HARvesting Crypto Volatility?

AlphabetIM Research, June 2023

1. Realized Forecasting with HAR models

Realized volatility, which measures the actual fluctuations in asset prices over a given time period, is a crucial variable for investors, risk managers, and… policymakers, especially on high volatility assets like crypto currencies. Traditional econometric models, such as autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) models, have been widely used for realized volatility forecasting. However, these models often fail to capture more complex dynamics observed in financial time series.

In recent years, heterogeneous autoregressive (HAR) models have gained significant attention as a more flexible and robust approach to forecasting realized volatility. HAR models take into account the heterogeneous nature of market participants’ trading behaviors and the varying time scales at which information is incorporated into prices. These models allow for the integration of multiple lagged volatility measures, capturing short-term, medium-term, and long-term effects on future volatility.

Initially, HAR models were applied to traditional financial markets such as stock markets. [1] Corsi, Audrino, and Reno (2010) introduced the HAR model for forecasting realized volatility, incorporating lagged realized volatility measures of different frequencies. Their study demonstrated the superior performance of HAR models compared to traditional ARCH and GARCH models in capturing the complex dynamics of volatility. The key innovation of the HAR model lies in its incorporation of multiple time windows of realized volatility measures. By considering volatility at different frequencies or time scales, the model accounts for the diverse behaviors of market participants and captures the varying speeds at which new information is incorporated into prices. This approach allows for a more comprehensive understanding of volatility dynamics and improves the accuracy of volatility forecasts. By capturing these different time horizons, the HAR model offers a more nuanced representation of market behavior and enhances the predictive power of volatility forecasting.

Following the success of HAR models in traditional markets, researchers extended their application to the emerging field of cryptocurrency volatility forecasting. The unique characteristics of cryptocurrencies, such as high volatility, liquidity constraints, and market inefficiencies, present distinct challenges for volatility modeling. [2] Bouri, Azzi, Dyhrberg, and Roubaud (2019) investigated the return-volatility relationship in the Bitcoin market, highlighting the importance of understanding volatility dynamics in cryptocurrencies. [3] Dastgir, Demir, and Vigne (2020) compared the performance of HAR models, GARCH models, and machine learning-based models for volatility forecasting in Bitcoin. Their study demonstrated the effectiveness of HAR models in capturing the volatility patterns specific to cryptocurrencies. Furthermore, Hu, Kuo, and Härdle (2021) focused specifically on realized cryptocurrency volatility forecasting, emphasizing the importance of modeling and predicting volatility in the cryptocurrency market.

These studies, along with others such as [4] Corsi and Marmi (2015), [5] Katsiampa (2017), and [6] Dyhrberg (2016), have contributed to the growing body of research on the application of HAR models to cryptocurrency volatility forecasting. By considering multiple frequencies of realized volatility and accounting for heterogeneous trading behaviors, HAR models offer valuable insights into the complex dynamics of cryptocurrency markets while keeping key characteristics of volatility such as clustering, slowly deacaying auti-correlation and leverage effect.

2. BTC Realized Volatility Stylized Facts

Like TradFi realized volatility, we expect similar phenomena for crypto like positive autocorrelation with a long term memory process exhibiting dependencies at multiple time scales. We also look for association of volatility burst to important absolute returns as volatility can occur on both sides of the market compared to equity markets.

To illustrate these stylized facts, we consider 30-days historical volatility for BTC sampled every 30 minutes from Sep 2019 to May 2023 (we take here data from Binance public api). We then draw the autocorrelationfunction with a lag up to 30 days. On the same chart we display the correlation of our 30d realized vol with |r|, which is the absolute value of our daily lag return.

Chart 1. Correlation between 30d BTC vol with lag-vol and lag-absolute return

It provides clear evidence of the existence of long memory in realized volatility with a slowly decaying autocorrelation. Moreover, the figure suggests that past absolute daily returns have a substantial influence on future volatilities. These shocks endure for an extended duration, indicating a persistence that can be interpreted as long memory. Note that we do not explore here the correlation with potential jumps, which obviously exist in crypto markets but needs additional modelization and our purpose is to keep a rather simple model with fewer parameters. In this perspective, we could have also distinguished between positive leverage effect and negative leverage effect, which are both correlated (resp. anti-correlated) to realized vol, but again we try to reduce number of variables. Nonetheless, we will study specially the impact of jumps in further research.

We conclude as [1] that the presence of long-range dependence in the data can be attributed to long-memory data generating process which can be achieved by aggregating only a few heterogeneous time scales. This perspective has been further explored by considering the concept of an additive cascade of realized volatility aggregated across different time horizons with HAR model.

3. HAR and LHAR for Realized Variance

This model can be estimated by OLS with the Newey-West covariance correction for serial correlation. Actually, instead of a classical OLS, we will prefer a WLS by weighting residual with the inverse of variance to take less consideration into extreme variance points. As an example, we can see below the regression for 1d Realized Variance as a function of 1h/1d/7d/30d RV (note that the sampling is 30 min such that var_48 corresponds to 1d RV):

Table 1. WLS regression for HAR model on Realized Variance

As expected, mid frequencies RV have more impact than 1h RV and longer period 30d RV. We give below the complete set of coefficients and models metrics for HAR models on 1d/7d/30d Realized Variance:

Table 2. Coefficients for HAR models on 1d/7d/30 RV

We note that R2 tends to decrease with the tenor suggesting that HAR model is better for shorter term volatility. As documented in the literature, longer tenors have more impact on short-term volatility than short tenors have on longer-term RV. In that sense, coefficients of 1d-RV are quite small compared to other tenors. A tenor seems also more impacted by the corresponding previous realized on the same tenor. If now we add the leverage effect, we get the following results:

Table 3. Coefficients for LHAR models on 1d/7d/30 RV

The leverage is increasing slightly the quality of regressions across tenors. And even if same commentaries as for HAR models remain true, the inclusion of previous absolute return dampens logically the coefficient of 1d-RV.

4. HAR and LHAR for log-Realized Variance

A common transformation is to use the log(RV) for the HRA model and prefer a regression in logs like in [1] and [7]. This will reduce the impact of jumps, which are not addressed explicitlyhere and smooth the regression. Consequently, we expect better R-squared from HAR on log-RV. We can see below both tables on HAR and LHAR with log Realize Variance:

Table 4. Coefficients for HAR and LHAR models on 1d/7d/30 log-RV

We first observe a significative increase in the quality of regression with higher R-squared and reduced RMSE. However, regression is getting better for longer tenor, which is in contrast with models on RV. Here, 1d-RV and 7d-RV are largely predominant for all tenors. As discussed in the previous section, we have a slight benefit from leverage inclusion and dominance of LHAR compared to HAR.

5. Volatility Risk Premium HARvesting

So HAR models and even more LHAR models on RV and log-RV seem interesting to forecast further realized volatility. Could we now use this prediction to harvest BTC volatility risk premium?

We consider the history of ATM implied volatility for BTC from 30 Apr 2022 to 30 Apr 2023 (data come from Deribit exchange) on 1d/7d/30d constant maturity for which we have interpolated among different tenors. We then look at subsequent realized volatility against these implied volatility and we call the difference: realized VRP. We illustrate below the evolution of this metric:

Chart 2. Realized VRP for 1d/7d/30d constant maturity vol selling

Realized VRP is positive on average around 5 vol points cross tenors. Median is much higher as mean is dampened by extremal negative points on crash like Luna in May22 and FTX event in Nov22 as we can see in the statistics summary below:

Table 5. Stats of Realized VRP

This indicates that vol selling remains quite profitable over this period of time even if shocks (on the downside but also on the upside) are frequents and can erase rapidly most of accumulated VRP.

To test the interest of our models, we focus on 7d-RV, which is closer to real PnL than 1d vol selling strategy embedding a much higher strike risk. It will also constitute higher reactivity indicators compared to 30d-RV. We also noted earlier that LHAR is slightly better than HAR model. On this tenor we then form a first indicator where we subtract to the implied variance our estimator of LHAR on RV called “RV LHAR Filter” and a second indicator where we make the difference between log(implied variance) and our estimator LHAR on log-RV and we call it “log-RV LHAR Filter”. Finally, we will filter our 7d systematic vol selling when indicatorsare greater than 1 standard deviation. We manage to increase average PnL and reduce extremal negative loses, validating the opportunity to use such predictive HAR models to harvest BTC volatility risk premium:

Table 6. LHAR estimator to filter systematic 7d volatility selling

Final Thoughts

In this new study, we have explored HAR models applied to BTC volatility, which exhibits the same patterns of clustering and leverage effect similar as what is observed on equity markets. It justifies this approach on RV or log-RV on different tenors where estimators seem to bring good quality forecasts. These predictions can eventually be useful to filer systematic volatility selling to time VRP harvesting between implied vol and subsequent realized, with encouraging results on the weekly tenor. This approach brings questions on term structure strategy that could be managed and improved using these estimators and that will be explored in further research.

As usual, we would be happy to hear your thoughts.

Questions and comments can be addressed to: contact@alphabetim.io

References

1. Corsi, F., Audrino, F., & Reno, R. (2010). HAR modeling for realized volatility forecasting. Journal of Financial Econometrics, 8(2), 304–335.
2. Bouri, E., Azzi, G., Dyhrberg, A.H., & Roubaud, D. (2019). On the return-volatility relationship in the Bitcoin market around the price crash of 2013. Finance Research Letters, 28, 160–164.
3. Dastgir, S., Demir, E., & Vigne, S.A. (2020). Volatility forecasting of Bitcoin: A comparative analysis of GARCH, HAR, and machine learning-based models. Journal of Risk and Financial Management, 13(10), 242.
4. Corsi, F., & Marmi, S. (2015). Bitcoin: More than a cryptocurrency. International Journal of Theoretical and Applied Finance, 18(6), 1550034.
5. Katsiampa, P. (2017). Volatility estimation for Bitcoin: A comparison of GARCH models. Economics Letters, 158, 3–6.
6. Dyhrberg, A.H. (2016). Bitcoin, gold and the dollar — A GARCH volatility analysis. Finance Research Letters, 16, 85–92.
7. Hu, Y., Kuo, C.-C., & Härdle, W.K. (2021). Realized cryptocurrency volatility forecasting. Journal of Risk and Financial Management, 14(2), 63.

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