Hurst Exponent as Implied by Option Prices
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Wolfgang Schadner
Abstract
This paper develops a framework to estimate the ex-ante Hurst exponent for financial returns. It builds on the statistical concept of variance scaling and uses the implied variance term-structure as its sole input. Hence, return persistence is quantified in a forward-looking manner. The linkage is derived in a non-parametric fashion, utilizing the stylized fact of long-range dependent volatility. On empirical data of the S&P 500 index I observe that investors believe in trending returns during bull markets and anti-persistence in bearish times. Deviations from complete randomness, specifically serial dependence, often reach economic significance. Therefore, the expected Hurst exponent is strongly fluctuating, which implies that return expectations are of non-linear dynamics. While heavy-tailed distributions are known to de-stabilize markets, I observe that the nonlinear behavior is the potentially greater amplifier of market meltdowns. Expected Hurst exponent is thus a valuable metric for understanding the stability of financial markets. After comparing expectations with realizations, I detect a predictive potential from ex-ante implied on future realized return persistence. This means that the degree of random walk becomes predictable, which rises a broad variety of economic questions. Implications are discussed in the context of investor behavior, market efficiency, the anatomy of meltdowns and investment opportunities.
A Appendix
A.1 Between Variance-Scaling and Auto-Correlation
The subsequent illustrates an easy example for the interaction between variance scaling and auto-correlation. Consider, two series X 1 and X 2. One can think of the two series as cumulative return. Each series starts at 0 and changes either by +1 or −1. In total, the series will have four steps, where the changes x are ordered as
Both of the above series have a mean of zero. Their cumulative aggregates X 1 and X 2 evolve as shown in Figure A.1 below.
Simplified example: trending increments cause the variance of the cumulative return to grow faster, while anti-persistence has the opposite effect.
Both X 1 and X 2 will start and end at a value of 0, hence have the same expected return of zero. However, the former has a positive auto-correlation and the latter a negative one. We now see how the trending increments cause a larger variance for X 1, while variance does not grow for the anti-persistent X 2. Therefore, we recognize that variance of X grows faster under positive auto-correlation, while it grows slower when negative. This concept is key within the fractal analysis of time-series. If we further consider a 50 % chance that the signs of x 1, x 2 are inverted, then the expected values of X 1 and X 2 are zero for all t (and not just for t = 4). Hence, the term-structure of expected values does not necessarily tell whether expected auto-correlation is positive or negative.
A.2 The Rolling-Window Bias
To emphasize that estimating local return persistence is an econometrically not-so-obvious task I set up a simulation example similar to Bloch (2014). The main difficulty in estimating local serial dependence, as described in the main article, is the trade-off between sample size and estimation robustness. To illustrate the potential pitfalls I simulate three sub-paths of fractal Gaussian noise (see Mandelbrot and Van Ness 1968), each with a sample size of 500, but with serial-dependence of H = 0.65, 0.35 and 0.65. These simulations represent a stock’s daily log-returns, which are stacked together to create the overall time-series (see Figure A.2). The simulated series thus starts with trending returns, then jumps to anti-persistence, and then back to trending. Now, we are particularly interested in how different approaches perform in recognizing the actual change in the persistence structure. For this horse race I will compare the MF-DFA as implemented within the paper to rolling-window routines of classic R/S (Hurst 1956) and to rolling one-lag Pearson’s correlations
Upper plot: simulation of three return series (fractal Gaussian noise) with Hurst exponents of 0.65, 0.35 and 0.65, stacked together to an overall price path. Lower plots: comparison between various approaches for estimating local persistence H t . The difficulty is to find a balance between numerical reliability (more observations) and locality (fewer observations). The MF-DFA approach turns out to be the only one to find that balance. Blue lines indicate the segment’s overall H.
For the MF-DFA I will use a window size of 5–21 days. To make the rolling-window R/S competitive in terms of locality, I halve the recommended observation number down to a window-size of 100.[8] This makes the estimated H t numerically less reliable, but we hope to faster observe the change in true H. For the rolling Pearson analysis I choose two window-sizes. First, a size of 100 observations, such that the estimated correlation coefficients are reliable. Second, a window of 21 observations, which will be similarly local as the MF-DFA method. All methods will report right-aligned H t estimates, hence incorporate historical returns only.
The various H
t
estimates are visualized within the lower plots of Figure A.2. Let us study the results method by method. The MF-DFA approach uses a small time-window, hence can be considered to reflect local behavior. We observe this statement to hold upon the simulated series as the H
t
estimate adopts quickly to the change from the trending to the anti-persistent regime, and at the change back to the trending regime. Note that it is natural for empirical mono-fractal sequences, such as the simulated sub-paths, to have small fluctuations in H
t
. Since MF-DFA’s H
t
estimates vary only modestly within each regime and are on average very close to the simulation threshold H, we can also assume them to be reliable. Therefore, MF-DFA has a very good performance in balancing the trade-off between reliability and locality. With an eye on the rolling R/S analysis, we shall observe that the very opposite is the case. Here, the H
t
deviates much stronger from the true H level. Hence, the R/S fails to provide reliable persistence estimates. Especially at the switch from H = 0.35 to H = 0.65 we can further observe that the rolling window method also takes a long time to recognize the true change in persistence. As a consequence, the rolling-window routine is also improper in terms of capturing local correlation behavior. A similarly long-passage of time can be recognized at the Pearson correlation
Comparison of MF-DFA to rolling-window methods: setting and associated problems.
| Method | Window-size | SSE | Problem |
|---|---|---|---|
| MF-DFA | 21 | 5.12 | – |
| Rolling R/S | 100 | 28.32 | Window too short for reliable H t estimates. Also, window too large to reflect current behavior. |
| Pearson | 100 | 10.88 | Window too large to reflect current behavior. |
| Pearson | 21 | 48.97 | Window too short for reliable H t estimates. |
A.3 Economic Significance
The economic significance of serial-dependence is assessed upon simulated series of mono-fractal sequences for different levels of ρ. Simulations are made for ρ ∈ [−0.25, 0.25] with an interim step size of 0.01. For each level of ρ I simulate 103 many mono-fractal series with a length of 1,024 observations/days. Each simulation is then back-tested for a correlation-bet. This strategy expects a positive return for t + 1 if {r t , ρ t > 0} (continuation) or if {r t , ρ t < 0} (reversal). The strategy invests 1 when expected returns are positive, and −1 if negative. Transaction costs are considered in the size of 1.25 basis points, following the report of CME Group (2016). The mean return of the simulations is set to zero and standard deviation to 0.2 (annual). The benchmark for the strategy will be a long-only investment, which has by construction a mean return of 0. Therefore, I will use the mean return to assess whether the strategy is able to create value or not. Given that the strategy is a pure correlation bet, its profitability will rise the greater the serial-dependence ρ deviates from 0. The output of this simulation analysis is summarized in Figure A.3. I define economic significance as the level of ρ from which on the correlation-bet out-performs the benchmark in 95 % of the cases. These levels of economic significance are found at ρ = ±0.10.
![Figure A.3:
Estimating the level of serial-dependence ρ from which on it is profitable to make auto-correlation bets. Each bar plot summarizes the strategy’s mean return for 103 many simulations at a specific level of ρ. The blue points indicate the lowest 5 % of returns. The analysis indicates that serial-dependence ρ is economically significant whenever it is outside the [−0.10, 0.10] range.](/document/doi/10.1515/snde-2025-0099/asset/graphic/j_snde-2025-0099_fig_008.jpg)
Estimating the level of serial-dependence ρ from which on it is profitable to make auto-correlation bets. Each bar plot summarizes the strategy’s mean return for 103 many simulations at a specific level of ρ. The blue points indicate the lowest 5 % of returns. The analysis indicates that serial-dependence ρ is economically significant whenever it is outside the [−0.10, 0.10] range.
A.4 Crash Density
This section provides a brief analysis of the linkage between the serial correlation structure and market crashes, visualized in Figure A.4. The procedure behind the figure is as follows. The first step is to compute the empirical 5 % Value-at-Risk (VaR) of the daily log-returns. Next, we are interested into the level of ρ t and the probability that the next day return will be worse than the overall VaR. To assess this density, ρ t is classified into intervals of length 0.04 with exception to the outer ones being broader to capture at least 50 observations. The ρ t intervals are thus {[−0.47, − 0.32), [−0.32, − 0.28), [−0.28, − 0.24), …, [0.44, 0.52]}. Then, for each interval, I have computed the density of next day returns being worse than the VaR. This allows the following interpretation. In the case that crashes would be independent of ρ t , then the density should be constant at 5 %. However, Figure A.4 contradicts this statement. We can observe a convex pattern: the lower ρ t , the greater the probability for the next day return to be worse than the overall VaR. This means that market crashes are more likely to occur for low levels of expected auto-correlation ρ t . Following this interpretation, the measure becomes an interesting metric for the stability of financial markets.
Density of the t + 1 return to be worse than the overall 5 % Value-at-Risk with respect to expected auto-correlation ρ t . The convex pattern can be interpreted as future crashes are more likely to occur when ρ t is low.
A.5 Implied versus Future-Realized Auto-Correlation
See Figure A.5
The S&P 500’s time-conditional serial-dependence as implied by option prices versus its future realization. The relationship appears to be well approximated under a linear model.
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