Forecasting Asset Return Volatility from Past Price Data

Forecasting Volatility With Price Thresholds: This paper develops a specification of return dynamics that incorporates price thresholds for volatility forecasting. The new forecasting approach is to specify a time-varying transition matrix of the Markov Switching (MS) model in which the relative position of an asset price to its threshold determines future volatility. This price threshold is defined by multiplying an empirically determined ratio to the observed moving average.Then, the approach relates the probabilities of the current price crossing over the price thresholds to the transition probabilities of the states. The model has a closed-form likelihood, and its parameters can be estimated by the maximum likelihood estimation.The paper evaluates the point forecast, interval forecast, and density forecast by the proposed model and the competing models, such as the constant transition probability MS model and GARCH (1,1) model, from the out-of-sample of CRSP S&P 500 return data. As a result, the proposed model outperforms competing models for forecasting the return distribution.

Forecasting Volatility With Multi-State Price Threshold: This paper extends the three-state Price Threshold Volatility forecasting model (PTV-3) developed by Kato (2021). The original PTV model assumes only a limited numberof states in the economy, which does not have enough volatility components to match the variety of magnitude of volatility in the real world. The new model extends the number of states to 2k+ 1 by expanding the time-varying transition probability matrixwith one single parameter added to the PTV model. This multi-state PTV model (PTV-M) is parsimonious and has a closed-form likelihood. Therefore, the maximum likelihood estimation can apply to its parameters. The paper evaluates its performance of the point forecast, the interval forecast, and the density forecast for various state numbers k based on the out-of-sample of CRSP S&P 500 return data. As a result, the increase in the number of states improves the return distribution and the performance of out-of-sample forecasting.