1.5.3 Maximum likelihood estimate of parameters

[ SLIDE method - lambda - GARCH - forecast - run || VIDEO modem - LAN - DSL ]

Even if a model provides an accurate description of statistical data, it is important that occasional outliners be discarded from the fit. Instead of minimizing a residual between the model and all the data points, a

The same method can be applied when the market increments are normally distributed with a variance n

where the second expression has been obtained after realizing that the maximum of a quantity coincides with the maximum of its logarithm and the minimum of the opposite. Model parameters, such as l in n

where the last approximation guarantees a positive diagonal (i=j) when using a Levenberg-Marquardt solver to locate the zeros of the non-linear function (1.5.3#eq.2a).

The EWMA model (1.5.2#eq.5) has one free parameter (p

are updated before the

The GARCH(1,1) model (1.5.2#eq.6) has three free parameters (p

Maximum likelyhood of the fit is achieved when all three components of the gradient are equal to zero, which defines the parameters using the same Levenberg-Marquardt algorithm to locate the zeros of (1.5.3#eq.2a).

The

It turns out parameter estimation is an important but rather delicate taskin the sense that the result depends strongly on the choice of the time window and abrupt changes in the price history can lead to significant changes in the model. This is of course what the estimation is meant to do, but it is important to make sure that the values that are predicted are not only mathematically correct, but also financially meaningful.

Knowing the present value of the variance model parameters, it is possible to forecast the financial risk into the future. Substituting the long term average w=V(1-a-b) into the recursive definition (1.5.2#eq.6), exercise 1.11 shows that the expected value k days into the future becomes

For the EWMA model a+b=1) so that the expected future variance is equal to the present value. For the GARCH model, a+b<1) the second term decreases in importance for an increasing number of days k, showing that the variance exhibits a

is then generally used to parametrize option pricing models.

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