166 Frequently Asked Questions In Quantitative Finance
Sinceα+β<1 this is exponentially decay of the aver-
age to its mean. A much nicer, more realistic, time
dependence than we get from the EWMA model.
In GARCH(p,q)the(p,q) refers to there beingppast
variances andqpast returns in the estimate:
vn=
(
1 −
∑q
i= 1
αi−
∑p
i= 1
βi
)
w 0 +
∑p
i= 1
βivn−i+
∑q
i= 1
αiR^2 n−i.
Why? Volatility is a required input for all classical
option-pricing models, it is also an input for many asset-
allocation problems and risk estimation, such asValue
at Risk. Therefore it is very important to have a method
for forecasting future volatility.
There is one slight problem with these econometric
models, however. The econometrician develops his
volatility models in discrete time, whereas the option-
pricing quant would ideally like a continuous-time
stochastic differential equation model. Fortunately,
in many cases the discrete-time model can be rein-
terpreted as a continuous-time model (there isweak
convergenceas the time step gets smaller), and so both
the econometrician and the quant are happy. Still, of
course, the econometric models, being based on real
stock price data, result in a model for therealand not
therisk-neutralvolatility process. To go from one to
the other requires knowledge of the market price of
volatility risk.
How? The parameters in these models are usually deter-
mined byMaximum Likelihood Estimationapplied to
the (log)likelihood function. Although this technique
is usually quite straightforward to apply there can be
