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# Forecasting Volatility

One of the most important variables in determining the prices of the RiskON and RiskOFF instruments is the volatility of the underlying assets, such as BTC or ETH. TRP has conducted a systematic study on various attributes of cryptocurrency return volatilities for the top 50 cryptocurrencies (see [Gosal, McMurran, and Ding 2022 ](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4954736)for details). Our research shows that most stylized facts observed in other financial market instruments, such as equities, bonds, and foreign exchange rates, also hold for 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](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4921210)) 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 long-term memory and short-term fluctuations in the volatility process.

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


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