volatility. It is not assumed that the explanatory variables follow Markov regime-switching
processes. Under the assumption that the error termsζt in implied volatility models are
normally distributed conditional upon the history 3t-1, the cumulative density function
depends on the regime indicator z . Given the above specifications of implied volatility, the
density function depends on the conditioning variables of past returns and realized volatility
as well. This conditional density of implied volatility can be obtained from the joint density
of implied volatility and state variable as follows.
f (vt | zt ;31-ι, &)=∑ f (vt | zt =i;31-ι, &) ∙ Prob(zt =i | 31-ι, &) (7)
where & represents the vector of model parameters. The unknown parameters in the implied
volatility models are estimated using maximum likelihood. Hamilton (1990) shows that the
maximum likelihood estimates of the transition probabilities can be expressed as
/V
Pj
∑ t=2Prob(zt =j, zt-ι =i |3 τ ;■& )
∑t=2prob(zt-1^iT3TI^0
(8)
√∖
where & denotes the maximum likelihood estimates of model parameters. This estimation
procedure is applied to each model specification for the levels and first differences in
implied volatility in the Japanese and US stock markets.
3. Index Description and Distributional Properties
The empirical evidence on the dynamics of implied volatility is based on
regime-switching tests using the new implied volatility index disseminated by the CBOE
and a similar index for Nikkei 225 index options traded on Osaka Securities Exchange.
Whereas the new VIX index is available from CBOE database, this study uses the Nikkei
implied volatility index introduced in Nishina, Maghrebi and Kim (2006) to measure
volatility expectations in the Japanese market. The new VIX index gathers consensus
information on options market’s expectations about future stock market volatility, without
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