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	4.63%	3.62%	3.63%	5.47%	4.38%	4.44%	38.38	41.90	36.74	43.77
Weekly	9.80%	10.73%	10.31%	12.94%	9.64%	9.76%	36.41	44.16	33.07	48.63
Monthly	19.79%	25.95%	20.27%	26.74%	20.07%	20.33%	32.81	49.01	26.84	59.91
Quarterly	31.81%	38.53%	34.17%	47.33%	34.76%	35.21%	28.32	56.77	20.01	80.37
Annual	—	—	—	—	69.53%	70.41%	20.01	80.37	9.98	161.09
Option Time Horizon	—	—	—	—	26.24%	26.58%	30.84	52.13	23.72	67.78

Calculation Notes: Computed at 2026-03-23T16:43 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $40.10 | Strike (K): $42.50 (first OTM call ≥ spot) | Observed Mid-Price: $3.3500 (Bid $2.6000 / Ask $4.1000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 69.53% (CRR binomial tree, American, n=20 steps) | BS IV: 70.38% (European Black-Scholes) | Yahoo IV (Annual): 70.41% → 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 12.05 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.78%	6	0.38%	0.38%
-13.78% to -10.31%	9	0.57%	0.95%
-10.31% to -6.83%	50	3.17%	4.13%
-6.83% to -3.36%	196	12.44%	16.57%
-3.36% to 0.11%	551	34.98%	51.56%
0.11% to 3.59%	520	33.02%	84.57%
3.59% to 7.06%	176	11.17%	95.75%
7.06% to 10.53%	38	2.41%	98.16%
10.53% to 14.01%	13	0.83%	98.98%
Greater than 14.01%	16	1.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.52%	4.75%	79.75%	68.27%
±2σ	-9.15%	9.38%	95.94%	95.45%
±3σ	-13.78%	14.01%	98.60%	99.73%

Descriptive Statistics

Mean	0.114%
Standard Error	0.117%
Median	0.009%
Mode	0.000%
Standard Deviation	4.631%
Sample Variance	21.445%
Kurtosis	12.05
Skewness	0.83
Range	72.680%
Minimum	-35.516%
Maximum	37.164%
Count	1575

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.189%	788	50.03%	1.596%
Negative	-2.985%	782	49.65%	-1.482%
Zero	0.000%	5	0.32%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	4.59%
20	1 Month	4.99%
60	1 Quarter	5.61%
250	1 Year	4.53%
750	3 Years	4.54%
1,250	5 Years	5.47%
2,500	10 Years	5.82%
5,000	20 Years	5.82%
12,500	50 Years	5.82%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.54. Further, an Excess Kurtosis of 24.55 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.26%	0	0.00%	0.00%
-5.26% to -2.55%	0	0.00%	0.00%
-2.55% to 0.17%	0	0.00%	0.00%
0.17% to 2.88%	239	15.16%	15.16%
2.88% to 5.59%	740	46.95%	62.12%
5.59% to 8.30%	370	23.48%	85.60%
8.30% to 11.01%	130	8.25%	93.85%
11.01% to 13.73%	47	2.98%	96.83%
13.73% to 16.44%	25	1.59%	98.41%
Greater than 16.44%	25	1.59%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.97%	9.21%	86.68%	68.27%
±2σ	-1.64%	12.82%	96.45%	95.45%
±3σ	-5.26%	16.44%	98.41%	99.73%

Descriptive Statistics

Mean	5.590%
Standard Error	0.091%
Median	4.639%
Mode	2.442%
Standard Deviation	3.616%
Sample Variance	13.077%
Kurtosis	24.55
Skewness	3.54
Range	45.080%
Minimum	0.950%
Maximum	46.030%
Count	1576

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.590%	1576	100.00%	5.590%
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 -0.13. Further, an Excess Kurtosis of 3.86 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.84%	14	0.89%	0.89%
-10.84% to -8.11%	13	0.82%	1.71%
-8.11% to -5.38%	47	2.98%	4.70%
-5.38% to -2.66%	220	13.96%	18.65%
-2.66% to 0.07%	506	32.11%	50.76%
0.07% to 2.79%	497	31.54%	82.30%
2.79% to 5.52%	177	11.23%	93.53%
5.52% to 8.24%	70	4.44%	97.97%
8.24% to 10.97%	22	1.40%	99.37%
Greater than 10.97%	10	0.63%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.57%	3.70%	76.14%	68.27%
±2σ	-7.20%	7.33%	95.18%	95.45%
±3σ	-10.84%	10.97%	98.48%	99.73%

Descriptive Statistics

Mean	0.066%
Standard Error	0.092%
Median	0.020%
Mode	-1.261%
Standard Deviation	3.634%
Sample Variance	13.209%
Kurtosis	3.86
Skewness	-0.13
Range	43.220%
Minimum	-22.948%
Maximum	20.272%
Count	1576

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.687%	791	50.19%	1.349%
Negative	-2.581%	783	49.68%	-1.282%
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 3.95 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -28.88%	3	0.92%	0.92%
-28.88% to -21.53%	4	1.22%	2.14%
-21.53% to -14.19%	10	3.06%	5.20%
-14.19% to -6.84%	44	13.46%	18.65%
-6.84% to 0.51%	102	31.19%	49.85%
0.51% to 7.86%	109	33.33%	83.18%
7.86% to 15.21%	43	13.15%	96.33%
15.21% to 22.55%	6	1.83%	98.17%
22.55% to 29.90%	3	0.92%	99.08%
Greater than 29.90%	3	0.92%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.29%	10.31%	78.59%	68.27%
±2σ	-19.09%	20.10%	95.41%	95.45%
±3σ	-28.88%	29.90%	98.17%	99.73%

Descriptive Statistics

Mean	0.509%
Standard Error	0.542%
Median	0.520%
Mode	-6.571%
Standard Deviation	9.797%
Sample Variance	95.981%
Kurtosis	3.95
Skewness	0.35
Range	85.773%
Minimum	-38.600%
Maximum	47.173%
Count	327

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.101%	172	52.60%	3.735%
Negative	-6.805%	155	47.40%	-3.226%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	10.09%
12	1 Quarter	13.97%
52	1 Year	10.76%
156	3 Years	10.92%
260	5 Years	12.94%
520	10 Years	13.79%
1,040	20 Years	13.79%
2,600	50 Years	13.79%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.55. Further, an Excess Kurtosis of 8.58 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.99%	0	0.00%	0.00%
-16.99% to -8.94%	0	0.00%	0.00%
-8.94% to -0.89%	0	0.00%	0.00%
-0.89% to 7.16%	44	13.41%	13.41%
7.16% to 15.21%	174	53.05%	66.46%
15.21% to 23.26%	66	20.12%	86.59%
23.26% to 31.31%	19	5.79%	92.38%
31.31% to 39.36%	11	3.35%	95.73%
39.36% to 47.41%	6	1.83%	97.56%
Greater than 47.41%	8	2.44%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.47%	25.94%	88.72%	68.27%
±2σ	-6.26%	36.67%	94.82%	95.45%
±3σ	-16.99%	47.41%	97.56%	99.73%

Descriptive Statistics

Mean	15.208%
Standard Error	0.593%
Median	11.675%
Mode	8.182%
Standard Deviation	10.733%
Sample Variance	115.202%
Kurtosis	8.58
Skewness	2.55
Range	71.151%
Minimum	3.252%
Maximum	74.403%
Count	328

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.208%	328	100.00%	15.208%
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 1.42. Further, an Excess Kurtosis of 6.23 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -30.34%	1	0.30%	0.30%
-30.34% to -22.60%	2	0.61%	0.91%
-22.60% to -14.87%	5	1.52%	2.44%
-14.87% to -7.13%	49	14.94%	17.38%
-7.13% to 0.61%	126	38.41%	55.79%
0.61% to 8.34%	94	28.66%	84.45%
8.34% to 16.08%	33	10.06%	94.51%
16.08% to 23.82%	9	2.74%	97.26%
23.82% to 31.55%	4	1.22%	98.48%
Greater than 31.55%	5	1.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.71%	10.92%	78.96%	68.27%
±2σ	-20.02%	21.24%	95.43%	95.45%
±3σ	-30.34%	31.55%	98.17%	99.73%

Descriptive Statistics

Mean	0.606%
Standard Error	0.570%
Median	-0.401%
Mode	0.000%
Standard Deviation	10.315%
Sample Variance	106.400%
Kurtosis	6.23
Skewness	1.42
Range	92.932%
Minimum	-31.890%
Maximum	61.042%
Count	328

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.183%	154	46.95%	3.842%
Negative	-6.206%	171	52.13%	-3.235%
Zero	0.000%	3	0.92%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.66. Further, an Excess Kurtosis of 1.67 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -57.43%	0	0.00%	0.00%
-57.43% to -42.59%	1	1.33%	1.33%
-42.59% to -27.75%	3	4.00%	5.33%
-27.75% to -12.91%	9	12.00%	17.33%
-12.91% to 1.93%	29	38.67%	56.00%
1.93% to 16.77%	19	25.33%	81.33%
16.77% to 31.61%	8	10.67%	92.00%
31.61% to 46.45%	4	5.33%	97.33%
46.45% to 61.29%	1	1.33%	98.67%
Greater than 61.29%	1	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.86%	21.71%	70.67%	68.27%
±2σ	-37.64%	41.50%	94.67%	95.45%
±3σ	-57.43%	61.29%	98.67%	99.73%

Descriptive Statistics

Mean	1.929%
Standard Error	2.285%
Median	-0.309%
Mode	-42.958%
Standard Deviation	19.786%
Sample Variance	391.493%
Kurtosis	1.67
Skewness	0.66
Range	115.151%
Minimum	-42.958%
Maximum	72.193%
Count	75

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	16.900%	37	49.33%	8.337%
Negative	-12.648%	38	50.67%	-6.408%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	27.36%
12	1 Year	20.74%
36	3 Years	23.73%
60	5 Years	26.74%
120	10 Years	29.81%
240	20 Years	29.81%
600	50 Years	29.81%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.66. Further, an Excess Kurtosis of 9.14 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.26%	0	0.00%	0.00%
-41.26% to -21.80%	0	0.00%	0.00%
-21.80% to -2.34%	0	0.00%	0.00%
-2.34% to 17.13%	14	18.42%	18.42%
17.13% to 36.59%	36	47.37%	65.79%
36.59% to 56.05%	16	21.05%	86.84%
56.05% to 75.52%	4	5.26%	92.11%
75.52% to 94.98%	3	3.95%	96.05%
94.98% to 114.44%	1	1.32%	97.37%
Greater than 114.44%	2	2.63%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.64%	62.54%	89.47%	68.27%
±2σ	-15.31%	88.49%	94.74%	95.45%
±3σ	-41.26%	114.44%	97.37%	99.73%

Descriptive Statistics

Mean	36.589%
Standard Error	2.977%
Median	29.968%
Mode	12.416%
Standard Deviation	25.951%
Sample Variance	673.464%
Kurtosis	9.14
Skewness	2.66
Range	153.869%
Minimum	12.416%
Maximum	166.285%
Count	76

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	36.589%	76	100.00%	36.589%
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 0.93. Further, an Excess Kurtosis of 2.11 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -59.25%	0	0.00%	0.00%
-59.25% to -44.05%	1	1.32%	1.32%
-44.05% to -28.84%	1	1.32%	2.63%
-28.84% to -13.64%	14	18.42%	21.05%
-13.64% to 1.57%	23	30.26%	51.32%
1.57% to 16.77%	24	31.58%	82.89%
16.77% to 31.98%	7	9.21%	92.11%
31.98% to 47.18%	4	5.26%	97.37%
47.18% to 62.39%	0	0.00%	97.37%
Greater than 62.39%	2	2.63%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-18.70%	21.84%	76.32%	68.27%
±2σ	-38.98%	42.11%	96.05%	95.45%
±3σ	-59.25%	62.39%	97.37%	99.73%

Descriptive Statistics

Mean	1.568%
Standard Error	2.325%
Median	0.309%
Mode	-45.441%
Standard Deviation	20.273%
Sample Variance	410.994%
Kurtosis	2.11
Skewness	0.93
Range	115.532%
Minimum	-45.441%
Maximum	70.091%
Count	76

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.900%	39	51.32%	8.159%
Negative	-13.539%	37	48.68%	-6.591%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.46. Further, an Excess Kurtosis of -0.46 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -91.67%	0	0.00%	0.00%
-91.67% to -67.81%	0	0.00%	0.00%
-67.81% to -43.95%	1	4.00%	4.00%
-43.95% to -20.10%	5	20.00%	24.00%
-20.10% to 3.76%	8	32.00%	56.00%
3.76% to 27.62%	6	24.00%	80.00%
27.62% to 51.48%	2	8.00%	88.00%
51.48% to 75.34%	3	12.00%	100.00%
75.34% to 99.19%	0	0.00%	100.00%
Greater than 99.19%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.05%	35.57%	72.00%	68.27%
±2σ	-59.86%	67.38%	100.00%	95.45%
±3σ	-91.67%	99.19%	100.00%	99.73%

Descriptive Statistics

Mean	3.762%
Standard Error	6.362%
Median	-5.969%
Mode	-52.909%
Standard Deviation	31.811%
Sample Variance	1011.929%
Kurtosis	-0.46
Skewness	0.46
Range	118.725%
Minimum	-52.909%
Maximum	65.816%
Count	25

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	33.550%	11	44.00%	14.762%
Negative	-19.643%	14	56.00%	-11.000%
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 1.03. Further, an Excess Kurtosis of 1.66 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -98.01%	0	0.00%	0.00%
-98.01% to -72.39%	0	0.00%	0.00%
-72.39% to -46.76%	0	0.00%	0.00%
-46.76% to -21.13%	6	23.08%	23.08%
-21.13% to 4.50%	7	26.92%	50.00%
4.50% to 30.13%	9	34.62%	84.62%
30.13% to 55.76%	3	11.54%	96.15%
55.76% to 81.39%	0	0.00%	96.15%
81.39% to 107.01%	1	3.85%	100.00%
Greater than 107.01%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-29.67%	38.67%	76.92%	68.27%
±2σ	-63.84%	72.84%	96.15%	95.45%
±3σ	-98.01%	107.01%	100.00%	99.73%

Descriptive Statistics

Mean	4.500%
Standard Error	6.701%
Median	4.163%
Mode	-40.844%
Standard Deviation	34.171%
Sample Variance	1167.653%
Kurtosis	1.66
Skewness	1.03
Range	146.879%
Minimum	-40.844%
Maximum	106.035%
Count	26

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	28.941%	14	53.85%	15.584%
Negative	-24.014%	12	46.15%	-11.083%
Zero	0.000%	0	0.00%	0.000%

BILL Deep Analytical Report