VXX is composed of a certain number of first month and 2nd month VIX futures.
Suppose you have :
n1: number of 1st month futures contracts
n2: number of 2nd month futures contracts
n1, n2 are chosen so that the average future lifetime of the VXX contracts is 1 month. The initial n1, n2 values you use can be chosen arbitrarily, they are later adjusted so that the calculated VXX matches the market one.
Lets call p1 and p2 the prices of the first and second month vix futures, which are readily available here from the CBOE where I originally gathered it. The processed data ready to use to calculate the VXX can be downloaded here (data since March 2004).
The price of the VXX is calculated as :
VXX = (p1*n1+p2*n2)/c
As of now
VIX 33.60 +1.73
VX Q1-CF 33.67 +1.57
VX U1-CF 27.65 +0.95
VX V1-CF 26.80 +0.85
VX X1-CF 26.20 +0.75
So by replacing you get VXX=33.4, the slight difference with the market value is simply due to daily demand/offer dynamics.
Note that n1, and n2 change every day
r: the number of remaining days until the expiration of the first future
Then n1 is reduced by n1/r every day, and with that amount you can buy more n2.
So if there are r days left to expiry of future 1 then as of tomorrow n2 will become:
and n1 = n1-n1/r
Therefore when there is backwardation the contract base (n1+n2) gets bigger every day because n1/r 1st month contracts are sold every day to buy more second month ones, but for every 1 contract sold of the 1st month you can buy p1/p2 contracts of the second month and p1/p2>1 by backwardation definition. When you have contango the opposite happens and n1+n2 decreases every day because p1/p2<1.
So the contract base expands in backwardation days giving the VXX quite a considerable tailwind and decreases in contango periods. Even though most of the time VIX futures are in contango, empirically backwardation periods are of higher strength than contango periods as you can see here. Strong persistent backwardation accompanied with increasing prices creates high VXX prices in a short period of time as evidenced by he graph available in this post: the great recession VXX values, making VXX a dangerous instrument to short.
The following implications are good things to understand and remember, they can be easily demonstrated. Based on how the VXX is priced you have the following consequences:
(For a financial math demonstration of these corollaries use the definition that VXX = (p1*n1+p2*n2)/c so if n2 is increasing/decreasing faster due to the presence of backwardation/contango then that affects over time the VXX value directly and more than if there was none or if it was smaller. Also use the fact that futures converge to spot prices.)
0) The Level of backwardation/contango affects the VXX valuation, the bigger the more influence.
1) Backwardation with increasing futures makes the VXX go up fast because not only the futures are growing, which directly influence it, but also because the number of contracts is increasing.
2) Backwardation with decreasing future prices will make the VXX fall or go up depending on the strength of each. But backwardation will act as a tailwind pushing the VXX price up (because the number of vix futures is increasing) even if VIX futures are falling.
3) Contango with increasing future prices will make the VXX fall or go up depending on the strength but contango will act as a headwind (because the number of vix futures is decreasing) pushing the VXX price down.
4) Contango with decreasing future prices will make the VXX fall faster because not only are the future prices decreasing but also the contract base is shrinking.
How and why c is used?:
Its because you start with an arbitrary number of contracts but then you get a huge VXX number or small, but definitely different than then market VXX, but proportional to it, that’s what matters.
You can use as initial value n1=100 and n2=0 just the day after the expiration date of a future. That will correspond to the day when all contracts are 1st month contracts which are exactly one month away.
You can start with c=1 and then chose another c to adjust it to the market data. Otherwise the VXX you calculated by starting with an arbitrary number of contracts will be proportional to the VXX values only but not like them. So you use c to adjust it. For that you take any date where you have a VXX market value like the previous closing date and you calculate c such that the calculated VXX matches the VXX of the previous VXX market close price
n1_ : number future 1 contracts previous day
n2_ : number future 2 contracts previous day
p1_ : close price future 1 contracts previous day
p2_ : close price future 1 contracts previous day
VXX_ : close price VXX market price contracts previous day
So when you divide by c you get that the calculated VXX matches exactly the market VXX on the previous day and all the other calculated VXX will be very similar if the calculations you did were right, if not there is a mistake :), but the calculations are easy because you know by how many contracts to decrease n1 every day (just the available number divided by the days left to expiration) and also how many to increase n2 (based on the future prices relation multiplied by the future 1 contracts sold).
You can like this basically create your own vxx calculation the hardest part is getting the futures data, but that is available in this blog. That can allow you make forecasts, like this one for example, based on futures “future” prices and see how they are influenced by the contango/backwardation periods. Or at least you will be better off to make decisions once you understand its dynamics.
If you did not understand anything just come over here and we discuss it over a caipirinha!
Hope it helped.
“But I don’t understand
how futures are priced when VIX falls
10% in a week?”
The futures are priced based on market expectations, demand and offer, so even though the VIX fell from 48 the 2nd month future was never too high and did not fall much:
Date/1st month future/2nd month future/VIX
08/08/2011 36.55 30.20 48
08/09/2011 30.50 25.40 35.06
08/10/2011 36.00 28.75 42.99
08/11/2011 35.15 27.65 39
08/12/2011 35.00 27.85 36.36
08/15/2011 32.10 26.70 31.87
As you can see the VIX fell from 48 to 31.87 but the 2nd month future just fell from 30.2 to 26.7, and its that future who determines more veavily the VXX price (because the 1st future is about to expire and VXX has little first month futures left)
Plus the contract base is expanding at around a 1% rate per day given backwardation (more 2nd months futures can be bought by selling 1st month ones), so the VXX price is boosted by that contract expansion.
“That’s what I’m saying…
30.2 to 26.7 = 11.67%
VXX isn’t down that much.”
Here is the data:
Date 1st mth 2nd mth Days left VXX (NAV) VXX (market) VIX #Contr 1m #Contr 2m Tot contracts
The day that the second future was @30.2 you had the 1st one @36.55 and the VXX @36.22
And yesterday with the 2nd future @26.7 and the 1st one @32.1 and the VXX @32.18
So actually the VXX fell from 36.22 to 32.18 just a bit less percentually than the 2nd future fall.
Not that the contract base augmented in total from 92.57 to 100 due to backwardation.