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.

v_t =w_i +δ_iv_t-1 +β_ir_t-1 +γ_ir_t²_-1 +ζ_t                                         (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 z_t depends only on
regime z_t-1 prevailing over the preceding period. This process is governed by the following
transition probability conditional on past information.

The typical transition probability is denoted by p_ij = Prob(z_t = j | z_t-1 = i)with∑ p_ij = 1.
The probability p_ii 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 p_ij = 1- p_ii .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 - p_ii)^-1 . The stochastic process z_t can be shown to follow an
autoregressive process

z_t = (1- p₁₁ )+ π z_t-1 +ς_t                                                    (6)

where π = p₁₁ + p₂₂ - 1 and ς_t 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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