hist-vol calc using price relatives – right vs wrong ways #Piroz

(See also post on var swap PnL)

You asked me how historical volatility is computed from daily closing prices. There’s an over-simplification in my answer.

Suppose we have daily closing prices

Day 1) p1 = $30.00
Day 2) p2 = $34.50
Day 3) p3 =…
Day 4) p4
Day 5) p5

First compute p2/p1 as PR2 (price relative over day 2), p3/p2 as PR3… p5/p4 as PR5. I was CORRECT here.

Then we SHOULD compute the Natural log of PR2, PR3, PR4 and PR5. These natural logs are known as “continuously-compounded-rate-of-return” or un-annualized “daily-realized-vol”, and are denoted r2, r3, r4, r5. I missed this step.
These r2, r3, r4, r5 look like low percentages like 8.2%, 3.1%, 4%, 15%…
Realized volatility is defined as standard deviation of these percentages, usually assuming zero mean. I was right at this step.
Since the r values look like percentages, so does their stdev.

There’s also a capital “R” denoting (PriceRelative-100%). Since PR2 = 115%, so R2 = 15%. R is widely known as “Return”, return on investment, or investment return etc.

R2 is always slightly above r2 = 13.9762%. For small price movements, R is a very close approximation of r.
My oversimplification was to misuse R in place of r.

I have since verified my calc with an ex-quant and developers at a volatility trading desk.

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