The value of financial assets vary on a daily basis. Investors need an indicator to quantify these moves that are often difficult to predict. Supply and demand are the two principal factors that affect changes in asset prices. In return, price moves reflect an amplitude of fluctuations which are the causes of proportional profits and losses. From an investor's perspective, the uncertainty surrounding such influences and fluctuations is called risk.

The price of an option depends on its underlying ability to move or not, or in other words, its ability to be volatile. The more likely it is to move, the more expensive its premium will be closer to expiration. Thus, computing how volatile an underlying asset is good for understanding how to price derivatives off of that asset.

**I – Measuring the Asset's Variation**

One way to measure an asset's variation would be to quantify the daily returns (percent move on a daily basis) of the asset. This brings us to define and discuss the concept of historical volatility.

**II – Definition **

Historical volatility is based on historical prices and represents the degree of variability in the returns of an asset. This number is without a unit and expressed as a percentage. (For more, see: *What Volatility Really Means*.)

**III – Computing the Historical Volatility **

If we call P (t), the price of a financial asset (foreign exchange asset, stocks, forex pair, etc.) at time t and P (t-1) the price of the financial asset at t-1, we define the daily return r (t) of the asset at time t by:

r (t) = ln (P (t) / P (t-1)) with Ln (x) = natural logarithm function.

The total return R at time t is thus:

R = r1 + r2 + r3 + 2 + ... +rt-1+ rt which is equivalent to:

R = Ln (P1 / P0) + ... Ln (Pt-1 / Pt-2) + Ln (Pt / Pt-1)

We have the following equality:

Ln (a) + Ln (b) = Ln (a*b)

So, this gives:

R = Ln [(P1 / P0* (P2 / P1)* ... (Pt / Pt-1]

R = Ln [(P1. P2 ... Pt-1. Pt) / (P0. P1. P2 ... Pt-2. Pt-1)]

And after simplification, we get R = Ln (Pt / P0).

The yield is usually computed as the difference of relative price changes. This means that if an asset has a price of P (t) at time t and P (t + h) at time t + h> t, r the return is:

r = (P (t + t) -P (t)) / P (t) = [P (t + h) / P (t)] – 1

When the return r is small, such as just a few percent, we have:

r ≈ Ln (1 + r)

We can substitute r with the logarithm of the current price since:

r ≈ Ln (1 + r)

r ≈ Ln (1 + ([P (t + h) / P (t)] - 1))

r ≈ Ln (P (t + h) / P (t))

From a series of closing prices for instance, it is enough to take the logarithm of the ratio of two consecutive prices to compute daily returns r (t).

Thus, one can also compute the total return R by using only the initial and final prices.

**▪ Annualized Volatility**

To fully appreciate the different volatilities over a period of a year, we multiply this volatility obtained above by a factor that accounts for the variability of the assets for one year.

To do this we use the variance. The variance is the square of the deviation from the average of daily returns for one day.

To compute the square number of the deviations from the average of the daily returns for 365 days, we will multiply the variance by the number of days (365). The annualized standard deviation is found by taking the square root of the result:

Variance = σ²daily = [Σ (r (t)) ² / (n - 1)]

For the annualized variance, if one assumes that the year is 365 days, and every day has the same daily variance σ²daily we get:

Annualized Variance = 365. σ²daily

Annualized Variance = 365. [Σ (r (t)) ² / (n - 1)]

Finally, as the volatility is defined as the square root of variance:

Volatility = √ (variance annualized)

Volatility = √ (365. Σ²daily)

Volatility = √ (365 [Σ (r (t)) ² / (n - 1)].)

### Simulation

■ The Data

We simulate from the Excel function =*RANDBETWEEN* a stock price that varies daily between 94 and 104.

Resulting in:

■ Computing the Daily Returns

In the E column, we enter "Ln (P (t) / P (t-1))."

**■ Computing the Square of Daily Returns**

In the G column, we enter "(Ln (P (t) / P (t-1)) ^2.”

**■ Computing the Daily Variance**

To compute the variance, we get the sum of the squares obtained and divide by the (number of days -1). So:

- In the F25 cell we get "= sum (F6: F19)."

- In the F26 cell is computed "= F25 / 18," since we have 19 -1 data points to be taken for this calculation.

■** Computing the Daily Standard Deviation**

To compute the standard deviation on a daily basis, we need to compute the square root of the daily variance. So:

- In the F28 cell is computed "= Square.Root(F26)."

- In the G29 cell F28 is shown as a percentage.

**■ Computing the Annualized Variance**

To compute the annualized variance from the daily variance, it is assumed that each day has the same variance, and we multiply the daily variance by 365 with weekends included. So:

- In the F30 cell we have "= F26* 365."

**■ Computing the Annualized Standard Deviation**

To compute the annualized standard deviation, we only need to compute the square root of the annualized variance. So:

- In the F32 cell we get "= ROOT (F30)."

- In the G33 cell F32 is shown as a percentage.

This square root of the annualized variance gives us the historical volatility.