regimes and examines the issue of whether the presence of leverage effects affects the
significance of mean reversion and/or the length of the memory process.
vt =wi +δivt-1 +βirt-1 +γirt2-1 +ζt (4)
The model parameters are assumed to evolve according to a first-order Markov
process, which is not path-dependent in the sense that the current regime zt depends only on
regime zt-1 prevailing over the preceding period. This process is governed by the following
transition probability conditional on past information.
Prob(zt |
= 1 | zt -1 |
=1)= p11 |
(5-1) |
Prob(zt |
= 2 | zt- |
=1)= p12 |
(5-2) |
Prob(zt |
= 1 | zt -1 |
= 2) = p21 |
(5-3) |
Prob(zt |
= 2 | zt- |
= 2) = p22 |
(5-4) |
The typical transition probability is denoted by pij = Prob(zt = j | zt-1 = i)with∑ pij = 1.
The probability pii in the transition matrix denotes the likelihood that implied volatility
remains in regime z = i given that the same regime prevailed at time t - 1. The probability
of a switch from regime z = i at time t -1 to regime z = j at time t is pij = 1- pii .The
Markov model allows for multiple switches between regimes and the dynamics of these
shifts depend on the conditional transition probabilities, with the average duration of a given
regime i expressed as (1 - pii)-1 . The stochastic process zt can be shown to follow an
autoregressive process
zt = (1- p11 )+ π zt-1 +ςt (6)
where π = p11 + p22 - 1 and ςt denotes innovation terms which are assumed to be
uncorrelated with lagged values of the state variable z . In the model specifications (1) to (4),
implied volatility is defined as a function of the history of stock market returns and realized
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