  In financial markets, low prices are generally associated with high
volatilities and vice-versa, this well known stylized fact usually being
referred to as leverage effect. We propose a local volatility model, given by a
stochastic differential equation with piecewise constant coefficients, which
accounts of leverage and mean-reversion effects in the dynamics of the prices.
This model exhibits a regime switch in the dynamics accordingly to a certain
threshold. It can be seen as a continuous-time version of the Self-Exciting
Threshold Autoregressive (SETAR) model. We propose an estimation procedure for
the volatility and drift coefficients as well as for the threshold level.
Parameters estimated on the daily prices of 348 stocks of NYSE and S\&P 500, on
different time windows, show consistent empirical evidence for leverageeffects.
Mean-reversion effects are also detected, most markedly in crisis periods.
