  This paper is concerned with particle filtering for $\alpha$-stable
stochastic volatility models. The $\alpha$-stable distribution provides a
flexible framework for modeling asymmetry and heavy tails, which is useful when
modeling financial returns. An issue with this distributional assumption is the
lack of a closed form for the probability density function. To estimate the
volatility of financial returns in this setting, we develop a novel auxiliary
particle filter. The algorithm we develop can be easily applied to any hidden
Markov model for which the likelihood function is intractable or
computationally expensive. The approximate target distribution of our auxiliary
filter is based on the idea of approximate Bayesian computation (ABC). ABC
methods allow for inference on posterior quantities in situations when the
likelihood of the underlying model is not available in closed form, but
simulating samples from it is possible. The ABC auxiliary particle filter
(ABC-APF) that we propose provides not only a good alternative to state
estimation in stochastic volatility models, but it also improves on the
existing ABC literature. It allows for more flexibility in state estimation
while improving on the accuracy through better proposal distributions in cases
when the optimal importance density of the filter is unavailable in closed
form. We assess the performance of the ABC-APF on a simulated dataset from the
$\alpha$-stable stochastic volatility model and compare it to other currently
existing ABC filters.
