Supplementary results for "Forecasting Realized Volatility with Option-Implied
Variance for Multiple Instruments in MQL5"

Gain = out-of-sample R-squared of HAR-IV minus that of plain HAR, in
percentage points. R-squared is measured against the expanding historical
mean. CW t = Clark-West statistic; p-values are one-sided. NW = Newey-West
lags. Non-overlapping = every h-th forecast only.

Contents
  S1. SPY with option-chain implied variance, all horizons
  S2. Four markets with Cboe indices, all horizons
  S3. Ten equities, both horizons, and the four exclusions
  S4. Option-chain data specification
  S5. Coefficients on the SPY development period
  S6. Trading rules: full specification and results
  S7. Variance risk premium diagnostics in the feed (corrected)


S1. SPY with option-chain implied variance
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Development forecasts January 2022 to June 2025 from data starting May 2019.
Holdout July 1, 2025, to September 4, 2026, opened once.

Sample       h   n    HAR(%)  HAR-IV(%)  Gain   CW t NW 2(h-1)  CW t NW 4(h-1)
Development  1  581   40.5    46.8      +6.3   7.59            7.59
Development  5  581   50.8    56.4      +5.6   4.91            4.90
Development 22  581   38.8    44.3      +5.4   2.68            2.52
Holdout      1  289   35.6    41.2      +5.6   5.17            5.17
Holdout      5  285   38.3    45.5      +7.2   3.68            4.19
Holdout     22  268   32.9    39.7      +6.9   2.30            2.19

Non-overlapping windows only:
Sample       h   n    Gain   CW t   p
Development  5  136   +4.6   3.74   0.0001
Development 22   39   +8.5   2.04   0.021
Holdout      5   51   +7.2   2.57   0.005
Holdout     22   13  +14.1   2.43   0.008

The 22-session horizon was declared underpowered before the holdout was
opened: the holdout holds only 13 non-overlapping 22-session periods. It is
reported and not used for any conclusion.


S2. Four markets with Cboe indices
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Funds paired with their indices, January 2015 to September 4, 2026. The last
60 percent of each sample is scored. Reproduced by har_iv_research.py.

Fund Index  h  scored HAR(%) HAR-IV(%) Gain  CW t   CW t NW4  non-overl. n  CW t   p
SPY  VIX    1  1748   50.8   54.1     +3.3  10.73  10.73     1748          10.73  <0.0001
SPY  VIX    5  1746   58.7   63.7     +5.1   6.67   6.27      350           5.10  <0.0001
SPY  VIX   22  1736   46.3   52.4     +6.2   4.88   4.60       79           2.08   0.019
GLD  GVZ    1  1748   30.2   33.7     +3.5   8.32   8.32     1748           8.32  <0.0001
GLD  GVZ    5  1746   47.2   54.9     +7.8   6.15   5.61      350           5.56  <0.0001
GLD  GVZ   22  1736   46.6   55.6     +9.0   4.75   4.84       79           3.13   0.0009
USO  OVX    1  1748   37.8   40.7     +2.9   7.77   7.77     1748           7.77  <0.0001
USO  OVX    5  1746   51.4   57.4     +5.9   7.14   6.86      350           6.22  <0.0001
USO  OVX   22  1736   44.4   51.7     +7.4   3.83   3.90       79           3.05   0.0011
QQQ  VXN    1  1748   49.8   52.5     +2.7  10.42  10.42     1748          10.42  <0.0001
QQQ  VXN    5  1746   60.4   64.8     +4.5   6.19   5.69      350           5.25  <0.0001
QQQ  VXN   22  1736   48.7   54.7     +6.0   4.18   3.91       79           2.74   0.0031

All twelve gains are positive, and every NW 2(h-1) and NW 4(h-1) p-value is
below 0.0001.


S3. Ten equities, weekly option chains
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Wednesdays, January 1, 2020, to September 4, 2026. The second half of each
series is scored. Weekly sampling means five-session windows do not overlap.
Reproduced by har_iv_chains.py.

Symbol  h   n    scored  HAR(%)  HAR-IV(%)  Gain    CW t   p
SPY     1  329   165     36.5    43.3       +6.8    4.12   <0.0001
SPY     5  328   164     48.2    56.3       +8.1    5.29   <0.0001
DIA     1  253   127     35.3    43.1       +7.7    2.85   0.0022
DIA     5  252   126     57.8    67.4       +9.6    3.78   0.0001
AAPL    1  333   167     30.5    36.6       +6.1    3.57   0.0002
AAPL    5  332   166     28.0    41.9      +13.9    4.66   <0.0001
MSFT    1  305   153     17.2    31.2      +14.0    4.75   <0.0001
MSFT    5  304   152     24.6    49.9      +25.3    6.98   <0.0001
JPM     1  303   152     36.3    40.7       +4.4    3.56   0.0002
JPM     5  303   152     42.7    53.9      +11.2    5.95   <0.0001
XOM     1  253   127     42.3    49.4       +7.1    3.86   0.0001
XOM     5  253   127     53.8    62.9       +9.1    7.36   <0.0001
NVDA    1  331   166      7.8    17.9      +10.1    4.05   <0.0001
NVDA    5  330   165     21.0    30.2       +9.2    3.26   0.0006
GOOGL   1  297   149     -2.8     8.6      +11.4    3.69   0.0001
GOOGL   5  296   148     -8.4    15.0      +23.4    6.04   <0.0001
AMZN    1  335   168      4.0    11.3       +7.3    3.49   0.0002
AMZN    5  334   167      9.8    29.1      +19.3    4.76   <0.0001
CAT     1  277   139     14.5    28.7      +14.1    4.34   <0.0001
CAT     5  277   139     18.8    42.9      +24.2    4.63   <0.0001

Excluded before any forecast was fitted (fewer than 120 usable Wednesdays):
XLF 65, XLE 99, XLK 23, KO 55 usable Wednesdays; median quoted spreads 21,
20, 25 and 25 percent of the option price.

Correlation between plain HAR R-squared and the gain across the ten symbols:
-0.74 at five sessions (two-sided p = 0.014, Student's t-test, 8 degrees of
freedom); -0.59 at one session (p = 0.07, not significant).


S4. Option-chain data specification
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Source       Dolt database post-no-preference/options, table option_chain,
             https://www.dolthub.com/repositories/post-no-preference/options
             read through the DoltHub SQL API, branch master:
             https://www.dolthub.com/api/v1alpha1/post-no-preference/options/master
Downloaded   September 6 to 8, 2026. har_iv_chains.py prints the database
             commit it reads and caches every day it downloads.
Query        SELECT date, expiration, strike, call_put, bid, ask, vol
             FROM option_chain WHERE date = '<day>' AND act_symbol = '<symbol>'
Probed       SPY DIA QQQ IWM EEM EFA TLT HYG XLF XLE XLK XLV GLD SLV USO VXX
             AAPL MSFT AMZN GOOGL META NVDA TSLA JPM BAC XOM JNJ PFE WMT KO
             DIS INTC CSCO T VZ GE BA CAT MCD NKE IBM (41)
Candidates   The 31 present on September 4, 2026: META MSFT XLK CAT XLV SPY
             AAPL TSLA XLE JPM GOOGL NVDA IBM MCD AMZN CSCO INTC XLF DIA GE
             WMT DIS BA KO JNJ VZ XOM BAC PFE T NKE
Fixed first  SPY DIA XLF XLE XLK AAPL MSFT JPM XOM KO
Added later  NVDA GOOGL AMZN CAT, from the same candidates, same code
Passed       SPY DIA AAPL MSFT JPM XOM NVDA GOOGL AMZN CAT
Excluded     XLF XLE XLK KO

Implied variance for one day (variance_for and svix_for_day):
  mid          (bid + ask) / 2
  quote filter bid > 0, mid >= 0.05, (ask - bid) / mid <= 0.25
  forward      put-call parity: median of K + (C - P) * exp(r t) over the 8
               strikes where C and P are closest; r = 0.04
  strikes      out-of-the-money puts below the forward, calls at or above it;
               at least 8 per expiry
  integration  implied volatility (the vol field) interpolated linearly in
               K / F on a grid from 1/1.6 to 1.6, 1,501 points (2,001 for the
               daily SPY series); Black-76 prices; variance =
               2 / (t * F^2) * integral of the OTM price over strike
  horizon      total variance interpolated linearly between the expiries
               around 30 calendar days, then annualized; at least 2 expiries
Sampling     Equities: Wednesdays. SPY series: every weekday from 2019; a day
             whose shared contracts are at least 98 percent unchanged from
             the previous kept day is a holiday repeat and is dropped (30 such
             days). 1,180 sessions remain.
Exclusion    fewer than 120 Wednesdays with a usable implied variance
Realized     daily bars from yfinance, auto-adjusted; overnight gap squared
             plus Garman-Klass; forward windows start the session after the
             quote date


S5. Coefficients on the SPY development period
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In-sample fit on the development period. The weight of the 22-session term
falls from 0.224 to 0.031 at one session once the implied term enters, and the
implied term takes 0.723; at five sessions the 22-session weight falls from
0.243 to 0.090. This is consistent with a thirty-day option price already
carrying a view of the coming month, but it is not proof: the implied term and
the 22-session term are correlated at 0.74, so how the weight is split between
them is partly a matter of collinearity. The design matrix has a condition
number of about 51. The tests evaluate the forecast, not the coefficients.


S6. Trading rules
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Data         SPY option-chain panel, HAR-IV forecasts at 22 sessions, refitted
             on an expanding window, dated January 3, 2022, to June 30, 2025
             (581 sessions).
Payoff       short volatility on a unit of variance notional over the next 22
             sessions: (IV^2 - RV^2) / IV^2. Approximates a variance swap or
             a continuously hedged short straddle. No commissions, bid-ask,
             margin, expiry selection, hedging error or roll.
Forecast     raw = exp(log forecast); smeared = raw times the mean of
             exp(training residuals) (Duan), computed in the training window.
Edge         (IV^2 - forecast) / IV^2.
Terciles     taken over the whole test period, for the control and the model
             rules alike.
Sharpe       annualized from non-overlapping 22-session blocks, averaged over
             the 22 possible starting points.
Break-even   the cost per period at which the rule earns zero; with one unit
             of turnover per active day it equals the mean payoff.

Rule                                              Model  Days  Mean(%)  Sharpe  Hit(%)  Worst(%)
A  short every day                                no     581    6.1     0.35   75.6   -633.6
D  short when IV is in its top third (control)    no     194    8.7     0.63   74.7   -386.9
B  short when edge > 0, raw                       yes    551    4.8     0.29   74.6   -633.6
C  short when edge is in its top third, raw       yes    194    9.1     0.49   69.6   -314.4
E  short if edge > 0, else long, raw              yes    581    3.0     0.22   71.1   -633.6
B  short when edge > 0, smeared                   yes     54   11.7     0.40   68.5   -260.7
C  short when edge is in its top third, smeared   yes    194    5.3     0.23   68.0   -361.3
E  short if edge > 0, else long, smeared          yes    581   -3.9    -0.23   27.9   -260.7

No model rule has a higher Sharpe ratio than the control. Two have a higher
mean payoff. The holdout contained no volatility stress and could not test
the question either way.


S7. Variance risk premium diagnostics in the feed
--------------------------------------------------------------------------
Fraction of dates on which the index variance exceeded the realized variance
of the fund over the next 21 sessions, t+1 to t+21. Measured on the fund's
own bars with the same estimator. Reference only; the terminal does not use
these figures.

Feeds rebuilt with the fixed script on October 30, 2026. Full window from
January 2, 2015 (2,953 dates); recent window from October 26, 2023, 735
dates (October 30, 2023, and 733 dates for SPY). Variance ratio = mean implied over mean realized
variance in the recent window.

Fund / index   Full window (%)   Recent window (%)   Change (points)   Variance ratio
SPY / VIX      84.7              86.9                +2.2              1.39
QQQ / VXN      81.3              79.6                -1.7              1.18
USO / OVX      79.5              78.1                -1.4              1.19
GLD / GVZ      82.8              69.8                -13.0             1.00

For comparison, the version that included the quote date in the window gave,
on the feeds of September 20, 2026: SPY 85.3 and 87.6, QQQ 81.4 and 80.2,
USO 80.8 and 79.8, GLD 82.9 and 68.3. The variance ratio does not use the
forward window, so the fix does not affect it; it moved by 0.01 only because
the data now runs to October 2026.
