DOCU - 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	3.57%	2.45%	2.88%	4.38%	3.26%	3.31%	46.53	49.66	45.04	51.30
Weekly	7.71%	8.14%	8.38%	10.72%	7.17%	7.29%	44.74	51.64	41.65	55.48
Monthly	13.89%	14.10%	14.61%	22.64%	14.92%	15.18%	41.41	55.81	35.67	64.79
Quarterly	31.52%	30.33%	32.09%	39.86%	25.84%	26.29%	37.12	62.25	28.67	80.60
Annual	—	—	—	—	51.69%	52.59%	28.67	80.60	17.10	135.16
Option Time Horizon	—	—	—	—	19.51%	19.85%	39.55	58.43	32.54	71.01

Calculation Notes: Computed at 2026-03-23T16:35 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $48.07 | Strike (K): $50.00 (first OTM call ≥ spot) | Observed Mid-Price: $3.0700 (Bid $2.7900 / Ask $3.3500) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 51.69% (CRR binomial tree, American, n=20 steps) | BS IV: 52.31% (European Black-Scholes) | Yahoo IV (Annual): 52.59% → 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 Left Skewed (Negative Asymmetry) with a Skewness factor of -0.65. Further, an Excess Kurtosis of 16.74 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.62%	10	0.50%	0.50%
-10.62% to -7.95%	18	0.91%	1.41%
-7.95% to -5.27%	64	3.22%	4.63%
-5.27% to -2.60%	237	11.94%	16.57%
-2.60% to 0.07%	662	33.35%	49.92%
0.07% to 2.75%	669	33.70%	83.63%
2.75% to 5.42%	240	12.09%	95.72%
5.42% to 8.10%	58	2.92%	98.64%
8.10% to 10.77%	11	0.55%	99.19%
Greater than 10.77%	16	0.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.49%	3.64%	78.69%	68.27%
±2σ	-7.06%	7.21%	96.27%	95.45%
±3σ	-10.62%	10.77%	98.69%	99.73%

Descriptive Statistics

Mean	0.075%
Standard Error	0.080%
Median	0.075%
Mode	0.000%
Standard Deviation	3.566%
Sample Variance	12.714%
Kurtosis	16.74
Skewness	-0.65
Range	70.081%
Minimum	-42.225%
Maximum	27.856%
Count	1985

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.422%	1013	51.03%	1.236%
Negative	-2.383%	967	48.72%	-1.161%
Zero	0.000%	5	0.25%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	6.23%
20	1 Month	5.35%
60	1 Quarter	4.72%
250	1 Year	3.86%
750	3 Years	3.57%
1,250	5 Years	4.38%
2,500	10 Years	4.54%
5,000	20 Years	4.54%
12,500	50 Years	4.54%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.08. Further, an Excess Kurtosis of 8.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.03%	0	0.00%	0.00%
-3.03% to -1.19%	0	0.00%	0.00%
-1.19% to 0.65%	0	0.00%	0.00%
0.65% to 2.48%	437	22.00%	22.00%
2.48% to 4.32%	777	39.12%	61.13%
4.32% to 6.16%	429	21.60%	82.73%
6.16% to 8.00%	190	9.57%	92.30%
8.00% to 9.84%	81	4.08%	96.37%
9.84% to 11.68%	37	1.86%	98.24%
Greater than 11.68%	35	1.76%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.87%	6.78%	79.66%	68.27%
±2σ	-0.58%	9.23%	95.57%	95.45%
±3σ	-3.03%	11.68%	98.24%	99.73%

Descriptive Statistics

Mean	4.323%
Standard Error	0.055%
Median	3.720%
Mode	2.469%
Standard Deviation	2.452%
Sample Variance	6.014%
Kurtosis	8.98
Skewness	2.08
Range	28.909%
Minimum	0.779%
Maximum	29.688%
Count	1986

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.323%	1986	100.00%	4.323%
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 0.16. Further, an Excess Kurtosis of 2.12 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.65%	9	0.45%	0.45%
-8.65% to -6.49%	23	1.16%	1.61%
-6.49% to -4.33%	76	3.83%	5.44%
-4.33% to -2.17%	268	13.49%	18.93%
-2.17% to -0.01%	642	32.33%	51.26%
-0.01% to 2.16%	602	30.31%	81.57%
2.16% to 4.32%	242	12.19%	93.76%
4.32% to 6.48%	81	4.08%	97.83%
6.48% to 8.64%	31	1.56%	99.40%
Greater than 8.64%	12	0.60%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.89%	2.88%	73.41%	68.27%
±2σ	-5.77%	5.76%	94.51%	95.45%
±3σ	-8.66%	8.64%	98.94%	99.73%

Descriptive Statistics

Mean	-0.006%
Standard Error	0.065%
Median	-0.084%
Mode	0.000%
Standard Deviation	2.883%
Sample Variance	8.312%
Kurtosis	2.12
Skewness	0.16
Range	27.741%
Minimum	-13.151%
Maximum	14.590%
Count	1986

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.175%	962	48.44%	1.054%
Negative	-2.068%	1018	51.26%	-1.060%
Zero	0.000%	6	0.30%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.29. 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 -22.76%	3	0.73%	0.73%
-22.76% to -16.98%	5	1.21%	1.94%
-16.98% to -11.20%	18	4.37%	6.31%
-11.20% to -5.42%	50	12.14%	18.45%
-5.42% to 0.36%	121	29.37%	47.82%
0.36% to 6.14%	144	34.95%	82.77%
6.14% to 11.92%	52	12.62%	95.39%
11.92% to 17.70%	12	2.91%	98.30%
17.70% to 23.48%	3	0.73%	99.03%
Greater than 23.48%	4	0.97%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.35%	8.07%	77.67%	68.27%
±2σ	-15.05%	15.77%	94.90%	95.45%
±3σ	-22.76%	23.48%	98.30%	99.73%

Descriptive Statistics

Mean	0.360%
Standard Error	0.380%
Median	0.790%
Mode	-0.682%
Standard Deviation	7.707%
Sample Variance	59.397%
Kurtosis	3.87
Skewness	-0.29
Range	73.676%
Minimum	-42.791%
Maximum	30.885%
Count	412

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.570%	222	53.88%	3.001%
Negative	-5.728%	190	46.12%	-2.642%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	12.02%
12	1 Quarter	12.08%
52	1 Year	9.44%
156	3 Years	8.68%
260	5 Years	10.72%
520	10 Years	10.87%
1,040	20 Years	10.87%
2,600	50 Years	10.87%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.93. 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 -12.78%	0	0.00%	0.00%
-12.78% to -6.67%	0	0.00%	0.00%
-6.67% to -0.56%	0	0.00%	0.00%
-0.56% to 5.54%	50	12.11%	12.11%
5.54% to 11.65%	220	53.27%	65.38%
11.65% to 17.76%	86	20.82%	86.20%
17.76% to 23.87%	29	7.02%	93.22%
23.87% to 29.98%	14	3.39%	96.61%
29.98% to 36.09%	7	1.69%	98.31%
Greater than 36.09%	7	1.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.51%	19.80%	88.14%	68.27%
±2σ	-4.64%	27.94%	96.13%	95.45%
±3σ	-12.78%	36.09%	98.31%	99.73%

Descriptive Statistics

Mean	11.654%
Standard Error	0.401%
Median	9.678%
Mode	6.579%
Standard Deviation	8.145%
Sample Variance	66.345%
Kurtosis	28.72
Skewness	3.93
Range	91.405%
Minimum	2.801%
Maximum	94.206%
Count	413

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.654%	413	100.00%	11.654%
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 2.26. Further, an Excess Kurtosis of 16.81 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.83%	0	0.00%	0.00%
-24.83% to -18.55%	4	0.97%	0.97%
-18.55% to -12.26%	12	2.91%	3.87%
-12.26% to -5.98%	44	10.65%	14.53%
-5.98% to 0.31%	172	41.65%	56.17%
0.31% to 6.59%	114	27.60%	83.78%
6.59% to 12.88%	42	10.17%	93.95%
12.88% to 19.16%	16	3.87%	97.82%
19.16% to 25.44%	5	1.21%	99.03%
Greater than 25.44%	4	0.97%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.07%	8.69%	81.11%	68.27%
±2σ	-16.45%	17.06%	95.40%	95.45%
±3σ	-24.83%	25.44%	99.03%	99.73%

Descriptive Statistics

Mean	0.307%
Standard Error	0.412%
Median	-0.746%
Mode	-5.543%
Standard Deviation	8.379%
Sample Variance	70.209%
Kurtosis	16.81
Skewness	2.26
Range	99.041%
Minimum	-23.534%
Maximum	75.507%
Count	413

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.615%	187	45.28%	2.995%
Negative	-4.935%	225	54.48%	-2.689%
Zero	0.000%	1	0.24%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.25. Further, an Excess Kurtosis of 0.85 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -40.48%	0	0.00%	0.00%
-40.48% to -30.06%	2	2.11%	2.11%
-30.06% to -19.64%	3	3.16%	5.26%
-19.64% to -9.23%	17	17.89%	23.16%
-9.23% to 1.19%	26	27.37%	50.53%
1.19% to 11.60%	30	31.58%	82.11%
11.60% to 22.02%	9	9.47%	91.58%
22.02% to 32.43%	4	4.21%	95.79%
32.43% to 42.85%	4	4.21%	100.00%
Greater than 42.85%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.70%	15.07%	76.84%	68.27%
±2σ	-26.59%	28.96%	93.68%	95.45%
±3σ	-40.48%	42.85%	100.00%	99.73%

Descriptive Statistics

Mean	1.186%
Standard Error	1.425%
Median	1.171%
Mode	-38.176%
Standard Deviation	13.887%
Sample Variance	192.845%
Kurtosis	0.85
Skewness	0.25
Range	76.838%
Minimum	-38.176%
Maximum	38.662%
Count	95

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.467%	49	51.58%	5.915%
Negative	-9.765%	46	48.42%	-4.728%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	22.98%
12	1 Year	17.78%
36	3 Years	19.15%
60	5 Years	22.64%
120	10 Years	23.03%
240	20 Years	23.03%
600	50 Years	23.03%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.76. Further, an Excess Kurtosis of 4.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.69%	0	0.00%	0.00%
-15.69% to -5.11%	0	0.00%	0.00%
-5.11% to 5.46%	0	0.00%	0.00%
5.46% to 16.04%	23	23.96%	23.96%
16.04% to 26.61%	36	37.50%	61.46%
26.61% to 37.19%	20	20.83%	82.29%
37.19% to 47.76%	9	9.38%	91.67%
47.76% to 58.34%	5	5.21%	96.88%
58.34% to 68.91%	1	1.04%	97.92%
Greater than 68.91%	2	2.08%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.51%	40.71%	82.29%	68.27%
±2σ	-1.59%	54.81%	95.83%	95.45%
±3σ	-15.69%	68.91%	97.92%	99.73%

Descriptive Statistics

Mean	26.612%
Standard Error	1.439%
Median	22.791%
Mode	10.514%
Standard Deviation	14.100%
Sample Variance	198.796%
Kurtosis	4.39
Skewness	1.76
Range	79.312%
Minimum	10.514%
Maximum	89.826%
Count	96

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	26.612%	96	100.00%	26.612%
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 1.11. Further, an Excess Kurtosis of 3.44 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.97%	0	0.00%	0.00%
-42.97% to -32.01%	0	0.00%	0.00%
-32.01% to -21.06%	5	5.21%	5.21%
-21.06% to -10.10%	13	13.54%	18.75%
-10.10% to 0.85%	34	35.42%	54.17%
0.85% to 11.81%	26	27.08%	81.25%
11.81% to 22.76%	13	13.54%	94.79%
22.76% to 33.72%	3	3.12%	97.92%
33.72% to 44.67%	0	0.00%	97.92%
Greater than 44.67%	2	2.08%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.75%	15.46%	80.21%	68.27%
±2σ	-28.36%	30.07%	95.83%	95.45%
±3σ	-42.96%	44.67%	97.92%	99.73%

Descriptive Statistics

Mean	0.853%
Standard Error	1.491%
Median	-1.101%
Mode	-27.710%
Standard Deviation	14.606%
Sample Variance	213.326%
Kurtosis	3.44
Skewness	1.11
Range	90.588%
Minimum	-27.710%
Maximum	62.878%
Count	96

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.557%	44	45.83%	5.755%
Negative	-9.051%	52	54.17%	-4.903%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 1.00. Further, an Excess Kurtosis of 2.26 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -89.51%	0	0.00%	0.00%
-89.51% to -65.88%	0	0.00%	0.00%
-65.88% to -42.24%	1	3.12%	3.12%
-42.24% to -18.60%	6	18.75%	21.88%
-18.60% to 5.04%	11	34.38%	56.25%
5.04% to 28.67%	8	25.00%	81.25%
28.67% to 52.31%	4	12.50%	93.75%
52.31% to 75.95%	1	3.12%	96.88%
75.95% to 99.58%	0	0.00%	96.88%
Greater than 99.58%	1	3.12%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-26.48%	36.55%	75.00%	68.27%
±2σ	-58.00%	68.07%	96.88%	95.45%
±3σ	-89.51%	99.58%	96.88%	99.73%

Descriptive Statistics

Mean	5.035%
Standard Error	5.571%
Median	-3.788%
Mode	-54.806%
Standard Deviation	31.516%
Sample Variance	993.270%
Kurtosis	2.26
Skewness	1.00
Range	161.804%
Minimum	-54.806%
Maximum	106.998%
Count	32

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	33.104%	14	43.75%	14.483%
Negative	-16.796%	18	56.25%	-9.448%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	27.12%
12	3 Years	35.14%
20	5 Years	39.86%
40	10 Years	42.88%
80	20 Years	42.88%
200	50 Years	42.88%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.82. Further, an Excess Kurtosis of 5.06 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -37.52%	0	0.00%	0.00%
-37.52% to -14.77%	0	0.00%	0.00%
-14.77% to 7.98%	0	0.00%	0.00%
7.98% to 30.73%	9	27.27%	27.27%
30.73% to 53.48%	10	30.30%	57.58%
53.48% to 76.22%	8	24.24%	81.82%
76.22% to 98.97%	4	12.12%	93.94%
98.97% to 121.72%	1	3.03%	96.97%
121.72% to 144.47%	0	0.00%	96.97%
Greater than 144.47%	1	3.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	23.14%	83.81%	87.88%	68.27%
±2σ	-7.19%	114.14%	96.97%	95.45%
±3σ	-37.52%	144.47%	96.97%	99.73%

Descriptive Statistics

Mean	53.475%
Standard Error	5.280%
Median	45.815%
Mode	10.514%
Standard Deviation	30.333%
Sample Variance	920.072%
Kurtosis	5.06
Skewness	1.82
Range	156.130%
Minimum	10.514%
Maximum	166.644%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	53.475%	33	100.00%	53.475%
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 1.49. Further, an Excess Kurtosis of 4.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -92.61%	0	0.00%	0.00%
-92.61% to -68.55%	0	0.00%	0.00%
-68.55% to -44.48%	1	3.03%	3.03%
-44.48% to -20.42%	6	18.18%	21.21%
-20.42% to 3.65%	10	30.30%	51.52%
3.65% to 27.71%	10	30.30%	81.82%
27.71% to 51.78%	4	12.12%	93.94%
51.78% to 75.84%	1	3.03%	96.97%
75.84% to 99.91%	0	0.00%	96.97%
Greater than 99.91%	1	3.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.44%	35.74%	81.82%	68.27%
±2σ	-60.53%	67.82%	96.97%	95.45%
±3σ	-92.61%	99.91%	96.97%	99.73%

Descriptive Statistics

Mean	3.649%
Standard Error	5.586%
Median	2.393%
Mode	-52.410%
Standard Deviation	32.087%
Sample Variance	1029.587%
Kurtosis	4.38
Skewness	1.49
Range	172.868%
Minimum	-52.410%
Maximum	120.458%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.420%	18	54.55%	13.320%
Negative	-21.275%	15	45.45%	-9.671%
Zero	0.000%	0	0.00%	0.000%

DOCU Deep Analytical Report