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Since futures trade on a contract by contract basis, how can you plot the price over an extended period of time? As a reference I have included a plot of VX futures (VIX CBOE volatility index) from investing.com: enter image description here

How do you determine when to roll from one contract to the other? Also is there not a price difference from the contract which is expiring and the one that is being rolled forward to? but yet there are no sudden price discontinuities in the charts.

Could anyone please explain what the method of extracting a continuous price series from a compilation of individual month by month contacts is?

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Note that the series you are showing is the historical spot index (what you would pay to be long the index today), not the history of the futures quotes. It's like looking at the current price of a stock or commodity (like oil) versus the futures price. The prompt futures quote will be different that the spot quote. If you graphed the history of the prompt future you might notice the discontinuity more.

How do you determine when to roll from one contract to the other?

Many data providers will give you a time series for the "prompt" contract history, which will automatically roll to the next expiring contract for you. Some even provide 2nd prompt, etc. time series. If that is not available, you'd have to query multiple futures contracts and interleave them based on the expiry rules, which should be publicly available.

Also is there not a price difference from the contract which is expiring and the one that is being rolled forward to?

Yes, since the time to delivery is extended by ~30 days when you roll to the next contract.

but yet there are no sudden price discontinuities in the charts.

Well, there are, but it could be indistinguishable from the normal volatility of the time series.

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