Hi
@hsmirror They both should count in the same direction where "+" is forward in time and "-" is backward in time: if S(i) is today then yesterday is S(i-1) and if σ^2(n) is today's (aka, updated) variance estimate then σ^2(n-1) is yesterday's (aka, most recent) variance, and we can refer to an estimate of the expected E[σ^2(n+t)] in +t days.
Today is Friday, say the closing prices for the week were (Mon = $10.00, 14, 16, 18, Fri = $20.00). Five prices gives us four returns, in sequence: ln(14/10) = 33.65%, ln(16/14) = 13.35%, ln(18/16) = 11.78%, and most recently ln(20/18) = 10.54%. Each return is ln[S(i)/(i-1)] or ln[S(n)/(n-1)] or ln(price_day/price_previous_day), or even ln[S(i+1)/(i)] because (i) is just an index.
That's a vector of returns ordered in time: (33.65%, 13.35%, 11.78%,
10.54%) where 10.54 is the most recent return. EWMA updates the most recent variance (i.e., computed at the end of yesterday after yesterday's price close); semantically, that might updating yesterday's (end of day) variance or today's (beginning of day) variance to today's (end of day) variance or even tomorrow's (beginning of day) variance. Maybe yesterday's variance was σ^2(n-1) = 15.55% such that the updated variance estimate is given by σ^2(n) = λ*σ^2(n-1) + (1-λ)*r^2(n-1) = λ*15.55% + (1-λ)*
10.54%^2.
I think I'm following Hull:
"Define σ(n) as the volatility of a market variable on day n, as estimated at the end of day n-1. The square of the volatility, σ^2(n), on day n is the variance rate. We described the standard approach to estimating σ(n) from historical data in Section 15.4. Suppose that the value of the market variable at the end of day i is S(i). The variable u(i) is defined as the continuously compounded return during day i (between the end of day i-1 and the end of day i): u(i) = ln[S(i)/S(i-1)]" -- Hull, John C.. Options, Futures, and Other Derivatives (p. 520). Pearson Education. Kindle Edition.
... if it helps, but I think the reason I don't get bogged down in this notional aspect is that I actually have to calculate it often. When you perform the calculation (given a price vector as input), you'll see that it's just a series (vector) of prices that determine the returns and EWMA is updating those returns. Prices are
points in time, the returns are immediately after each price but really describe a function
during the day (between two points in time), and the volatility is "just a statistic" (itself unobservable) that summarizes a vector of returns, EWMA being one of an infinite variety. Hope that helps. If you click thru to the actual YouTube comments, you'll see that I do have a mistake in the XLS, but I think it's different than your point (although it wouldn't help clarify your point either). I hope that's helpful,
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