### 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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