Threshold Local Volatility Models for Leverage and Mean Reversion

Summary

The document presents a continuous-time local volatility model with piecewise constant drift and volatility coefficients. A price threshold switches the dynamics between regimes, allowing the model to represent both leverage effects, where lower prices are associated with higher volatility, and mean-reverting behavior. The approach is described as a continuous-time analogue of a self-exciting threshold autoregressive model.

An estimation procedure is proposed for the coefficients and threshold. Applied to daily prices for NYSE and S&P 500 stocks across multiple time windows, the estimates show consistent evidence of leverage effects and detect mean reversion most clearly during crisis periods. The reported evidence is empirical and based on historical samples; the document gives no forecasting or trading performance results, and does not specify robustness checks or implementation details.

Key ideas

• A price threshold determines which drift and volatility regime applies.
• The model uses piecewise constant coefficients in a continuous-time price process.
• The framework aims to capture leverage and mean-reversion effects together.
• Historical estimates show leverage evidence, with mean reversion more evident in crises.

Tags

# A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data


# A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.