The conclusion of the new article [1] is consistent with the previous one [2]; that is, the volatilities of mean-reverting assets are smaller than those of assets that follow the GBM process. The reverse applies to trending assets.

In this post, we are going to investigate whether the mean-reverting/trending property of an asset has any impact on a trading strategy’s PnL volatility.

To do this, we first generate asset prices using Monte Carlo simulations. We evolve the asset prices in both mean-reverting and trending regimes for 500 days. We then apply a simple trading system to the simulated asset prices. The trading system is as follows,

*Go LONG when the Relative Strength Index <40, SHORT when the Relative Strength Index >70*

The picture below shows the Autocorrelation Functions (ACF) of the asset returns. Panels (a) and (b) present ACFs of the trending and mean-reverting assets respectively. It’s clear that the assets are trending and mean-reverting at lag 3, respectively.

The picture below shows the simulated equity curves of the trading strategy applied to the trending (a) and mean-reverting (b) assets. The starting capital is $100 in both cases.

Visually, we do not observe any difference in terms of PnL dispersion. Indeed, the standard deviation of the terminal wealth at day 500 is $18.9 in the case of the trending asset (a), and $17.3 in the case of the mean-reverting asset (b). Is the difference statistically significant? We don’t think so.

This numerical experiment shows that the PnL volatility of a trading strategy has little to do with the underlying asset’ mean-reverting/trending property. Maybe it depends more on the strategy itself? (Note that in this example, we utilize a mean-reverting strategy). What would happen at the portfolio level?

**References**

[1] L. Middleton, J. Dodd, S. Rijavec, *Trading styles and long-run variance of asset prices*, 2021, arXiv:2109.08242

[2] Liao, S.L. and Chen, C.C. (2006), *The valuation of European options when asset returns are autocorrelated*, Journal of Futures Markets, 26, 85-102.

While the VIX index measures the volatility risks, VVIX measures the volatility-of-volatility risks. Its calculation methodology is similar to the VIX’s except that instead of using SPX options it uses VIX options.

To study the relationship between these 2 indices, we first calculated the rolling 20-days correlation of the VIX and VVIX returns from January 2007 to March 2020. The median value of correlation is 0.807 and 25% quantile is 0.66

The figure below presents the rolling 20-days VIX/VVIX correlation for the last 2 years. We also superimposed SPY on the chart. We observe that the correlation is usually high but there are periods where it decreases significantly. The current period is one of those.

The next figure shows the scatter plot of VVIX returns vs. VIX returns. It’s observed that there is a significant population where VIX and VVIX returns are of opposite signs. We subsequently calculated the number of instances where VIX and VVIX move in the opposite direction. This indeed happens 22% of the time.

Some implications of this study are:

- Although the correlation between VIX and VVIX appears to be high, there is a significant number of instances where VIX and VVIX move in the opposite direction. So it’s fair to say that VVIX follows separate price dynamics which is different from the VIX. In other words, VVIX prices in different risks.
- Long VIX options or SPX back spreads are not always a good hedge for an equity portfolio. The hedge can break down.
- At times it’s cheaper to hedge a long equity portfolio using SPX options; at times it’s cheaper using VIX options.
- The speed of VIX mean reversion is greater when VIX is high as compared to when VIX is low.

Recall that the VIX index

- Is a model-independent measure of volatility,
- It contains a basket of options, including out-of-the-money options. Therefore it incorporates the skew effect to some degree.

Plot below shows RV as a function of the VIX index.

We observe that a high VIX index will usually lead to a higher realized volatility. The correlation between RV and the VIX is 0.6397.

For traders who manage fixed-strike options, the use of option-specific implied volatilities, in conjunction with the VIX index, should be considered. In this example, we calculate the one-month at-the-money implied volatility using SPY options. Unlike the VIX index, the fixed-strike volatilities are model-dependent. To simplify, we use the Black-Scholes model to determine the fixed-strike, fixed-maturity implied volatilities. The constant-maturity, floating-strike implied volatilities are then calculated by interpolation.

Plot below shows RV as a function of ATM IV.

We observe similar behaviour as in the previous plot. However, the correlation (0.5925) is smaller. This is probably due to the fact that ATM IV does not include the skew.

In summary:

- There are differences between the VIX index and at-the-money implied volatility.
- Higher implied volatilities (as measured by the VIX or ATM IV) will usually lead to higher RV.

**Footnotes**

[1] In this example, forward realized volatility is historical volatility shifted by one month.

]]>Note that within the usual BSM framework, the underlying asset is assumed to follow a GBM process. So to answer the above question, it’d be useful to use a different process to model the asset price.

We found an interesting article on this subject [1]. Instead of using GBM, the authors used a process where the asset returns are auto-correlated and then developed a closed-form formula to price the options. Specifically, they assumed that the underlying asset follows an MA(1) process,

where β represents the impact of past shocks and *h* is a small constant. We note that and in case β=0 the price dynamics becomes GBM.

After applying some standard pricing techniques, a closed-form option pricing formula is derived which is similar to BSM except that the variance (and volatility) contains the autocorrelation coefficient,

From the above equation, it can be seen that

- When the underlying asset is mean reverting, i.e. β<0, which is often the case for equity indices, the MA(1) volatility becomes smaller. Therefore if we use BSM with σ as input for volatility, it will overestimate the option price.
- Conversely, when the asset is trending, i.e. β>0, BSM underestimates the option price.
- Time to maturity, τ, also affects the degree of over- underpricing. Longer-dated options will be affected more by the autocorrelation factor.

**References**

[1] Liao, S.L. and Chen, C.C. (2006), *Journal of Futures Markets*, 26, 85-102.

*We find that the distributions for ∆t ≤4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent α≈ 3, well outside the stable Levy regime 0 < α <2. .. For time scales longer than ∆t ≈4 days, our results are consistent with slow convergence to Gaussian behavior.*

Basically, the paper says that the equity indices exhibit fatter tails in shorter time frames, from 1 to 4 days. We apply this idea to our breakout system. We’d like to see whether the 4-day rule manifests itself in this simple strategy. To do so, we use the same entry rule as before, but with a different exit rule. The entry and exit rules are as follows,

*Short at the close when Close of today < lowest Close of the last 10 days*

*Cover at the close T days after entry (T=1,2,… 10)*

The system was backtested on SPY from 1993 to the present. Graph below shows the average trade PnL as a function of number of days in the trade,

We observe that if we exit this trade within 4 days of entry, the average loss (i.e. the cost of hedging) is in the range of -0.2% to -0.4%, i.e. an average of -0.29% per trade. From day 5, the loss becomes much larger (more than double), in the range of -0.6% to -0.85%. The smaller average loss incurred during the first 4 days might be a result of the fat-tail behaviour.

This test shows that there is some evidence that the scaling behaviour demonstrated in Ref [1] still holds true today, and it manifested itself in this system. More rigorous research should be conducted to confirm this.

** References**

[1] Gopikrishnan P, Plerou V, Nunes Amaral LA, Meyer M, Stanley HE, *Scaling of the distribution of fluctuations of financial market indices*, Phys Rev E, **60,** 5305 (1999).

To do so, we first calculate the returns of the VIX index. We then determine the quantiles of the return distribution. The table below summarizes the results.

Quantile | 50% | 75% | 85% | 95% |

Volatility spike | -0.31% | 3.23% | 5.68% | 10.83% |

We next calculate the returns of the VIX after a significant volatility spike. We choose round-number spikes of 3% and 6%, which roughly correspond to the 75% and 85% quantiles, respectively. Finally, we count the numbers of occurrences of negative VIX returns, i.e. instances where it decreases to below its initial value before the spike.

Tables below present the numbers of occurrences 1, 5, 10 and 20 days out. As in a previous study, we divide the volatility environment into 2 regimes: low (VIX<=20) and high (VIX>20). We used data from January 1990 to December 2017.

VIX spike > 3% | |||

Days out |
All cases |
VIX<=20 |
VIX>20 |

1 | 56.1% | 54.9% | 58.1% |

5 | 59.7% | 58.4% | 61.8% |

10 | 60.3% | 57.0% | 65.8% |

20 | 61.6% | 57.0% | 69.5% |

VIX spike > 6% | |||

Days out |
All cases |
VIX<=20 |
VIX>20 |

1 | 58.2% | 56.9% | 60.3% |

5 | 62.5% | 62.0% | 63.3% |

10 | 64.0% | 61.7% | 67.6% |

20 | 65.9% | 61.4% | 73.2% |

We observe the followings,

- The greater the spike, the stronger the mean reversion. For example, for all volatility regimes (“all cases”), 10 days after the initial spike of 3%, the VIX decreases 60% of the time, while after a 6% volatility spike it decreases 64% of the time,
- The mean reversion is stronger in the high volatility regime. For example, after a volatility spike of 3%, if the VIX was initially low (<20), then after 10 days it reverts 57% of the time, while if it was high (>20) it reverts 66% of the time,
- The longer the time frame (days out), the stronger the mean reversion.

The implication of this study is that

- After a volatility spike, the risk of a long volatility position, especially if VIX options are involved, increases. We would better off reducing our vega exposure or consider taking profits, at least partially,
- If we don’t have a position prior to a spike, we then can take advantage of its quick mean reversion by using bounded-risk options positions.

*Short at the close when Close of today < lowest Close of the last 10 days*

*Cover at the close when Close of today > lowest Close of the last 10 days*

The Table below presents results for SPY from 1993 to the present. We performed the tests for 2 different volatility regimes: low (VIX<=20) and high (VIX>20). Note that we have tested other lookback periods and VIX filters, but obtained qualitatively the same results.

Number of Trades | Winner | Average trade PnL | |

All | 455 | 24.8% | -0.30% |

VIX<=20 | 217 | 23.5% | -0.23% |

VIX>20 | 260 | 26.5% | -0.37% |

It can be seen that the average PnL for all trades is -0.3%, so overall shorting SPY is a losing trade. This is not surprising, since in the short term the SP500 exhibits a strong mean reverting behavior, and in a long term it has a positive drift.

We still expected that when volatility is high, the SP500 would exhibit some momentum characteristics and short selling would be profitable. The result indicates the opposite. When VIX>20, the average trade PnL is -0.37%, which is higher (in absolute value) than the average trade PnL for the lower volatility regime and all trades combined (-0.23% and -0.3% respectively). This result implies that the mean reversion of the SP500 is even stronger when the VIX is high.

The average trade PnL, however, does not tell the whole story. We next look at the maximum favorable excursion (MFE). Table below summarizes the results

Average | Median | Max | |

VIX<=20 | 0.83% | 0.44% | 10.59% |

VIX>20 | 1.62% | 0.73% | 24.25% |

Despite the fact that the short SPY trade has a negative expectancy, both the average and median MFEs are positive. This means that the short SPY trades often have large unrealized gains before they are exited at the close. Also, as volatility increases, the average, median and largest MFEs all increase. This is consistent with the fact that higher volatility means higher risks.

The above result implies that during a sell-off, a long equity portfolio can suffer a huge drawdown before the market stabilizes and reverts. Therefore, it’s prudent to hedge long equity exposure, especially when volatility is high.

An interesting, related question arises: should we use options or futures to hedge, which one is cheaper? Based on the average trade PnL of -0.37% and gamma rent derived from the lower bound of the VIX, a back of the envelope calculation indicated that hedging using futures appears to be cheaper.

]]> |
4% down |
4% down and bullish |

From 1970 | 40 | 5 |

On average, a 4% down day occurred each 1.2 years, which is probably not a rare occurrence.

We next counted the number of days when the SP500 dropped 4% or more during a bull market. We defined the bull market as *price > 200-Day simple moving average*. Since 1970 there have been 5 occurrences, i.e. on average once every 10 years. We don’t know whether this qualifies as a black swan event, but a drop of more than 4% during a bull market is indeed very rare.

The table below shows the dates of such occurrences. It’s interesting to note that before the February 5 event, the last two 4% drops when *price> 200-day SMA* occurred around the dot-com period.

Date |
%change |

September 11, 1986 | -4.8 |

October 13, 1989 | -6.1 |

October 27, 1997 | -6.9 |

April 14, 2000 | -5.8 |

February 5, 2018 | -4.1 |

]]>

However, if we look at the correlation between SPX daily returns and VIX changes, it’s more or less in line with the long term average of -0.79. So the divergence was not significant.

The implied volatility (VIX) actually tracked the realized volatility (not shown) quite well. The latter happened to increase when the market has moved to the upside since the beginning of the year.

]]>There exist a variety of techniques for calculating the Hurst exponent, see e.g. the Wikipedia page. We prefer the method presented in reference [1] as it could be related to the variance of a Weiner process which plays an important role in the options pricing theory. When H=0.5, the underlying is said to be following a random walk (GBM) process. When H<0.5, the underlying is considered mean reverting, and when H>0.5 it is considered trending.

Table below presents the Hurst exponents for SPX, VIX and VXX. The data used for SPX and VIX is the same as in the previous post. The data for VXX is from Feb 2009 to the present. We display Hurst exponents for 2 different ranges of lags: short term (5-20 days) and long term (200-250 days).

Lag (days) |
SPX |
VIX |
VXX |

5-20 | 0.45 | 0.37 | 0.46 |

200-250 | 0.51 | 0.28 | 0.46 |

We observe that SPX is mean reverting in a short term (average H=0.45) while trending in a long term (average H=0.51). This is consistent with our experience.

The result for spot VIX (non tradable) is interesting. It’s mean reverting in a short term (H=0.37) and strongly mean reverting in a long term (H=0.28).

As for VXX, the result is a little bit surprising. We had thought that VXX should exhibit some trendiness in a certain timeframe. However, VXX is mean reverting in both short- and long-term timeframes (H=0.46).

Knowing whether the underlying is mean reverting or trending can improve the efficiency of the hedging process.

**References**

[1] T. Di Matteo *et al.* Physica A 324 (2003) 183-188