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Is it possible to get a calculation of IV from the volatility on components of the options chain?

EG I have this data:

+--------+-----------------+---------+------------+--------+
| symbol | underlyingPrice | ivMean  | Expiration | strike |
+--------+-----------------+---------+------------+--------+
| aapl   | 141.4           | 13.83   | Aug2015    | 140    |
| aapl   | 141.4           | 13.43   | Aug2015    | 142.5  |
| aapl   | 141.4           | 13.12   | Aug2015    | 145    |
+--------+-----------------+---------+------------+--------+

Can I reach a conclusion about the 30-day IV of this underlying without doing the whole VIX-style calculation?

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This is a somewhat ill-posed question. The "components" in your question are not components, they are just different options and all have different implied volatilities - all for the same underlying.

If you are looking to get single number volatility a-la VIX without the whole VIX calculation, you should use ATM (at-the-money) implied volatility, which is implied volatility of options closest to the underlying price.

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If you want a quick back of the envelop number, just use the IV of the nearest ATM option.

Still the best would be to get the IV which is quite simple. However, three data points are definitely not enough to do this.

Here's a matlab code that allows you to do it (credits to: volopta.com):

% Variance swap calculation using the replication algorithm of
% Demeterfi, Derman, Kamal, and Zou (1999).
% Example using SPX options on 2/28/2011. Linear interpolation of implied
% volatility to create a continuum of IV along a fine grid.
% This example ignores dividends on the S&P 500.
% By Fabrice Douglas Rouah

clc; clear;

% Select the maturity (t = 1 through 9).
t = 5;

% Auxiliary Function
f = @(S, Sb, T) 2/T*((S - Sb) / Sb - log(S/Sb));

% Input the SPX spot price on 2/28/2011 and define the cutoff
S  = 1327.22;
Sb = S;

% Input the implied volatility surface for SPX on 2/28/2011
IV = [ ...
0.66249 0.46257 0.35304 0.24611 0.14728 0.10517 0.10375 0.10375 0.10375 0.10375;...
0.48812 0.35515 0.28751 0.22307 0.16360 0.12388 0.11477 0.11477 0.11477 0.11477;...
0.41881 0.31711 0.26753 0.22104 0.17847 0.14578 0.12511 0.12266 0.12266 0.12266;...
0.38830 0.30064 0.25893 0.22012 0.18468 0.15602 0.13465 0.12496 0.12491 0.12491;...
0.37170 0.29163 0.25422 0.21965 0.18821 0.16231 0.14205 0.12896 0.12559 0.12555;...
0.32688 0.27052 0.24523 0.22215 0.20128 0.18354 0.16873 0.15639 0.14591 0.13695;...
0.31740 0.27327 0.25354 0.23552 0.21915 0.20442 0.19115 0.17915 0.16815 0.15797;...
0.32155 0.28460 0.26801 0.25280 0.23892 0.22632 0.21485 0.20440 0.19474 0.18575;...
0.33407 0.30398 0.29032 0.27771 0.26612 0.25551 0.24580 0.23693 0.22870 0.22103];

% Input the strikes (ATM is column 5)
Strike = [0.5 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5].*S;

%% Input the maturities and the date for 2/28/2011
Today = '02/28/11';
Mat  = {'03/28/11'; '05/31/11'; '08/29/11'; '11/28/11'; '02/28/12'; '02/28/14'; '02/29/16'; '02/28/18'; '03/01/21'};
Time = (datenum(Mat) - datenum(Today))/365;

% Input the discount factors based on the 2/28/2011 yield curve
Dates = {...
'03/03/11'; '03/05/11'; '03/09/11'; '03/16/11'; '03/23/11'; '04/02/11'; '05/02/11'; ...
'06/02/11'; '06/16/11'; '09/15/11'; '12/21/11'; '03/21/12'; '06/21/12'; '09/20/12'; ...
'12/19/12'; '03/19/13'; '06/20/13'; '09/19/13'; '12/18/13'; '03/18/14'; '03/02/15'; ...
'03/01/16'; '03/01/17'; '03/01/18'; '03/01/19'; '03/02/20'; '03/01/21'; '03/01/22'; ...
'03/01/23'; '03/02/26'; '03/03/31'; '03/03/36'; '03/01/41'; '03/01/51'; '03/01/61'};

Years = (datenum(Dates) - datenum(Today))/365;

DF = [...
0.99998500014869800 0.99997500044912600 0.99995111308640100 0.99991222912297800 0.99987334818351100 ...
0.99979198571696100 0.99954964292588800 0.99923668663870500 0.99910794722545700 0.99818605657608200 ...
0.99697717643439500 0.99544225977217400 0.99323396957980800 0.99017081381990600 0.98625297354899800 ...
0.98154157399382700 0.97598196677327000 0.96990691843316500 0.96333781311287600 0.95622588310726600 ...
0.92563237401242000 0.88771864100926600 0.84756561049366600 0.80726790504638900 0.76781566721062900 ...
0.72959238518582100 0.69284469670556100 0.65686678080626400 0.62404851186287800 0.53360097103237400 ...
0.41623864372750800 0.32729883450264000 0.25866419647973800 0.16729143399603000 0.12188702483655400];

% Extract the interest rate for the selected maturity
T  = Time(t);
df = interp1(Years,DF,T);
r  = -log(df)/T;;

%% Select and interpolate the strikes and implied volatilies
% Select the implied volatility for OTM calls and puts
CallV = IV(t,5:10);
PutV  = IV(t,1:5);

% Select OTM strikes for calls, create fine grid, and interpolate the IV
CallK = Strike(5:10);
CallKI = [CallK(1)+0.1:0.1:CallK(end)];
CallVI = interp1(CallK,CallV,CallKI,'linear');

% Select OTM strikes for puts, create fine grid, and interpolate the IV
PutK = Strike(1:5);
PutKI = [PutK(1):0.1:PutK(end)-0.1];
PutVI = interp1(PutK,PutV,PutKI,'linear');

%% Do the required calculations on calls.  
% Rename CallK and CallV for convenience.
n = length(CallVI);
K = CallKI;
V = CallVI;

for i=1:n-1
    Temp(i) = (f(K(i+1), Sb, T) - f(K(i), Sb, T)) / (K(i+1) - K(i));
    if i==1 
        CallWeight(1) = Temp(1);
    end
    CallValue(i) = BSPrice(S, K(i), r, T, V(i), 'Call');
    if i>1
        CallWeight(i) = Temp(i) - Temp(i-1);
    end;
    CallContrib(i) = CallValue(i)*CallWeight(i);
end
Pi1 = sum(CallContrib);

%% Do the calculations on puts. Flip the Vectors for Convenience
n = length(PutVI);
K = fliplr(PutKI);
V = fliplr(PutVI);

for i=1:n-1
    Temp2(i) = (f(K(i+1), Sb, T) - f(K(i), Sb, T)) / (K(i) - K(i+1));
    if i==1
        PutWeight(1) = Temp2(1);
    end;
    PutValue(i) = BSPrice(S, K(i), r, T, V(i), 'Put');
    if i>1
        PutWeight(i) = Temp2(i) - Temp2(i-1);
    end
    PutContrib(i) = PutValue(i) * PutWeight(i);
end
Pi2 = sum(PutContrib);

% Total cost of the portfolio
Pi_CP = Pi1 + Pi2;

%% Results of the replication
% Estimate of fair variance
Kvar = 2/T*(r*T - (S/Sb*exp(r*T) - 1) - log(Sb/S)) + exp(r*T)*Pi_CP;

% Estimate of fair volatility
Kvol = sqrt(Kvar)*100;

% Estimates from Numerix
Numerix = [...
17.4346541536 19.5067089310 20.9119726457 21.3518611465 21.7475857520 ...
21.6785038366 21.4961533188 21.4257801140 20.8642345295];

disp(char('For maturity on '))
disp(Mat(t))
disp(char('Matlab fair volatility'))
disp(Kvol)
disp(char('Numerix fair volatility'))
disp(Numerix(t))
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