Relationship between Asset Volatilities and Spread Volatility - I
Oct 16, 2007
If there are two assets, asset 1 and asset 2 with prices of and ?respectively then the spread of the prices between the two assets would be given by:
Now if we calculate the volatility of asset 1 and asset 2 then we should be able to calculate the volatility of the spread between asset 1 and 2. All we need to do is to calculate the correlation between the two assets (asset 1 and asset 2) and use the formula:
In fact, the above formula holds only if we work with the price volatilities of the assets. That is, the spread volatility given by the above formula holds only if we take the two asset price time series as such. If we work with percentage volatilities, i.e. calculate the volatility on a time series of geometric returns of the assets and their spreads, i.e. the price relatives (natural log of the asset price today divided by the asset price yesterday) the above formula breaks down. Let's see what happens.
Consider a time series of two asset prices, price of asset 1 and the price of asset 2 over ten trading days. Next to that we consider the spread between asset 1 and asset 2 over the same period of ten trading days.
We first calculate the return volatility (i.e. the percentage volatility) of asset 1 and asset 2 independently. For this we take the price relative of the two asset price time series above (for the prices of asset 1 and asset 2) and then take the standard deviation of that series. Remember that price relative (or the geometric return measure) between the price of an asset in period t and the previous period is given by:
This gives a value of 1.645% as the daily volatility of asset 1 and 13.32% as the daily volatility of asset 2 (this number is extremely high, but ignore any real life implications here since these numbers are dummy numbers). Calculating the correlation between the two return series we get a value of -0.119 (minus 0.119).
Now, as we mentioned at the start of this article, there is an exact relationship between the asset prices of asset 1 and asset 2 and the price of the spread between the two given by formula (1). The spread between asset 1 and 2 is simply the difference of the price of asset 1 and asset 2. And given this price relationship we should be able to calculate the standard deviation of the spread between asset 1 and 2 price by equation (2) above.
If we use the two percentage volatilities and the correlation between the (geometric) returns asset 1 and 2 we get the percentage spread volatility as 13.62%:
This should be equivalent to (and exactly equal) to the percentage volatility of the spread between asset 1 and asset 2 (given their exact relationship) if we estimate that independently of the above formula.
We can simply take the natural log of the spread today divided by spread yesterday and get the geometric return series (price relatives) of the spread; and then calculate the volatility of that series (i.e. the standard deviation). The standard deviation (volatility) of the time series of geometric returns of the spread comes out to 3.09%.
There is a huge difference between the values of 3.09% and 13.62%.
However, if we take the price volatilities, that is calculate the standard deviation of the pure price series (without converting them into price relatives or geometric returns) then the two volatilities come out to be same.
Note that the correlation, when calculated on the price series of the assets (instead of the geometric return series of the assets), comes out to -0.27 instead of -0.119.
Now if we calculate the standard deviation of the pure spread series (difference in the price of asset 1 and 2) as shown in the data above we get a value of 3.209. Therefore, volatility turns out to be the same no matter which way we calculate if we work with the price series only.
To be continued .............................
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