Calculating the VIX in Excel
Tom Arnold The Robins School of Business Department of Finance University of Richmond Richmond, VA 23173 O: 804-287-6399 F: 804-289-8878
[email protected] John H. Earl, Jr. The Robins School of Business Department of Finance University of Richmond Richmond, VA 23173 O: 804-289-8589 F: 804-289-8878
[email protected] December 14, 2007 Preliminary, do not cite without permission Key words: VIX, options, Excel JEL: G13
Electronic copy available at: http://ssrn.com/abstract=1103971 http://ssrn.com/abstract=1103971
Calculating the VIX in Excel Abstract:
The VIX has become a popular volatility index that is based on a weighted average of S&P 500 options that straddle a 30-day maturity. This manner of calculating the VIX emerged in September of 2003 and is documented with an example by the CBOE. In this paper, the calculation of the VIX is reproduced in an Excel template to automate and to some degree simplify the calculation. Further, one can also apply other option series to calculate a VIX-type analysis for the underlying security which is of great benefit because the calculation is independent of option pricing model biases.
1 Electronic copy available at: http://ssrn.com/abstract=1103971
INTRODUCTION:
In September of 2003, the CBOE introduced a new manner of calculating the VIX volatility index that was independent of the biases of any particular option pricing model – the motivating factor for changing the VIX (see CBOE documentation, 2003). In this paper, an Excel template is developed that mimics the example of the VIX calculation documented by the CBOE in 2003. The template is more flexible in the information it requires the user to supply and in some cases, simplifies the calculation demonstrated in the CBOE document. However, the greater benefit of the template, is that a VIX-type calculation can be produced for any option series creating a forward looking gauge of the underlying security that is independent of option pricing model biases. This benefit can be of use to investors and researchers alike as a means of capturing risk assessment based on a view of the future rather than estimated from past data. The Excel template is developed in the next section. In the following section, a VIX-type of analysis is performed using an equity option series to demonstrate how the calculation can be applied more broadly. Section three concludes the paper.
SECTION 1: An Excel Template for the VIX Index
The VIX index is calculated using two maturity series from S&P 500 options that straddle a 30-day maturity with the shorter-term options having a maturity in excess of eight days. The length (i.e. the number of strikes included) of the option series is dependent on the out-of-the-money (call or put) option at a given strike having a non-zero price (based on the mid-quote). To help develop the Excel template the option series supplied in the CBOE example are used:
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Table 1: CBOE Example of S&P 500 Option Series Near Term Options (16 days to maturity) Next Term options (44 days to maturity) Strike: Call: Put: Strike: Call: Put: 775 125.48 0.11 775 128.78 2.72 800 100.79 0.41 800 105.85 4.76 825 76.70 1.30 825 84.14 8.01 850 54.01 3.60 850 64.13 12.97 875 34.05 8.64 875 46.38 20.18 900 18.41 17.98 900 31.40 30.17 925 8.07 32.63 925 19.57 43.31 950 2.68 52.23 950 11.00 59.70 975 0.62 75.16 975 5.43 79.10 1000 0.09 99.61 1000 2.28 100.91 1025 0.01 124.52 1025 0.78 124.38 Note: It should not be implied from this example that each option series must have t he same number of strike prices. The number of strike prices for an option series of a given maturity is determined by the strike of the out-of-the-money option having a positive option price (based on midquote) associated with it. When the out-of-the-money option does not have a positive price, the option is effectively eliminated from the calculation (via providing zero contribution to the calculation) and the series ends when two successive out-of-the-money strikes yield option prices of zero. Further, the actual levels of the strike prices may differ between the different maturity series.
The template begins with user supplied information, including the option pricing series with associated time to maturity and a risk-free interest rate (set to 1.1625% APR in this example, however, an appropriate T-Bill rate would be recommended in practice). Some preliminary calculations are performed concerning the time to maturity based on minutes (the current time is assumed to be 8:30 AM CST) and the difference between call and put prices taken as an absolute value. Figure 1: Initial Excel Template for VIX A 1 2 3 4 5 6 7 8 9 10 11
B
C
D
E
F
G
Time to Maturity: Current Time:
8:30:00
Option 1:
16
days
0.04110
years
15.00
days
Option 2:
44
days
0.11781
years
43.00
days
Risk-free Rate:
1.1625%
annual
Near-Term
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12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Difference: $ 125.37 $ 100.38 $ 75.40 $ 50.41 $ 25.41 $ 0.43 $ 24.56 $ 49.55 $ 74.54 $ 99.52 $ 124.51
Strike: Call 775 $ 125.48 800 $ 100.79 825 $ 76.70 850 $ 54.01 875 $ 34.05 900 $ 18.41 925 $ 8.07 950 $ 2.68 975 $ 0.62 1000 $ 0.09 1025 $ 0.01
Put $ 0.11 $ 0.41 $ 1.30 $ 3.60 $ 8.64 $ 17.98 $ 32.63 $ 52.23 $ 75.16 $ 99.61 $ 124.52
Far-term Difference: $ 126.06 $ 101.09 $ 76.13 $ 51.16 $ 26.20 $ 1.23 $ 23.74 $ 48.70 $ 73.67 $ 98.63 $ 123.60
Strike: Call 775 $ 128.78 800 $ 105.85 825 $ 84.14 850 $ 64.13 875 $ 46.38 900 $ 31.40 925 $ 19.57 950 $ 11.00 975 $ 5.43 1000 $ 2.28 1025 $ 0.78
Put $ 2.72 $ 4.76 $ 8.01 $ 12.97 $ 20.18 $ 30.17 $ 43.31 $ 59.70 $ 79.10 $ 100.91 $ 4.38
Cell D5: = (1440 - (HOUR($B$3)*60+MINUTE(B$3$)+SECOND(B$3$)/60)+ 510)/(1440*365) + (B5 2)/365...Note: 1440 is the number of minutes in a day and 510 is the number of minutes to 8:30 AM which is the time the option expires on its expiration date. Cell F5: = D5*365 Cell D7: copy cell D5 Cell F7: copy cell F5 CellAD13: =ABS(C13 - D13) Cells A14 through A23: copy cell A13 Cells A27 through A37: copy cell A13
Below this information, the at-the-money strike is defined for eac h option maturity based on which pair of call and put option prices have the least amount of difference (in absolute value) between them. These two sets of put and call at-the-money options are then used to imply the forward price for the S&P 500 Index based on put-call forward parity for each maturity. Figure 2: Forward Index Level Calculation Portion of Excel Template for VIX
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A 39 40 41 42 43 44 45 46 47
B
C
D
E
900
Call $ 18.41
Put $ 17.98
Level: $ 900.43
Ref-Strike: 900
900
Call $ 31.40
Put $ 30.17
Level: $ 901.23
Ref-Strike: 900
F
G
Index Level: Near-Term: Strike:
Far-Term: Strike:
Cell A43: =VLOOKUP(MIN(A13:A23),A13:D23,2,FALSE) Cell B43: =VLOOKUP(MIN(A13:A23),A13:D23,3,FALSE) Cell C43: =VLOOKUP(MIN(A13:A23),A13:D23,4,FALSE) Cell D43: =A43+EXP(B9*D5)*(B43 - C43)…Note: this is an application of put-call forward parity to imply the forward price based on the near-term options. Cell E43: = IF(B43>=C43,A43,LOOKUP((A43-0.01),B13:B23)…Note: should the forward level have been below 900, the LOOKUP function would return 875 as the reference strike price. The reference strike price will be used later in the VIX calculation. Cell A47: =VLOOKUP(MIN(A27:A37),A27:D37,2,FALSE) Cell B47: =VLOOKUP(MIN(A27:A37),A27:D37,3,FALSE) Cell C47: =VLOOKUP(MIN(A27:A37),A27:D37,4,FALSE) Cell D47: =A47+EXP(B9*D7)*(B47 - C47)…Note: this is an application of put-call forward parity to imply the forward price based on the far-term options. Cell E47: = IF(B47>=C47,A47,LOOKUP((A47-0.01),B27:B37)…Note: should the forward level have been below 900, the LOOKUP function would return 875 as the reference strike price. The reference strike price will be used later in the VIX calculation.
Preliminary calculations for the VIX Index are performed for eac h strike price: ♦
If the strike is above the “reference strike” (see Figure 2), use the call price
♦
If the strike is below the “reference strike” (see Figure 2), use the put price
♦
If the strike equals the “reference strike” (see Figure 2), use the average of the call and put prices
Each option price will be weighted based on the average of the difference between the two surrounding strike prices or in the case of two surrounding strikes not being available, the difference between the given option’s strike price and the strike price of the next option closest in strike price. These “differences” are defined as “Delta-Strike” in the spreadsheet and will be appreciated by the risk-free rate and divided by the square of the option’s strike price to create the “weight” associated with each contributing option
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price. Why squared values in the weight? The goal is to produce a variance calculation which is a quadratic computation. Figure 3: Weighting of Option Price Contributions to Calculate the VIX A 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
B
C
D
E
F
G
Near-Term Difference: $ 125.37 $ 100.38 $ 75.40 $ 50.41 $ 25.41 $ 0.43 $ 24.56 $ 49.55 $ 74.54 $ 99.52 $ 124.51
Strike: 775 800 825 850 875 900 925 950 975 1000 1025
Call $ 125.48 $ 100.79 $ 76.70 $ 54.01 $ 34.05 $ 18.41 $ 8.07 $ 2.68 $ 0.62 $ 0.09 $ 0.01
Put $ 0.11 $ 0.41 $ 1.30 $ 3.60 $ 8.64 $ 17.98 $ 32.63 $ 52.23 $ 75.16 $ 99.61 $124.52
DeltaStrike:
C/PNear: VIX-Near: 25 0.11 0.00000458 25 0.41 0.00001602 25 1.3 0.00004777 25 3.6 0.00012463 25 8.64 0.00028226 25 18.195 0.00056184 25 8.07 0.00023591 25 2.68 0.00007427 25 0.62 0.00001631 25 0.09 0.00000225 25 0.01 0.00000024
DeltaStrike:
C/PFar: VIX-Far: 25 2.72 0.00011337 25 4.76 0.00018619 25 8.01 0.00029462 25 12.97 0.00044940 25 20.18 0.00065984 25 30.785 0.00095146 25 19.57 0.00057259 25 11 0.00030513 25 5.43 0.00014300 25 2.28 0.00005708 25 0.78 0.00001859
Far-term Difference: $ 126.06 $ 101.09 $ 76.13 $ 51.16 $ 26.20 $ 1.23 $ 23.74 $ 48.70 $ 73.67 $ 98.63 $ 123.60
Strike: 775 800 825 850 875 900 925 950 975 1000 1025
Call $ 128.78 $ 105.85 $ 84.14 $ 64.13 $ 46.38 $ 31.40 $ 19.57 $ 11.00 $ 5.43 $ 2.28 $ 0.78
Put $ 2.72 $ 4.76 $ 8.01 $ 12.97 $ 20.18 $ 30.17 $ 43.31 $ 59.70 $ 79.10 $100.91 $ 4.38
Cell E13: =IF(B13=MIN($B$13:$B$23),B14 - B13, IF(B13=MAX($B$13:$B$23), B13 - B12,(B14 B12)/2))…Note: the nested or iterated “IF” state ment controls for strikes that occur at the upper and lower bound. Cell F13: =IF(B13=$B$43,(C13+D13)/2,IF(B13>$E$43,C13,D13))…Note: the nested or iterated “IF” statement determines which option is to be used or when to use an average of the call and put option prices. Cell G13: = E13/B13^2*EXP($B$9*$D$5)*F13…Note: the weight is applied to the appreciated contributing option price (designated as C/P Near). Cells E14 through E23: copy cell E13 Cells F14 through F23: copy cell F13 Cells G14 through G23: copy cell G13
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Cell E27: =IF(B27=MIN($B$27:$B$37),B28 – B27, IF(B27=MAX($B$27:$B$37), B27 – B26,(B28 – B26)/2))…Note: the nested or iterated “IF” state ment controls for strikes that occur at the upper and lower bound. Cell F27: =IF(B27=$B$47,(C27+D27)/2,IF(B27>$E$47,C27,D27))…Note: the nested or iterated “IF” statement determines which option is to be used or when to use an average of the call and put option prices. Cell G27: = E27/B27^2*EXP($B$9*$D$7)*F27…Note: the weight is applied to the appreciated contributing option price (designated as C/P Near). Cells E28 through E37: copy cell E27 Cells F28 through F37: copy cell F27 Cells G28 through G37: copy cell G27 General Note: these calculations are performed wit h greater precision than the calculations performed in the CBOE VIX document which will lead to very slight variation in the final VIX calculation.
With the option prices weighted, a final VIX calculation can be performed. Figure 4: Final Calculations for the VIX A 39 40 41 42 43 44 45 46 47 48 49
B
C
D
E
F
G
Index Level: Near-Term: Strike:
Call 900 $ 18.41
Put $ 17.98
Level: $ 900.43
RefStrike: Variance: Term1: 900 0.06647723 0.00126840
Call 900 $ 31.40
Put $ 30.17
Level: $ 901.23
RefStrike: Variance: Term2: 900 0.06366825 0.00401820
Far-Term: Strike:
VIX Calculation:
25.36
Cell F43: = (2/D5)*SUM(G13:G23) - ((D43/E43 - 1)^2)/D5…Note: this performs a variance-type of calculation for the near-term options. Cell G43: =D5*F43*((F7 - 30)/(F7 - F5))…Note: the variance calculation is weighted to be “centered” on 30 days. Cell F47: = (2/D7)*SUM(G27:G37) - ((D47/E47 - 1)^2)/D7…Note: this performs a variance-type of calculation for the far-term options. Cell G47: = D7*F47*((30 - F5)/(F7 - F5)) …Note: the variance calculation is weighted to be “centered” on 30 days. Cell B49: = SQRT((G43 + G47)*365/30)*100…Note: this is the VIX Index and is the square root of the weighted variance calculations from the two option series scaled on an annual basis. General Note: these calculations are performed wit h greater precision than the calculations performed in the CBOE VIX document which will lead to very slight variation in the final VIX calculation.
To summarize the VIX Index calculation, it is a weighted average of volatility based on a series option prices and implied forward prices. The calculation is centered on 30 days and assuming two appropriate option series exist, a similar calculation could be produced for longer or shorter than a 30 day period (adjust cells G43, G47, and B49
7
accordingly). Further, a similar calculation can be produced for option series other than the S&P 500, which is demonstrated in the next section of the paper. SECTION 2: VIX-type Analysis of an Equity Security
Because the VIX calculation is not biased due to a particular option pricing model, the calculation has value beyond S&P 500 options. For example, a VIX-type calculation using two series of equity options could prove to be valuable as a forecast or a technical indicator. A bias will exist in that equity options are American (S&P 500 options used for the VIX are European), but if the forecasted period avoids dividend events or other disbursement events, the bias should be minimal. Finding a series of equity options is not difficult, particularly with practitioner software. However, for this example, delayed quote data from the CBOE website is used (www.cboe.com, using “option chains” from the “quotes” section). The option quotes (time stamped at 1:09 EST or 12:09 CST) are for 9-day and 37-day options on 3M (ticker: MMM). Table 2 reflects the data collected. Table 2: 3M Option Quotes (midpoint) from www.cboe.com Near Term Options (9 days to maturity) Next Term options (37 days to maturity) Strike: Call: Put: Strike: Call: Put: 75 11.75 0.05 75 na na 80 6.90 0.08 80 7.70 0.73 85 2.40 0.60 85 3.80 1.80 90 0.18 3.40 90 1.05 4.05 95 0.05 8.30 95 na na Note: Some of the quote midpoints have been rounded to the nearest penny.
Figure 5 reflects the VIX-analysis of the 3M options (Note: the risk-free rate is simply assumed to be 1.1625% APR for convenience). The Excel template is only altered to reflect the smaller amount of option data. Adjustments are made to the various cell ranges within cell formulas to accommodate the smaller data set (one can do this by
8
putting the data into the previous spreadsheet and then deleting entire rows that are not necessary; Excel adjusts the cell ranges automatically in this instance). Figure 5: VIX analysis of 3M A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
B
C
D
E
F
G
Time to Maturity: Current Time:
12:09:00
Option 1:
9
days
0.02150
years
7.85
days
Option 2:
37
days
0.09821
years
35.85
days
Risk-free Rate:
1.1625%
annual
Near-Term Difference: $ 11.70 $ 6.83 $ 1.80 $ 3.23 $ 8.25
Strike:
DeltaStrike:
75 80 85 90 95
Call $ 11.75 $ 6.90 $ 2.40 $ 0.18 $ 0.05
Put $ 0.05 $ 0.08 $ 0.60 $ 3.40 $ 8.30
5 5 5 5 5
C/P-Near: VIX-Near: 0.05 0.00004446 0.08 0.00006252 1.5 0.00103832 0.18 0.00011114 0.05 0.00002771
80 85 90
Call $ 7.70 $ 3.80 $ 1.05
Put $ 0.73 $ 1.80 $ 4.05
5 5 5
C/P-Far: VIX-Far: 0.73 0.00057096 2.8 0.00193993 1.05 0.00064889
Put $ 0.60
Level: $ 86.80
RefStrike: Variance: Term1: 85 0.0985814 0.00044269
Level: $ 87.00
RefStrike: Variance: Term2: 85 0.058695 0.00456069
Far-term Difference: $ 6.98 $ 2.00 $ 3.00
Strike:
DeltaStrike:
Index Level: Near-Term: Strike:
Call 85 $ 2.40
Far-Term: Strike:
Call 85 $ 3.80
VIX Calculation:
Put $ 1.80
24.67
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The value of 24.67 is above the actual VIX level at the time (VIX level of 22.50), which appears to be appropriate given that an individual equity security should be more volatile than the market as a whole. SECTION 3: Conclusion
Calculating the VIX index using an Excel template is not overly difficult considering the spreadsheet skills expected within the financial analysis profession. However, developing the template is instructive for understanding how the VIX is calculated and with the template in hand, the same index can be computed for different time horizons (not demonstrated in this paper) and for different underlying securities. When applied to equity options, the index calculation can be used as a forward looking risk measure, a technical indicator, or in conjunction with other analysis (especially if the analysis is based on past data only).
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REFERENCES:
Chicago Board Options Exchange (9/18/03). “VIX: CBOE Volatility Index.” (www.cboe.com/micro/vix/vixwhite.pdf)
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