Weighted historical volatility


The weighted historical volatility is similar to the weighted average, where we define a series of conditions and different associated weights that we will apply to the observations of the sample.

In other words, we assign more or less weight to certain observations in the sample following a given criterion. In this way, we will only give relevance to the observations that are important for our study.

Weighted historical volatility formula

Superscript i represents the criterion that we want to apply in the weighting. The subscript t represents the observation we are using.

  • pit is the weight of criterion i for observation t, where pi1, pi2,…, piN
  • zt: is the profitability of the observation t.
  • zt: is the profitability of the observation t.
  • z–: is the average value of returns.

To adjust the parameter p in reality it would have to be between 0 and 1. However, it can be simplified and larger natural numbers used as in the example. When we want to adjust the parameter pto reality in a much more precise way, we will use the ARCH and GARCH models.

Weighted historical volatility example

We use the same example of the quote for AlpineSki exposed in the concept of historical volatility. We find two weighting conditions:

  • Depending on the weather forecast: We will assign more weight to the months that have the most similar environmental conditions.
  • Temporal effect: Since we want to estimate future volatility in the short and medium term, we will assign more weight to the closest observations and less weight to the more distant observations.

So, since we have two criteria: time and time effect, we can calculate:

  • Historical volatility weighted by time.
    • Superscript i: weather.
  • Historical volatility weighted by the time effect.
    • Superscript i: temporary effect.

Time-weighted historical volatility

Investors are concerned about the volatility of the stock over the next year. The weather forecasts are heavy rains and low temperatures.

Apart from the returns, we have the temperatures. We are going to use time as a variable to assign the weights. So based on the weather forecast, we will assign more weight to the cold months and less weight to the warmer months.

Assigning more weight to the returns of the cold months and less weight to the returns of the warmer months, we obtain a volatility of 4.931%.

So we went from a historical volatility of 6.98% to a time-weighted historical volatility of 4.93%. Given the weather forecast, it would be more appropriate to inform investors of 4.93% volatility rather than 6.98% volatility.

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