Forecast Volatility

One of the most important variables in determining the Risk-on and Risk-off instruments prices is the volatility of the underlying instrument, such as BTC or ETH. In The Risk Protocol, we have carried out a systematic study on various attributes of cryptocurrency return volatilities for the top 50 cryptocurrencies (see Gosal, McMurran, and Ding 2022 for details). Our research shows that most stylized facts exhibited in other financial market instruments, such as equities, bonds, foreign exchange rates, etc., also exist in cryptocurrency returns.

For BTC and ETH, there are two significant features of return volatility that are commonly found in other financial data:

1. The leverage effect, which states that future volatility is usually higher if the underlying price goes down than when it goes up by the same amount (see Black 1976).

2. The long-memory property of speculative returns (see Ding, Granger, and Engle 1993), which shows that autocorrelations in absolute or squared returns decay hyperbolically instead of exponentially. The long-memory property is especially evident in high-frequency financial data.

To capture these stylized facts in the return and volatility generating process, we use the two-component GARCH-GJR specification (see Ding, Gosal, and McMurran 2024) to model the volatility process. The conditional variance equation is specified as follows:

$$\begin{align*} \sigma_t^2 &= \frac{\alpha_0}{(1-\beta_1)(1-\beta_2)} + \frac{\alpha_1 + \varphi_1 S_{t-1}}{1-\beta_1 B} \varepsilon_{t-1}^2 + \frac{\alpha_2 + \varphi_2 S_{t-1}}{1-\beta_2 B} \varepsilon_{t-1}^2 \\ &= \alpha_0 + \left[ (\alpha_1 + \alpha_2) + (\varphi_1 + \varphi_2)S_{t-1} \right] \varepsilon_{t-1}^2 \\ &\quad - \left[ (\alpha_1 \beta_2 + \alpha_2 \beta_1) + (\beta_2 \varphi_1 + \beta_1 \varphi_2)S_{t-2} \right] \varepsilon_{t-2}^2 + (\beta_1 + \beta_2)\sigma_{t-1}^2 - \beta_1 \beta_2 \sigma_{t-2}^2 \end{align*} (1)$$

$$\text{where } \varepsilon_t \text{ is the de-meaned returns (residuals) with conditional mean 0 and standard deviation of } \\ \sigma_t, \text{and } S_{t-k} = \begin{cases} 1 & \text{if } \varepsilon_{t-k} < 0, \\ 0 & \text{if } \varepsilon_{t-k} \geq 0 \end{cases} \quad \text{for } k=1,2$$

is the GJR term (see Glosten, Jaganathan, and Runkle 1993) in the conditional variance equation to capture the asymmetric leverage effect of the volatility process. The two volatility components capture both the long-term memory and short-term fluctuation in the volatility process.

Our research shows that the two-component GARCH-GJR model captures the volatility dynamics in cryptocurrencies quite well. Among various GARCH family models, the two-component GARCH-GJR model performed best in terms of both the Bias-stat and Q-stat in backtesting. It also outperformed implied volatility overall.