Historical volatility

Volatility is the magnitude of movement in an asset's price, up or down. It is the one option-pricing input that cannot be looked up, so it has to be estimated, and there are exactly two ways to do it.

Historical volatility looks at what the asset has already done. Take the percentage price changes over some past window — the workbook's illustration is the weekly percentage changes in the Nifty over the past one year — and compute their standard deviation. That number tells you how volatile the index was.

implied-volatility looks at what the market thinks it will do, by running an option pricing model backwards from the traded premium.

The two are different numbers, and the gap between them is where option traders live.

Historical volatility is a plain statistical exercise:

1. Take a series of closing prices over the chosen window — weekly, in the workbook's example, over one year.
2. Convert to percentage changes, one per period.
3. Compute the standard deviation of those changes.

That gives volatility per period. The market convention is to restate it per annum by multiplying by the square root of the number of periods in a year — √52 for weekly data, √252 for daily. The workbook stops at the standard deviation of the weekly changes and does not set out the annualisation step; know the convention, but answer what is asked.

The limitation is structural rather than technical. Historical volatility is backward-looking by construction. It cannot know about a court verdict due next week, a policy meeting, or an earnings release. The workbook's own example is a pharmaceutical stock that has been range-bound for a year at a historical volatility of about 15 per cent, while a pending court judgment means the coming week could easily run at 70 or 80 per cent. Historical volatility has no way of seeing that; the option premium does.

Periodic return  rₜ = (Pₜ − Pₜ₋₁) ÷ Pₜ₋₁

                       √( Σ (rₜ − r̄)² ÷ (n − 1) )
Historical volatility = ──────────────────────────   per period

Annualised HV = Periodic HV × √(periods per year)

Six weekly closing levels of an index, and the weekly returns they imply:

Mean return = (2.0 − 1.5 + 0.5 − 2.5 + 1.0 + 0.5) ÷ 6 = 0.0%

Squared deviations = 4.00 + 2.25 + 0.25 + 6.25 + 1.00 + 0.25 = 14.00
Variance           = 14.00 ÷ 5 = 2.80
Weekly HV          = √2.80 = 1.67%
Annualised         = 1.67% × √52 = 12.1%

Historical volatility is 12.1% a year.

Now put money on the difference. A trader prices a one-month option off that 12.1% and finds the market quoting the same contract at an implied volatility of 19%. On the workbook's vega convention, with a vega of 0.80 and a premium of Rs 124.50 at 12.1%:

Gap in volatility = 19.0 − 12.1 = 6.9 percentage points
Premium uplift    = 0.80% × 6.9 = 5.5%
Market premium    ≈ 124.50 × 1.055 = Rs 131.35
Per contract (lot 50) = Rs 6,568 against Rs 6,225

Rs 343 a lot of pure expectation, sitting on top of what the past would justify. The buyer is paying for a future the history does not contain; the writer is selling it. Whether that Rs 343 is expensive or cheap is not a statistical question at all — it depends on whether anything is actually due to happen before expiry.

Chapter 16.9 (Implied volatility of an option) introduces historical volatility as the first of the two ways to measure volatility, using the weekly-Nifty illustration, and then sets it against implied volatility with the pharmaceutical court-case example. Expect a definition-contrast question — which measure is computed from past prices and which from the option price — and a "when do traders sell options" question, whose answer is when implied volatility is high relative to historical.

• Historical is computed from the underlying's prices; implied is computed from the option's price. Reversing the two is the standard error on this pair.
• Historical volatility is backward-looking and blind to known future events. The workbook's 15% versus 70–80% pharma example exists to make exactly that point.
• Volatility is direction-free. A share falling 3% a week is precisely as volatile as one rising 3% a week.
• The window is a choice, not a fact. One-year weekly and three-month daily data on the same index give different numbers, and neither is wrong.
• Annualisation uses the square root of time, not time. Weekly volatility × 52 is a common and badly wrong shortcut.
• Higher volatility raises calls and puts alike. It is never a directional signal.