QS - Detail Report

Quick Overview

Timeframe	C-C DoR σ (Historical)	H-L DoR σ (Historical)	O-C DoR σ (Historical)	5Y ATRP (Avg)	Binomial IV (Period)	Yahoo IV (Period)	1σ Lower (68.27%)	1σ Upper	2σ Lower (95.45%)	2σ Upper
Daily	6.78%	4.85%	4.99%	7.21%	5.26%	5.30%	6.53	7.25	6.19	7.64
Weekly	17.39%	18.70%	15.16%	17.61%	11.58%	11.67%	6.13	7.73	5.46	8.67
Monthly	45.88%	60.06%	27.02%	38.05%	24.12%	24.30%	5.41	8.76	4.25	11.14
Quarterly	75.70%	200.95%	51.38%	75.23%	41.77%	42.09%	4.53	10.45	2.98	15.86
Annual	—	—	—	—	83.54%	84.18%	2.98	15.86	1.29	36.58
Option Time Horizon	—	—	—	—	27.31%	27.52%	5.24	9.04	3.98	11.88

Calculation Notes: Computed at 2026-02-22T14:24 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $6.88 | Strike (K): $7.50 (first OTM call ≥ spot) | Observed Mid-Price: $0.5250 (Bid $0.4200 / Ask $0.6300) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 83.54% (CRR binomial tree, American, n=20 steps) | BS IV: 84.22% (European Black-Scholes) | Yahoo IV (Annual): 84.18% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.97. Further, an Excess Kurtosis of 28.72 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.17%	3	0.22%	0.22%
-20.17% to -15.08%	6	0.43%	0.65%
-15.08% to -9.99%	27	1.95%	2.60%
-9.99% to -4.90%	166	11.99%	14.60%
-4.90% to 0.18%	574	41.47%	56.07%
0.18% to 5.27%	429	31.00%	87.07%
5.27% to 10.36%	118	8.53%	95.59%
10.36% to 15.45%	29	2.10%	97.69%
15.45% to 20.54%	13	0.94%	98.63%
Greater than 20.54%	19	1.37%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.60%	6.97%	82.95%	68.27%
±2σ	-13.38%	13.75%	96.10%	95.45%
±3σ	-20.17%	20.54%	98.41%	99.73%

Descriptive Statistics

Mean	0.184%
Standard Error	0.182%
Median	-0.370%
Mode	0.000%
Standard Deviation	6.784%
Sample Variance	46.022%
Kurtosis	28.72
Skewness	2.97
Range	128.241%
Minimum	-40.841%
Maximum	87.400%
Count	1384

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.774%	633	45.74%	2.183%
Negative	-3.781%	732	52.89%	-2.000%
Zero	0.000%	19	1.37%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.08%
20	1 Month	6.67%
60	1 Quarter	5.98%
250	1 Year	7.96%
750	3 Years	6.75%
1,250	5 Years	7.21%
2,500	10 Years	7.69%
5,000	20 Years	7.69%
12,500	50 Years	7.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.22. Further, an Excess Kurtosis of 19.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -7.07%	0	0.00%	0.00%
-7.07% to -3.43%	0	0.00%	0.00%
-3.43% to 0.21%	2	0.14%	0.14%
0.21% to 3.85%	201	14.51%	14.66%
3.85% to 7.49%	655	47.29%	61.95%
7.49% to 11.13%	325	23.47%	85.42%
11.13% to 14.77%	116	8.38%	93.79%
14.77% to 18.40%	40	2.89%	96.68%
18.40% to 22.04%	20	1.44%	98.12%
Greater than 22.04%	26	1.88%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.63%	12.34%	87.29%	68.27%
±2σ	-2.22%	17.19%	95.96%	95.45%
±3σ	-7.07%	22.04%	98.12%	99.73%

Descriptive Statistics

Mean	7.487%
Standard Error	0.130%
Median	6.207%
Mode	5.263%
Standard Deviation	4.852%
Sample Variance	23.540%
Kurtosis	19.98
Skewness	3.22
Range	63.163%
Minimum	0.102%
Maximum	63.265%
Count	1385

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.487%	1385	100.00%	7.487%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Left Skewed (Negative Asymmetry) with a Skewness factor of -0.56. Further, an Excess Kurtosis of 4.01 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -14.65%	14	1.01%	1.01%
-14.65% to -10.90%	13	0.94%	1.95%
-10.90% to -7.16%	49	3.54%	5.49%
-7.16% to -3.41%	161	11.62%	17.11%
-3.41% to 0.33%	439	31.70%	48.81%
0.33% to 4.08%	456	32.92%	81.73%
4.08% to 7.83%	184	13.29%	95.02%
7.83% to 11.57%	45	3.25%	98.27%
11.57% to 15.32%	18	1.30%	99.57%
Greater than 15.32%	6	0.43%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.66%	5.33%	76.46%	68.27%
±2σ	-9.65%	10.32%	94.58%	95.45%
±3σ	-14.65%	15.32%	98.56%	99.73%

Descriptive Statistics

Mean	0.334%
Standard Error	0.134%
Median	0.415%
Mode	0.000%
Standard Deviation	4.994%
Sample Variance	24.942%
Kurtosis	4.01
Skewness	-0.56
Range	54.236%
Minimum	-32.973%
Maximum	21.263%
Count	1385

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.624%	747	53.94%	1.955%
Negative	-3.649%	615	44.40%	-1.620%
Zero	0.000%	23	1.66%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.79. Further, an Excess Kurtosis of 16.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -51.05%	1	0.35%	0.35%
-51.05% to -38.00%	1	0.35%	0.69%
-38.00% to -24.96%	2	0.69%	1.39%
-24.96% to -11.91%	34	11.81%	13.19%
-11.91% to 1.13%	135	46.88%	60.07%
1.13% to 14.18%	76	26.39%	86.46%
14.18% to 27.22%	27	9.38%	95.83%
27.22% to 40.27%	5	1.74%	97.57%
40.27% to 53.31%	2	0.69%	98.26%
Greater than 53.31%	5	1.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.26%	18.52%	82.99%	68.27%
±2σ	-33.66%	35.92%	96.18%	95.45%
±3σ	-51.05%	53.31%	97.92%	99.73%

Descriptive Statistics

Mean	1.131%
Standard Error	1.025%
Median	-1.067%
Mode	-56.632%
Standard Deviation	17.393%
Sample Variance	302.505%
Kurtosis	16.49
Skewness	2.79
Range	183.905%
Minimum	-56.632%
Maximum	127.273%
Count	288

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.154%	128	44.44%	5.846%
Negative	-8.541%	159	55.21%	-4.715%
Zero	0.000%	1	0.35%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	16.76%
12	1 Quarter	14.44%
52	1 Year	19.59%
156	3 Years	16.72%
260	5 Years	17.61%
520	10 Years	19.63%
1,040	20 Years	19.63%
2,600	50 Years	19.63%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.21. Further, an Excess Kurtosis of 24.86 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -34.42%	0	0.00%	0.00%
-34.42% to -20.39%	0	0.00%	0.00%
-20.39% to -6.37%	0	0.00%	0.00%
-6.37% to 7.66%	14	4.84%	4.84%
7.66% to 21.69%	175	60.55%	65.40%
21.69% to 35.71%	70	24.22%	89.62%
35.71% to 49.74%	16	5.54%	95.16%
49.74% to 63.77%	6	2.08%	97.23%
63.77% to 77.79%	3	1.04%	98.27%
Greater than 77.79%	5	1.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.98%	40.39%	90.31%	68.27%
±2σ	-15.72%	59.09%	96.54%	95.45%
±3σ	-34.42%	77.79%	98.27%	99.73%

Descriptive Statistics

Mean	21.687%
Standard Error	1.100%
Median	16.751%
Mode	20.259%
Standard Deviation	18.702%
Sample Variance	349.777%
Kurtosis	24.86
Skewness	4.21
Range	159.593%
Minimum	0.508%
Maximum	160.101%
Count	289

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.687%	289	100.00%	21.687%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.79. Further, an Excess Kurtosis of 17.40 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.18%	2	0.69%	0.69%
-44.18% to -32.81%	3	1.04%	1.73%
-32.81% to -21.44%	6	2.08%	3.81%
-21.44% to -10.07%	32	11.07%	14.88%
-10.07% to 1.29%	104	35.99%	50.87%
1.29% to 12.66%	99	34.26%	85.12%
12.66% to 24.03%	31	10.73%	95.85%
24.03% to 35.40%	9	3.11%	98.96%
35.40% to 46.76%	1	0.35%	99.31%
Greater than 46.76%	2	0.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.86%	16.45%	80.28%	68.27%
±2σ	-29.02%	31.61%	96.19%	95.45%
±3σ	-44.18%	46.76%	98.62%	99.73%

Descriptive Statistics

Mean	1.293%
Standard Error	0.892%
Median	1.015%
Mode	-7.143%
Standard Deviation	15.157%
Sample Variance	229.733%
Kurtosis	17.40
Skewness	1.79
Range	182.793%
Minimum	-55.911%
Maximum	126.882%
Count	289

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.517%	153	52.94%	5.568%
Negative	-9.151%	135	46.71%	-4.275%
Zero	0.000%	1	0.35%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.25. Further, an Excess Kurtosis of 25.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -132.35%	0	0.00%	0.00%
-132.35% to -97.94%	0	0.00%	0.00%
-97.94% to -63.53%	0	0.00%	0.00%
-63.53% to -29.12%	6	9.09%	9.09%
-29.12% to 5.29%	39	59.09%	68.18%
5.29% to 39.70%	13	19.70%	87.88%
39.70% to 74.11%	6	9.09%	96.97%
74.11% to 108.52%	1	1.52%	98.48%
108.52% to 142.93%	0	0.00%	98.48%
Greater than 142.93%	1	1.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-40.59%	51.17%	87.88%	68.27%
±2σ	-86.47%	97.05%	98.48%	95.45%
±3σ	-132.35%	142.93%	98.48%	99.73%

Descriptive Statistics

Mean	5.289%
Standard Error	5.647%
Median	-4.991%
Mode	-48.194%
Standard Deviation	45.880%
Sample Variance	2105.004%
Kurtosis	25.39
Skewness	4.25
Range	346.499%
Minimum	-48.194%
Maximum	298.305%
Count	66

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	39.833%	25	37.88%	15.088%
Negative	-15.775%	41	62.12%	-9.800%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	27.55%
12	1 Year	49.39%
36	3 Years	37.73%
60	5 Years	38.05%
120	10 Years	48.59%
240	20 Years	48.59%
600	50 Years	48.59%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.90. Further, an Excess Kurtosis of 17.41 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -122.86%	0	0.00%	0.00%
-122.86% to -77.82%	0	0.00%	0.00%
-77.82% to -32.78%	0	0.00%	0.00%
-32.78% to 12.27%	1	1.49%	1.49%
12.27% to 57.31%	43	64.18%	65.67%
57.31% to 102.35%	19	28.36%	94.03%
102.35% to 147.39%	1	1.49%	95.52%
147.39% to 192.43%	1	1.49%	97.01%
192.43% to 237.47%	0	0.00%	97.01%
Greater than 237.47%	2	2.99%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.75%	117.36%	94.03%	68.27%
±2σ	-62.80%	177.42%	97.01%	95.45%
±3σ	-122.86%	237.48%	97.01%	99.73%

Descriptive Statistics

Mean	57.307%
Standard Error	7.337%
Median	38.941%
Mode	2.156%
Standard Deviation	60.056%
Sample Variance	3606.728%
Kurtosis	17.41
Skewness	3.90
Range	361.449%
Minimum	2.156%
Maximum	363.605%
Count	67

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	57.307%	67	100.00%	57.307%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.17. Further, an Excess Kurtosis of 0.66 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -76.80%	0	0.00%	0.00%
-76.80% to -56.54%	1	1.49%	1.49%
-56.54% to -36.28%	5	7.46%	8.96%
-36.28% to -16.01%	9	13.43%	22.39%
-16.01% to 4.25%	19	28.36%	50.75%
4.25% to 24.51%	17	25.37%	76.12%
24.51% to 44.78%	13	19.40%	95.52%
44.78% to 65.04%	2	2.99%	98.51%
65.04% to 85.30%	1	1.49%	100.00%
Greater than 85.30%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-22.77%	31.27%	70.15%	68.27%
±2σ	-49.79%	58.29%	97.01%	95.45%
±3σ	-76.80%	85.30%	100.00%	99.73%

Descriptive Statistics

Mean	4.250%
Standard Error	3.301%
Median	4.091%
Mode	-74.617%
Standard Deviation	27.018%
Sample Variance	729.976%
Kurtosis	0.66
Skewness	-0.17
Range	155.569%
Minimum	-74.617%
Maximum	80.952%
Count	67

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.090%	40	59.70%	12.591%
Negative	-21.493%	26	38.81%	-8.341%
Zero	0.000%	1	1.49%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.21. Further, an Excess Kurtosis of 5.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -213.52%	0	0.00%	0.00%
-213.52% to -156.75%	0	0.00%	0.00%
-156.75% to -99.97%	0	0.00%	0.00%
-99.97% to -43.20%	2	9.09%	9.09%
-43.20% to 13.58%	13	59.09%	68.18%
13.58% to 70.36%	3	13.64%	81.82%
70.36% to 127.13%	3	13.64%	95.45%
127.13% to 183.91%	0	0.00%	95.45%
183.91% to 240.68%	0	0.00%	95.45%
Greater than 240.68%	1	4.55%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-62.12%	89.28%	81.82%	68.27%
±2σ	-137.82%	164.98%	95.45%	95.45%
±3σ	-213.52%	240.68%	95.45%	99.73%

Descriptive Statistics

Mean	13.580%
Standard Error	16.139%
Median	-17.112%
Mode	-60.781%
Standard Deviation	75.700%
Sample Variance	5730.498%
Kurtosis	5.63
Skewness	2.21
Range	331.544%
Minimum	-60.781%
Maximum	270.763%
Count	22

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	74.866%	9	40.91%	30.627%
Negative	-28.849%	13	59.09%	-17.047%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	127.68%
12	3 Years	86.70%
20	5 Years	75.23%
40	10 Years	118.80%
80	20 Years	118.80%
200	50 Years	118.80%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.11. Further, an Excess Kurtosis of 18.24 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -454.76%	0	0.00%	0.00%
-454.76% to -304.05%	0	0.00%	0.00%
-304.05% to -153.34%	0	0.00%	0.00%
-153.34% to -2.62%	0	0.00%	0.00%
-2.62% to 148.09%	17	73.91%	73.91%
148.09% to 298.80%	5	21.74%	95.65%
298.80% to 449.51%	0	0.00%	95.65%
449.51% to 600.22%	0	0.00%	95.65%
600.22% to 750.93%	0	0.00%	95.65%
Greater than 750.93%	1	4.35%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-52.86%	349.04%	95.65%	68.27%
±2σ	-253.81%	549.99%	95.65%	95.45%
±3σ	-454.76%	750.93%	95.65%	99.73%

Descriptive Statistics

Mean	148.088%
Standard Error	41.901%
Median	100.827%
Mode	38.248%
Standard Deviation	200.949%
Sample Variance	40380.503%
Kurtosis	18.24
Skewness	4.11
Range	986.582%
Minimum	38.248%
Maximum	1024.830%
Count	23

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	148.088%	23	100.00%	148.088%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.65. Further, an Excess Kurtosis of 0.96 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -140.67%	0	0.00%	0.00%
-140.67% to -102.14%	0	0.00%	0.00%
-102.14% to -63.60%	1	4.35%	4.35%
-63.60% to -25.07%	3	13.04%	17.39%
-25.07% to 13.46%	7	30.43%	47.83%
13.46% to 52.00%	8	34.78%	82.61%
52.00% to 90.53%	2	8.70%	91.30%
90.53% to 129.07%	1	4.35%	95.65%
129.07% to 167.60%	1	4.35%	100.00%
Greater than 167.60%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-37.91%	64.84%	69.57%	68.27%
±2σ	-89.29%	116.22%	95.65%	95.45%
±3σ	-140.67%	167.60%	100.00%	99.73%

Descriptive Statistics

Mean	13.465%
Standard Error	10.713%
Median	20.963%
Mode	-72.731%
Standard Deviation	51.379%
Sample Variance	2639.807%
Kurtosis	0.96
Skewness	0.65
Range	220.241%
Minimum	-72.731%
Maximum	147.510%
Count	23

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	41.068%	15	65.22%	26.783%
Negative	-38.292%	8	34.78%	-13.319%
Zero	0.000%	0	0.00%	0.000%

QS Deep Analytical Report