BILL - 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	4.65%	3.64%	3.64%	5.51%	4.84%	4.86%	44.02	48.49	41.94	50.90
Weekly	9.81%	10.79%	10.33%	13.02%	10.66%	10.71%	41.53	51.40	37.33	57.18
Monthly	19.88%	26.05%	20.41%	26.84%	22.19%	22.29%	37.01	57.68	29.64	72.00
Quarterly	31.74%	38.53%	34.24%	47.33%	38.43%	38.61%	31.46	67.85	21.42	99.64
Annual	—	—	—	—	76.86%	77.22%	21.42	99.64	9.93	214.90
Option Time Horizon	—	—	—	—	25.12%	25.24%	35.94	59.40	27.95	76.36

Calculation Notes: Computed at 2026-02-22T14:25 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $46.20 | Strike (K): $47.00 (first OTM call ≥ spot) | Observed Mid-Price: $4.3500 (Bid $2.8000 / Ask $5.9000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 76.86% (CRR binomial tree, American, n=20 steps) | BS IV: 76.88% (European Black-Scholes) | Yahoo IV (Annual): 77.22% → 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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.83. Further, an Excess Kurtosis of 12.01 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.82%	6	0.39%	0.39%
-13.82% to -10.34%	9	0.58%	0.97%
-10.34% to -6.85%	48	3.09%	4.05%
-6.85% to -3.36%	195	12.55%	16.60%
-3.36% to 0.12%	543	34.94%	51.54%
0.12% to 3.61%	516	33.20%	84.75%
3.61% to 7.10%	171	11.00%	95.75%
7.10% to 10.58%	37	2.38%	98.13%
10.58% to 14.07%	13	0.84%	98.97%
Greater than 14.07%	16	1.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.53%	4.77%	79.73%	68.27%
±2σ	-9.17%	9.42%	96.01%	95.45%
±3σ	-13.82%	14.07%	98.58%	99.73%

Descriptive Statistics

Mean	0.124%
Standard Error	0.118%
Median	0.014%
Mode	0.000%
Standard Deviation	4.649%
Sample Variance	21.610%
Kurtosis	12.01
Skewness	0.83
Range	72.680%
Minimum	-35.516%
Maximum	37.164%
Count	1554

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.205%	779	50.13%	1.607%
Negative	-2.993%	770	49.55%	-1.483%
Zero	0.000%	5	0.32%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.26%
20	1 Month	7.75%
60	1 Quarter	4.83%
250	1 Year	4.53%
750	3 Years	4.55%
1,250	5 Years	5.51%
2,500	10 Years	5.83%
5,000	20 Years	5.83%
12,500	50 Years	5.83%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.53. Further, an Excess Kurtosis of 24.30 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.31%	0	0.00%	0.00%
-5.31% to -2.59%	0	0.00%	0.00%
-2.59% to 0.14%	0	0.00%	0.00%
0.14% to 2.87%	237	15.24%	15.24%
2.87% to 5.59%	728	46.82%	62.06%
5.59% to 8.32%	367	23.60%	85.66%
8.32% to 11.05%	127	8.17%	93.83%
11.05% to 13.78%	46	2.96%	96.78%
13.78% to 16.50%	25	1.61%	98.39%
Greater than 16.50%	25	1.61%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.96%	9.23%	86.75%	68.27%
±2σ	-1.68%	12.87%	96.40%	95.45%
±3σ	-5.31%	16.50%	98.39%	99.73%

Descriptive Statistics

Mean	5.594%
Standard Error	0.092%
Median	4.639%
Mode	2.442%
Standard Deviation	3.636%
Sample Variance	13.219%
Kurtosis	24.30
Skewness	3.53
Range	45.080%
Minimum	0.950%
Maximum	46.030%
Count	1555

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.594%	1555	100.00%	5.594%
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 -0.14. Further, an Excess Kurtosis of 3.87 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.86%	14	0.90%	0.90%
-10.86% to -8.12%	12	0.77%	1.67%
-8.12% to -5.39%	46	2.96%	4.63%
-5.39% to -2.66%	219	14.08%	18.71%
-2.66% to 0.07%	496	31.90%	50.61%
0.07% to 2.80%	493	31.70%	82.32%
2.80% to 5.53%	174	11.19%	93.50%
5.53% to 8.26%	69	4.44%	97.94%
8.26% to 10.99%	22	1.41%	99.36%
Greater than 10.99%	10	0.64%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.57%	3.71%	76.08%	68.27%
±2σ	-7.21%	7.35%	95.24%	95.45%
±3σ	-10.86%	11.00%	98.46%	99.73%

Descriptive Statistics

Mean	0.069%
Standard Error	0.092%
Median	0.028%
Mode	-1.261%
Standard Deviation	3.642%
Sample Variance	13.263%
Kurtosis	3.87
Skewness	-0.14
Range	43.220%
Minimum	-22.948%
Maximum	20.272%
Count	1555

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.683%	784	50.42%	1.353%
Negative	-2.596%	769	49.45%	-1.284%
Zero	0.000%	2	0.13%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.35. Further, an Excess Kurtosis of 3.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -28.89%	3	0.93%	0.93%
-28.89% to -21.53%	4	1.24%	2.17%
-21.53% to -14.17%	10	3.10%	5.26%
-14.17% to -6.81%	42	13.00%	18.27%
-6.81% to 0.56%	104	32.20%	50.46%
0.56% to 7.92%	105	32.51%	82.97%
7.92% to 15.28%	43	13.31%	96.28%
15.28% to 22.64%	6	1.86%	98.14%
22.64% to 30.00%	3	0.93%	99.07%
Greater than 30.00%	3	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.26%	10.37%	78.95%	68.27%
±2σ	-19.07%	20.18%	95.36%	95.45%
±3σ	-28.89%	30.00%	98.14%	99.73%

Descriptive Statistics

Mean	0.555%
Standard Error	0.546%
Median	0.471%
Mode	-6.571%
Standard Deviation	9.814%
Sample Variance	96.307%
Kurtosis	3.98
Skewness	0.35
Range	85.773%
Minimum	-38.600%
Maximum	47.173%
Count	323

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.183%	169	52.32%	3.758%
Negative	-6.719%	154	47.68%	-3.203%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	20.18%
12	1 Quarter	12.51%
52	1 Year	10.85%
156	3 Years	11.00%
260	5 Years	13.02%
520	10 Years	13.82%
1,040	20 Years	13.82%
2,600	50 Years	13.82%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.53. Further, an Excess Kurtosis of 8.45 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.12%	0	0.00%	0.00%
-17.12% to -9.03%	0	0.00%	0.00%
-9.03% to -0.94%	0	0.00%	0.00%
-0.94% to 7.15%	43	13.27%	13.27%
7.15% to 15.24%	172	53.09%	66.36%
15.24% to 23.34%	67	20.68%	87.04%
23.34% to 31.43%	17	5.25%	92.28%
31.43% to 39.52%	11	3.40%	95.68%
39.52% to 47.61%	6	1.85%	97.53%
Greater than 47.61%	8	2.47%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.46%	26.03%	88.58%	68.27%
±2σ	-6.33%	36.82%	94.75%	95.45%
±3σ	-17.12%	47.61%	97.53%	99.73%

Descriptive Statistics

Mean	15.245%
Standard Error	0.599%
Median	11.676%
Mode	8.182%
Standard Deviation	10.789%
Sample Variance	116.397%
Kurtosis	8.45
Skewness	2.53
Range	71.151%
Minimum	3.252%
Maximum	74.403%
Count	324

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.245%	324	100.00%	15.245%
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 1.43. Further, an Excess Kurtosis of 6.30 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -30.42%	1	0.31%	0.31%
-30.42% to -22.67%	2	0.62%	0.93%
-22.67% to -14.92%	5	1.54%	2.47%
-14.92% to -7.17%	49	15.12%	17.59%
-7.17% to 0.57%	123	37.96%	55.56%
0.57% to 8.32%	95	29.32%	84.88%
8.32% to 16.07%	31	9.57%	94.44%
16.07% to 23.82%	9	2.78%	97.22%
23.82% to 31.57%	4	1.23%	98.46%
Greater than 31.57%	5	1.54%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.76%	10.90%	79.63%	68.27%
±2σ	-20.09%	21.24%	95.37%	95.45%
±3σ	-30.42%	31.57%	98.15%	99.73%

Descriptive Statistics

Mean	0.574%
Standard Error	0.574%
Median	-0.396%
Mode	0.000%
Standard Deviation	10.331%
Sample Variance	106.731%
Kurtosis	6.30
Skewness	1.43
Range	92.932%
Minimum	-31.890%
Maximum	61.042%
Count	324

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.089%	153	47.22%	3.820%
Negative	-6.260%	168	51.85%	-3.246%
Zero	0.000%	3	0.93%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.63. Further, an Excess Kurtosis of 1.62 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -57.50%	0	0.00%	0.00%
-57.50% to -42.59%	1	1.35%	1.35%
-42.59% to -27.68%	3	4.05%	5.41%
-27.68% to -12.77%	9	12.16%	17.57%
-12.77% to 2.14%	28	37.84%	55.41%
2.14% to 17.05%	19	25.68%	81.08%
17.05% to 31.96%	8	10.81%	91.89%
31.96% to 46.87%	4	5.41%	97.30%
46.87% to 61.78%	1	1.35%	98.65%
Greater than 61.78%	1	1.35%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.74%	22.02%	70.27%	68.27%
±2σ	-37.62%	41.90%	94.59%	95.45%
±3σ	-57.50%	61.78%	98.65%	99.73%

Descriptive Statistics

Mean	2.141%
Standard Error	2.311%
Median	0.220%
Mode	-42.958%
Standard Deviation	19.881%
Sample Variance	395.250%
Kurtosis	1.62
Skewness	0.63
Range	115.151%
Minimum	-42.958%
Maximum	72.193%
Count	74

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.006%	37	50.00%	8.503%
Negative	-12.724%	37	50.00%	-6.362%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	26.70%
12	1 Year	21.19%
36	3 Years	24.01%
60	5 Years	26.84%
120	10 Years	29.98%
240	20 Years	29.98%
600	50 Years	29.98%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.64. Further, an Excess Kurtosis of 9.03 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.34%	0	0.00%	0.00%
-41.34% to -21.80%	0	0.00%	0.00%
-21.80% to -2.27%	0	0.00%	0.00%
-2.27% to 17.27%	14	18.67%	18.67%
17.27% to 36.81%	35	46.67%	65.33%
36.81% to 56.35%	16	21.33%	86.67%
56.35% to 75.89%	4	5.33%	92.00%
75.89% to 95.43%	3	4.00%	96.00%
95.43% to 114.97%	1	1.33%	97.33%
Greater than 114.97%	2	2.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.76%	62.87%	89.33%	68.27%
±2σ	-15.29%	88.92%	94.67%	95.45%
±3σ	-41.34%	114.97%	97.33%	99.73%

Descriptive Statistics

Mean	36.813%
Standard Error	3.008%
Median	29.973%
Mode	12.416%
Standard Deviation	26.052%
Sample Variance	678.708%
Kurtosis	9.03
Skewness	2.64
Range	153.869%
Minimum	12.416%
Maximum	166.285%
Count	75

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	36.813%	75	100.00%	36.813%
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 0.95. Further, an Excess Kurtosis of 2.07 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -59.80%	0	0.00%	0.00%
-59.80% to -44.49%	1	1.33%	1.33%
-44.49% to -29.19%	1	1.33%	2.67%
-29.19% to -13.88%	14	18.67%	21.33%
-13.88% to 1.42%	23	30.67%	52.00%
1.42% to 16.73%	23	30.67%	82.67%
16.73% to 32.03%	7	9.33%	92.00%
32.03% to 47.34%	4	5.33%	97.33%
47.34% to 62.64%	0	0.00%	97.33%
Greater than 62.64%	2	2.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-18.98%	21.83%	76.00%	68.27%
±2σ	-39.39%	42.24%	96.00%	95.45%
±3σ	-59.80%	62.64%	97.33%	99.73%

Descriptive Statistics

Mean	1.423%
Standard Error	2.356%
Median	0.180%
Mode	-45.441%
Standard Deviation	20.407%
Sample Variance	416.434%
Kurtosis	2.07
Skewness	0.95
Range	115.532%
Minimum	-45.441%
Maximum	70.091%
Count	75

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	16.085%	38	50.67%	8.150%
Negative	-13.635%	37	49.33%	-6.727%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.41. Further, an Excess Kurtosis of -0.47 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -90.88%	0	0.00%	0.00%
-90.88% to -67.08%	0	0.00%	0.00%
-67.08% to -43.28%	1	4.00%	4.00%
-43.28% to -19.48%	6	24.00%	28.00%
-19.48% to 4.32%	6	24.00%	52.00%
4.32% to 28.13%	7	28.00%	80.00%
28.13% to 51.93%	2	8.00%	88.00%
51.93% to 75.73%	3	12.00%	100.00%
75.73% to 99.53%	0	0.00%	100.00%
Greater than 99.53%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-27.41%	36.06%	72.00%	68.27%
±2σ	-59.15%	67.80%	100.00%	95.45%
±3σ	-90.88%	99.53%	100.00%	99.73%

Descriptive Statistics

Mean	4.324%
Standard Error	6.347%
Median	-1.273%
Mode	-52.909%
Standard Deviation	31.736%
Sample Variance	1007.178%
Kurtosis	-0.47
Skewness	0.41
Range	118.725%
Minimum	-52.909%
Maximum	65.816%
Count	25

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.339%	12	48.00%	15.043%
Negative	-20.613%	13	52.00%	-10.719%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	31.47%
12	3 Years	43.57%
20	5 Years	47.33%
40	10 Years	50.91%
80	20 Years	50.91%
200	50 Years	50.91%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.51. Further, an Excess Kurtosis of 2.00 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -45.00%	0	0.00%	0.00%
-45.00% to -16.10%	0	0.00%	0.00%
-16.10% to 12.79%	0	0.00%	0.00%
12.79% to 41.69%	4	15.38%	15.38%
41.69% to 70.59%	12	46.15%	61.54%
70.59% to 99.48%	7	26.92%	88.46%
99.48% to 128.38%	0	0.00%	88.46%
128.38% to 157.28%	1	3.85%	92.31%
157.28% to 186.17%	2	7.69%	100.00%
Greater than 186.17%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	32.06%	109.12%	84.62%	68.27%
±2σ	-6.47%	147.64%	92.31%	95.45%
±3σ	-45.00%	186.17%	100.00%	99.73%

Descriptive Statistics

Mean	70.586%
Standard Error	7.556%
Median	59.756%
Mode	19.494%
Standard Deviation	38.529%
Sample Variance	1484.492%
Kurtosis	2.00
Skewness	1.51
Range	152.086%
Minimum	19.494%
Maximum	171.580%
Count	26

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	70.586%	26	100.00%	70.586%
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 1.07. Further, an Excess Kurtosis of 1.70 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -98.75%	0	0.00%	0.00%
-98.75% to -73.08%	0	0.00%	0.00%
-73.08% to -47.40%	0	0.00%	0.00%
-47.40% to -21.72%	6	23.08%	23.08%
-21.72% to 3.96%	8	30.77%	53.85%
3.96% to 29.64%	8	30.77%	84.62%
29.64% to 55.31%	2	7.69%	92.31%
55.31% to 80.99%	1	3.85%	96.15%
80.99% to 106.67%	1	3.85%	100.00%
Greater than 106.67%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-30.28%	38.20%	76.92%	68.27%
±2σ	-64.52%	72.43%	96.15%	95.45%
±3σ	-98.75%	106.67%	100.00%	99.73%

Descriptive Statistics

Mean	3.958%
Standard Error	6.714%
Median	-2.400%
Mode	-40.844%
Standard Deviation	34.237%
Sample Variance	1172.163%
Kurtosis	1.70
Skewness	1.07
Range	146.879%
Minimum	-40.844%
Maximum	106.035%
Count	26

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.607%	13	50.00%	15.304%
Negative	-22.692%	13	50.00%	-11.346%
Zero	0.000%	0	0.00%	0.000%

BILL Deep Analytical Report