Volatility Forecasting

Volatility forecasting is the task of predicting the future volatility of an asset or portfolio, that is, the likely size of its price movements over a coming period. The forecast says nothing about the direction of the price. It is one of the central problems of quantitative finance, because volatility is an input to option prices, risk limits, margin requirements and position sizes.

Volatility is far easier to forecast than returns. Periods of large price moves tend to be followed by more large moves, and calm periods by calm, a pattern called volatility clustering. Volatility also drifts back toward a long-run average over time. These two regularities give forecasting models something to work with, although volatility itself cannot be observed directly and must be estimated even after the fact.

• Historical averages. The simplest forecast is the standard deviation of returns over a recent window, such as the past 30 or 90 days. It weights all days in the window equally and jumps when a large return enters or leaves.
• Exponential weighting. Recent returns are given more weight than older ones, with weights declining smoothly. This method was popularized by the RiskMetrics system published by J.P. Morgan in the 1990s.
• GARCH models. GARCH adds a pull toward a long-run level, so that forecasts for distant horizons converge to an average. Variants allow falls in price to raise volatility more than rises.
• Realized-volatility models. Daily realized volatility measured from intraday data is forecast directly, typically from its own daily, weekly and monthly averages.
• Implied volatility. The implied volatility of traded options is the market's own forecast for the life of the option.
• Machine learning. Tree-based models and neural networks combine many inputs, including volume, news and the volatility of related assets.

The exponentially weighted moving average updates the variance forecast each day:

• $\sigma_t^2$ is the forecast variance for day $t$.
• $\sigma_{t-1}^2$ is the previous day's forecast.
• $r_{t-1}$ is the return on the previous day.
• $\lambda$ is the decay factor, between 0 and 1. RiskMetrics used 0.94 for daily data.

Suppose yesterday's forecast volatility was 1.0 percent, a variance of 1.0 in percent squared, and yesterday's return was -3 percent. With $\lambda = 0.94$, the new variance is $0.94 \times 1.0 + 0.06 \times 9 = 0.94 + 0.54 = 1.48$. The new daily volatility is the square root, about 1.22 percent. Multiplying by the square root of 252 gives an annual figure of about 19.3 percent, up from 15.9 percent the day before. A single large move raises the forecast at once, and its influence then fades by 6 percent a day.

Option prices reflect the views of traders with money at stake and incorporate information that past returns cannot, such as a scheduled earnings release or election. Studies generally find that implied volatility is among the best single predictors of future volatility. It is, however, biased upward on average. Option sellers demand compensation for risk, so implied volatility tends to exceed the volatility that follows, a gap known as the volatility risk premium. Forecasters often combine implied volatility with model-based estimates and correct for the bias.

The full volatility surface also yields forecasts for future periods, known as forward volatility.

Judging a volatility forecast is harder than judging a price forecast, because the true value is never seen. Forecasts are compared with a proxy, usually realized volatility computed from intraday returns, which is far less noisy than a single squared daily return. Accuracy is scored with measures of average error, and competing models are tested on data not used to fit them, as in any exercise in time series forecasting.

A recurring finding is that simple models are hard to beat. A basic GARCH or exponentially weighted model often performs about as well as far more elaborate ones, particularly at short horizons.

• Option pricing and trading. A trader compares a forecast with implied volatility to decide whether options are cheap or expensive, the basis of volatility arbitrage.
• Risk measurement. Value at risk and margin models scale potential losses by forecast volatility.
• Position sizing. Strategies using volatility targeting set each position size in inverse proportion to forecast volatility.
• Portfolio construction. Forecasts of volatility and correlation are inputs to optimizers and to risk-based allocation methods.

Models built on past returns react only after volatility has changed. They cannot anticipate a shock, and they overstate risk for some time after an isolated jump. Volatility also shifts between regimes in ways that a model fitted to one period may not capture. Forecast errors are largest exactly when they matter most, at the onset of a crisis. Longer horizons are harder still: beyond a few months, most models offer little improvement on the long-run average. Forecasts of the variance of a whole portfolio additionally require correlations, which are less stable than individual volatilities.