### Asymmetric Power Multiplicative Error Model (APMEM)

Both $VIX and implied volatility have been shown to be inefficient estimations of realized volatility, both demonstrating an upward biased forecast. For reasons largely related to current market conditions and the anchoring heuristic, I think the APMEM model makes the most sense … when it comes to volatility forecasting.

That volatility only increases significantly in the face of losses is known as asymmetric volatility. This is partly explained when as price drops, transactions become more leveraged thereby causing the volatility of returns to rise.

The Asymmetric Power Multiplicative Error Model (APMEM) is a volatility forecasting model that attempts to work around the inefficient and upwardly biased estimations of realized vol characteristic of the $VIX and implied volatility. APMEM is based on the anchoring heuristic described by Kahneman.

### Forecast Accuracy

Volatility models are widely used by market practitioners and are considered a fairly good indication of future volatility. However, these models rely on the mean-reverting properties of volatility. Therefore, generally, when the current level of volatility is higher than the long term average, these models will predict lower volatility in the future, and vice versa when the current level of volatility is below average. Furthermore, as you forecast further into the future, in general the forecasts for these models will converge on the long term average level of volatility.

What’s even more interesting is analyzing the properties of the specific asset in question implied by the model — for example the persistence of a given asset’s volatility, which can tell you how the volatility of the asset changes with different returns.

The “log return” is defined as:

ln(p(t)) – ln(p(t-1))

Where p(t) is today’s price, p(t-1) is yesterday’s price, and ln is the natural logarithm function. It can be shown that, for small changes in price, the log returns are a close approximation of simple returns. In econometrics, we typically use log returns because they have more favorable mathematical properties than simple returns and, for small time periods, are generally close to simple returns.

Stephen Harlin

### References

1. Brownlees et al. A Practical Guide to Volatility Forecasting through Calm and Storm (2011)

2. Kişinbay T. Predictive Ability of Asymmetric Volatility Models At Medium-Term Horizons (2003)

3. Mensi, W., Nekhili, R., Vo, X.V., Suleman, T., & Kang, S.H. (2020). Asymmetric volatility connectedness among U.S. stock sectors. The North American Journal of Economics and Finance, 101327.

4. Qin, Y., Hong, K., Chen, J., & Zhang, Z. (2020). Asymmetric effects of geopolitical risks on energy returns and volatility under different market conditions. Energy Economics, 90, 104851.

5. Umar, M., Mirza, N., Rizvi, S.K., & Furqan, M. (2021). Asymmetric volatility structure of equity returns. The Quarterly Review of Economics and Finance.