ARE VOLATILITY EXPECTATIONS CHARACTERIZED BY REGIME SHIFTS? EVIDENCE FROM IMPLIED VOLATILITY INDICES



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)withpij = 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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