there are no signs of weakness or retracement whatsoever meaning that many traders purchased a lot of options in order to protect their portfolios and are likely to keep doing so in the next hours.

At the moment buying VIX calls and going long VIX futures seems to be the only reasonable thing to do.We expect the VIX Index to retest the 40% level by Friday.

### VIX Index Volatility Forecast (02/08/2011)

The VIX Index opened at 19.35% on Monday, rose to 20.23% on Tuesday, touched 22.98% on Wednesday, achieved 23.74% on Thursday and closed at 25.25% on Friday.

The volatility is now 8.1% (28% monthly) and the TGARCH plot is showing a volatility curve which has now achieved, like for the VXN Index, one of the highest points over the last 30 trading months. Also, if we exclude the big market crashes occurred on April-May 2010 and May 2011 we can easily notice that the actual readings are amongst the highest ever achieved by the VIX in the examined period and the probability of a “mean reverting journey” is very high.

The great increase in the VIX fluctuations is clearly associated to the US debt ceiling dispute going on during these hours. Furthermore, the upcoming days will see quite a few macroeconomics news released by governmental authorities such as NFP, Initial Jobless Claims, Unemployment change, etc whose side effect is to keep traders and investors worried.

The HyperVolatility team is bearish the VIX Index because, even in this case, the mean reverting effect of the S&P500 implied volatility index should manifest itself . Specifically, we are expecting the VIX Index to retest the 18% area by Friday, macroeconomics news permitting.