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Nov 3

Long Term Volatility Forecast

Posted by abiao at 07:24 | Paper Review | Comments(0) | Reads(10178)
A summary of an excellent paper Long term volatility forecast, by Ederington, Louis H. and Guan, Wei, forthcoming at Journal of Financial and Quantitative Analysis (JFQA).

Motivation: Option pricing models and longer-term value-at-risk (VaR) models generally require volatility forecasts over horizons considerably longer than the data frequency. For example, we may be interested in the 10-day VaR of our portfolio, or the model price of options we hold with 30 days time to maturity, for both cases we need to forecast a longer term volatility than one day.

Existing methods & shortcoming: several widely used GARCH-type time-series models, such as GARCH, EGARCH, GJR model, estimated from daily or higher frequency data are used to forecast volatility, however, a long term volatility forecast is generally obtained by successive forward substitution in which the volatility forecast for period t+1 is used together with the model parameters to forecast volatility for period t + 2, the forecast for t + 2 is used to forecast volatility for period t + 3, etc. These are then combined to obtain the “integrated volatility” forecast for the interval from t + 1 through t + N. In other words, today’s volatility receives the same weighting relative to volatility a week ago in forecasting volatility a month from now as it does in forecasting volatility tomorrow. One way to avoid this problem is to match the data frequency to the forecast horizon. For example, if the goal is to forecast volatility over the next month, one could use monthly data to estimate the GARCH model and forecast volatility for month t + 1. But if the forecast horizon is long, the number of observations is sharply reduced and poses a big challenge on data.

Argument: Suppose at the end of trading on a Tuesday, you are forecasting volatility for: i) tomorrow (Wednesday), and ii) Wednesday a week or month forward. Given evidence on volatility persistence, Tuesday’s volatility should be more important than Monday’s in predicting tomorrow’s volatility. But is it much more important than Monday’s volatility in predicting volatility a week or month forward? The successive substitution procedure preserves the relative importance of recent and older observations regardless of the forecast horizon, while the authors hypothesize that differences in relative importance between recent and past observations should decline as the forecast horizon lengthens.

New model: One model in which the relative importance of older and more recent observations varies with the forecast horizon is the absolute restricted least squares (ARLS) model of Ederington and Guan (2005).
long term volatility forecast ARLS model
here ASD(s)t is the standard deviation of returns from t+1 to t+s, r is return. Compared with a simple GARCH model, ARLS allows different coefficient beta for different forecasting horizon s (keep in mind in GARCH, beta is the same), thus is more flexible and overcomes the shortcomings of above mentioned GARCH type models.

Results: below is a sample 10-day volatility forecast, ARLS has the lowest RMSE for almost all assets.
long term volatility forecast performance

Conclusion: long term volatility forecast is an endless project, I personally like this model very much due to its good performance, easy to implement, and on top of these, straightforward idea.

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