- First off, measure teh change in asset value from period to period. Calculate the continuously compounded return over successive days as:

u_i = ln(S_i / (S_i - 1)

where S_i = asset price at time i

-- The mean return of these individual returns is given by:

mean = 1/m.sum(u_n-1)

-- the variance is given by averaging u_i²

The simplest interpretation of the EWMA model is that the day-n volatility estimate is calculated as  a function of the volatility calculated as of day n-1 and the most recent squared return. Depending on the weighting term (upside down Y), which ranges between zero and one, the previous volatility and most recent squared returns will have different impacts.

With the EWMA model the weights are assumed to decline exponentially back through time. The calculation for variance in the model is given by:

variance = weight on previous estimate*variance_n-1 + (1-weight on previous volatility estimate)*u²_n-1

remember that u = ln(S_i / S_i-1)

With EWMA, higher values of λ will tend to... 

low values of λ will tend to... 

minimise the effect of daily percentage returns.

low values will tend to increase the effect of daily percentage returns on the current volatility estimate.

- it's used for estimating volatility.

- A GARCH(1,1) model incorporates the most recent estimates of variance and squared return and also a variable that accounts for a long-run average level of variance.

p = the number of lagged terms on historical returns squared

q = the number of lagged terms on historical volatility

σ²_n = ω + αu²_n-1 + β.σ²_n-1

where:

α = weighting on previous period's return

β = weighting on the previous volatility estimate

ω = weighted long-run variance = λ.V_L

V_L = ω / 1-α-β

1

also λ is not negative

Maximum likelihood estimators select values of model parameters that maximise the likelihood that the observed data will occur in a sample.

It requires formulating an expression or function for the underlying probability distribution of the data and then searching for the parameters that maxmise the value generated by the expression.