Figure 1: Implied Volatilities by Moneyness and Maturity in Data
Panel A: IV by Moneyness
Panel B: IV by Maturity
Notes: This figure depicts the average implied volatilities (IV) by
moneyness and maturity groups over time from January 2018 to August
2023. On each date, three groups are classified such that the moneyness
or maturity is less than or equal to its 20th percentile (blue solid
line), between the 20th and 80th percentiles (red dashed line), and
greater than or equal to the 80th percentile (black dotted line). Panel
A shows IV by moneyness and Panel B shows IV by maturity. All IVs are
expressed in percentage points.
This figure contains two line charts showing time series of average implied volatilities (IV) by moneyness and maturity groups from January 2018 to August 2023. Panel A shows IV by moneyness with three lines representing different moneyness groups: less than or equal to 20th percentile (blue solid line), between 20th-80th percentiles (red dashed line), and greater than or equal to 80th percentile (black dotted line). The blue solid line represents OTM calls, while the black dotted line represents deep OTM puts. Average IV increases and becomes more volatile as moneyness increases toward deeper OTM puts. Panel B shows IV by maturity, where shorter-maturity IVs (blue solid line) display much more volatile patterns than longer-maturity IVs (red dashed and black dotted lines), though the overall IV levels remain similar across maturity dimensions. Note: All IVs are expressed in percentage points.
Figure 2: Autocorrelation Function of Implied Volatilities
Panel A: IV Level by Moneyness
Panel B: IV Level by Maturity
Panel C: ΔIV by Moneyness
Panel D: ΔIV by Maturity
Notes: This figure depicts the autocorrelation functions (ACF) of
implied volatilities (IVs) and their first differences up to 22 lags.
Sample ACFs are computed using the average IVs within three groups
classified such that the moneyness or maturity is less than or equal to
its 20th percentile (blue bar), between the 20th and 80th percentiles
(green bar), and greater than or equal to the 80th percentile (orange
bar). Panels A and B plot the ACF of IVs by moneyness and maturity
groups. Panels C and D plot the ACF of daily changes (first differences)
in IVs. The red dashed lines represent the 95% confidence intervals.
This figure presents four bar charts showing autocorrelation functions (ACF) of implied volatilities up to 22 lags. Panels A and B show ACFs of IV levels by moneyness and maturity groups, respectively. Panels C and D show ACFs of daily changes in IVs by moneyness and maturity. Each panel displays three differently colored bars (blue, green, orange) at each lag representing groups classified by the 20th and 80th percentiles of moneyness or maturity. While ACFs of IV levels look similar across moneyness groups (Panel A), they increase with maturity (Panel B). For daily changes, ACFs vary across moneyness levels (Panel C), with OTM call IVs (blue bars) showing more negative first-order and positive second-order autocorrelation, while ACFs appear similar across maturity groups (Panel D). Red dashed lines represent 95% confidence intervals.
Figure 3: Daily Out-of-Sample RMSEs for SHAR and Tree-SHAR Across Forecast
Horizons
Panel A: h = 1
Panel B: h = 5
Panel C: h = 22
Notes: This figure plots the daily out-of-sample RMSEs of
implied-volatility forecasts for the SHAR and Tree-SHAR models across
forecast horizons \(h=1\), \(5\), and \(22\). Panel A reports the one-day-ahead
horizon, Panel B reports the five-day-ahead horizon, and Panel C reports
the 22-day-ahead horizon. The blue solid line reports the daily RMSE of
the baseline SHAR model, while the red dashed line reports the daily
RMSE of the Tree-SHAR model. Daily RMSEs are computed across all options
observed on each forecast target date and are expressed in percentage
points of implied volatility.
This figure contains three line charts showing daily out-of-sample Root Mean Squared Errors (RMSEs) of implied volatility forecasts for the SHAR model (blue solid line) and Tree-SHAR model (red dashed line) across different forecast horizons. Panel A shows one-day-ahead forecasts (h=1), Panel B shows five-day-ahead forecasts (h=5), and Panel C shows 22-day-ahead forecasts (h=22). In all three panels, the Tree-SHAR error line consistently falls below the SHAR error line throughout the evaluation period, with the difference most pronounced during high-volatility periods (notably around 2020). The gap between the models is particularly visible when daily RMSEs spike. All RMSEs are expressed in percentage points of implied volatility.
Figure 4: Estimated Tree-SHAR Partition on January 4, 2018: h = 1
Panel A: Estimated Tree Structure
Panel B: Induced Moneyness–Maturity Partition
Notes: This figure illustrates the estimated Tree-SHAR partition for
the one-day-ahead forecast horizon on January 4, 2018, using AHBS-fitted
implied-volatility surfaces. Panel A reports the estimated regression
tree. Each internal node shows the selected splitting variable and
threshold, while each terminal leaf reports the number of option
observations assigned to that leaf and the corresponding sum of squared
errors (SSE). Panel B maps the same terminal leaves into the
moneyness–maturity space. The figure shows how the Tree-SHAR algorithm
partitions the option surface into economically interpretable regions
and estimates separate local HAR coefficients within each region.
This figure illustrates the Tree-SHAR partition for one-day-ahead forecasting on January 4, 2018. Panel A displays a decision tree with 6 terminal leaves created through splits on moneyness and maturity. The root node contains all 269,566 observations with SSE of 58.34. The first split separates deep OTM calls (moneyness < 0.96, Leaf 1, 10% of observations). Subsequent splits create regions based on maturity and moneyness thresholds, with each leaf showing observation count and sum of squared errors. Panel B maps these 6 terminal leaves onto the moneyness-maturity space, with moneyness (0.8-1.6) on the x-axis and maturity (40-240 days) on the y-axis. The visualization demonstrates how the algorithm partitions the option surface into economically interpretable regions, with distinct areas for short-dated options, long-dated options, OTM calls, and OTM puts.
Figure 5: Estimated Tree-SHAR Partition on March 17, 2020: h = 1
Panel A: Estimated Tree Structure
Panel B: Induced Moneyness–Maturity Partition
Notes: This figure illustrates the estimated Tree-SHAR partition for
the one-day-ahead forecast horizon on March 17, 2020, using AHBS-fitted
implied-volatility surfaces. Panel A reports the estimated regression
tree. Each internal node shows the selected splitting variable and
threshold, while each terminal leaf reports the number of option
observations assigned to that leaf and the corresponding sum of squared
errors (SSE). Panel B maps the same terminal leaves into the
moneyness–maturity space. Compared with Figure 4, the partition highlights how
the tree structure adapts during the COVID-19 stress period by isolating
more extreme regions (Leaf 1: deep OTM calls) of the moneyness–maturity
surface.
This figure shows the Tree-SHAR partition during the COVID-19 stress period on March 17, 2020. Panel A presents a decision tree with 6 terminal leaves but with different structure than in Figure 4. The root node contains 456,654 observations with SSE of 170.61. The tree now isolates deep OTM calls at a higher moneyness threshold (moneyness < 0.92 for Leaf 1), creating a more extreme partition. Panel B maps these leaves onto the moneyness-maturity space, showing how the tree structure adapts during market stress. Compared to Figure 4, the partition shows more heterogeneous dynamics in OTM calls and near-the-money options, but more homogeneous dynamics in short-term OTM puts. The moneyness cutoffs shift leftward overall, reflecting changing market conditions during the COVID-19 crisis.