When to use stdev of prices and when to use stdev of returns as the volatility

[JiriPik]

Hello:
I have seen an inconsistent usage of ways to calculate volatility, some authors use stdev of natural logarithms of returns, some use stdev of returns, some use stdev of price series.

Can you confirm when to use what method for calculating volatility? Most importantly in the VAR / hypothesis testing, CAPM, options pricing etc?

Can your recommend any papers outlining these?

 

[David Harper CFA FRM]

Hi Jiri,

There are many previous forum threads already on this (there is a search box in the upper left); i don't have a paper to recommend. I assume you mean w.r.t. FRM? Briefly b/c I am repeating/summarizing myself from prior threads:

• Exam probably needs to specify (this is a sub-class of a broader point on compound frequency: GARP has explicitly agree that compound freq needs to be specified)
• The advantage of continuously compounded log returns, LN(S1/S0), is they are time-additive; e.g., 2-day continuous return = sum of two 1-day continuous returns. Because EXP(x)*EXP(y)= EXP(x+y).
• Simple (arithmetic) returns, St/S0-1, add cross-sectionally across the portfolio assets (adding the daily returns of assets in a portfolio) but not over time
• In general, it's hard to go wrong with volatility/VaR based on StdDev of series of log returns, LN(S1/S0), because this is technically consistent with our very common underyling assumption that PRICE LEVELS are lognormal (i.e., log return normal --> prices levels are lognormal). When in doubt, this IMO is a safe choice.
• But in practice, especially with VaR, it's also common to just use the simple returns. Even Hull "simplifies" to this return as an approximation on the idea that, for daily returns, they aren't materially different
• I think the capm approaches, being single-period and not concerned with time additivity, tend to use the simple/arithmetic returns (I can't recall if Elton & Gruber ever explicitly address this?). capm gets a little religious on this and i'm too tired to go chase down the threads we have on this
• The volatility of a series of PRICES would be atypical for us. Typically, we are dealing in RETURNS and the volatility of returns, where are default assumption is the log returns LN(S1/S0) are normal, consistent with GBM into Black-Scholes.
• But GARP has definitely employed the simple returns in VaR.
• In short, good to know why each approach has a pro/con. For the exam, they will have to specify but it's unlike this will make any difference w.r.t. daily returns.

Thanks, David