Estimation in Financial Models - RiskLab

1.2 Discrete time
In the previous section we dealt with continuous time stochastic volatility
models based on systems of stochastic dierential equations. In the discrete
time approach we consider time series models which are systems of stochastic
dierence equations. In analogy to the continuous time approach we assume
the stock price S n , S n > 0, to follow the process
S n = S n ( n + n z n ); (1.6)
with and dependent on time and z n N (0; 1). We consider the logarithm
of the stock price H n ln S n . With the notation
we derive the process
H n = y 1 + :::+ y n ;
H n = y n =( n , 2 n
2 )+ nz n : (1.7)
We will concentrate on three dierent discrete stochastic volatility models.
First we consider a rather simple model in which the conditional variance of
the time series fh n g follows a logarithmic AutoRegressive (log-AR) process
(see Jacquier, Polson and Rossi [42]).
I. log-AR(p)/stochastic volatility:
v n = 0 + 1 v n,1 + :::+ p v n,p + ~z n ; (1.8)
with v n =ln 2 n and (z n ; ~z n ) independent N (0; 1).
The famous AutoRegressive Conditional Heteroskedastic (ARCH) model proposed
by Engle [22] will be the second model to look at. The key insight offered
by the ARCH model lies in the distinction between the conditional and
the unconditional second order moments. While the unconditional covariance
matrix for the variables of interest may be time invariant, the conditional
variances and covariances often depend non-trivially on the past.
II. ARCH(q):
2 n = 0 + 1 y 2 n,1 + :::+ qy 2 n,q : (1.9)
A long lag length q and a large number of parameters are often needed in empirical
applications of ARCH(q). To avoid this problem Bollerslev introduced
the Generalized ARCH (GARCH) model [12].
III. GARCH(p; q):
2 n = 0 + 1 y 2 n,1
+ :::+ q y 2 n,q + 1 2 n,1
+ :::+ p 2 n,p: (1.10)
For a discussion of AR, ARCH and GARCH models see section 3.2.
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