CZR - 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	3.70%	3.56%	3.18%	4.61%	3.17%	3.20%	25.84	27.53	25.03	28.41
Weekly	7.64%	11.41%	8.93%	10.60%	6.97%	7.05%	24.87	28.60	23.20	30.66
Monthly	17.07%	62.75%	25.84%	22.27%	14.51%	14.67%	23.07	30.83	19.95	35.65
Quarterly	26.61%	152.53%	34.73%	37.54%	25.13%	25.42%	20.74	34.29	16.13	44.09
Annual	—	—	—	—	50.26%	50.83%	16.13	44.09	9.76	72.88
Option Time Horizon	—	—	—	—	18.97%	19.19%	22.06	32.24	18.25	38.98

Calculation Notes: Computed at 2026-03-23T16:36 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $26.67 | Strike (K): $27.00 (first OTM call ≥ spot) | Observed Mid-Price: $1.9300 (Bid $1.6400 / Ask $2.2200) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 50.26% (CRR binomial tree, American, n=20 steps) | BS IV: 50.25% (European Black-Scholes) | Yahoo IV (Annual): 50.83% → 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 Fairly Symmetrical with a Skewness factor of 0.43. Further, an Excess Kurtosis of 20.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.96%	18	0.62%	0.62%
-10.96% to -8.19%	18	0.62%	1.25%
-8.19% to -5.42%	73	2.53%	3.77%
-5.42% to -2.64%	318	11.01%	14.78%
-2.64% to 0.13%	1053	36.45%	51.23%
0.13% to 2.90%	987	34.16%	85.39%
2.90% to 5.68%	295	10.21%	95.60%
5.68% to 8.45%	73	2.53%	98.13%
8.45% to 11.22%	33	1.14%	99.27%
Greater than 11.22%	21	0.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.57%	3.83%	80.72%	68.27%
±2σ	-7.26%	7.53%	95.98%	95.45%
±3σ	-10.96%	11.22%	98.65%	99.73%

Descriptive Statistics

Mean	0.131%
Standard Error	0.069%
Median	0.036%
Mode	0.000%
Standard Deviation	3.698%
Sample Variance	13.675%
Kurtosis	20.46
Skewness	0.43
Range	81.662%
Minimum	-37.505%
Maximum	44.157%
Count	2889

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.513%	1452	50.26%	1.263%
Negative	-2.350%	1391	48.15%	-1.131%
Zero	0.000%	46	1.59%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.14%
20	1 Month	6.15%
60	1 Quarter	5.41%
250	1 Year	4.67%
750	3 Years	4.14%
1,250	5 Years	4.61%
2,500	10 Years	4.76%
5,000	20 Years	4.78%
12,500	50 Years	4.78%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.56. Further, an Excess Kurtosis of 55.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.00%	0	0.00%	0.00%
-6.00% to -3.33%	0	0.00%	0.00%
-3.33% to -0.66%	0	0.00%	0.00%
-0.66% to 2.01%	212	7.34%	7.34%
2.01% to 4.67%	1666	57.65%	64.98%
4.67% to 7.34%	701	24.26%	89.24%
7.34% to 10.01%	180	6.23%	95.47%
10.01% to 12.68%	60	2.08%	97.54%
12.68% to 15.35%	20	0.69%	98.24%
Greater than 15.35%	51	1.76%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.12%	8.23%	91.49%	68.27%
±2σ	-2.44%	11.79%	96.71%	95.45%
±3σ	-6.00%	15.35%	98.24%	99.73%

Descriptive Statistics

Mean	4.674%
Standard Error	0.066%
Median	3.854%
Mode	3.226%
Standard Deviation	3.558%
Sample Variance	12.658%
Kurtosis	55.27
Skewness	5.56
Range	59.830%
Minimum	0.498%
Maximum	60.328%
Count	2890

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.674%	2890	100.00%	4.674%
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 1.95. Further, an Excess Kurtosis of 27.71 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.47%	16	0.55%	0.55%
-9.47% to -7.09%	24	0.83%	1.38%
-7.09% to -4.70%	82	2.84%	4.22%
-4.70% to -2.32%	354	12.25%	16.47%
-2.32% to 0.07%	1029	35.61%	52.08%
0.07% to 2.45%	919	31.80%	83.88%
2.45% to 4.84%	324	11.21%	95.09%
4.84% to 7.22%	95	3.29%	98.37%
7.22% to 9.61%	25	0.87%	99.24%
Greater than 9.61%	22	0.76%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.11%	3.25%	78.34%	68.27%
±2σ	-6.29%	6.43%	95.99%	95.45%
±3σ	-9.47%	9.61%	98.69%	99.73%

Descriptive Statistics

Mean	0.068%
Standard Error	0.059%
Median	0.000%
Mode	0.000%
Standard Deviation	3.180%
Sample Variance	10.112%
Kurtosis	27.71
Skewness	1.95
Range	65.283%
Minimum	-18.603%
Maximum	46.680%
Count	2890

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.286%	1400	48.44%	1.107%
Negative	-2.085%	1441	49.86%	-1.040%
Zero	0.000%	49	1.70%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.44. 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 -22.32%	7	1.17%	1.17%
-22.32% to -16.59%	3	0.50%	1.67%
-16.59% to -10.86%	15	2.50%	4.17%
-10.86% to -5.12%	64	10.67%	14.83%
-5.12% to 0.61%	221	36.83%	51.67%
0.61% to 6.34%	184	30.67%	82.33%
6.34% to 12.07%	79	13.17%	95.50%
12.07% to 17.81%	15	2.50%	98.00%
17.81% to 23.54%	7	1.17%	99.17%
Greater than 23.54%	5	0.83%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.04%	8.25%	78.17%	68.27%
±2σ	-14.68%	15.90%	95.33%	95.45%
±3σ	-22.32%	23.54%	98.00%	99.73%

Descriptive Statistics

Mean	0.609%
Standard Error	0.312%
Median	0.411%
Mode	-4.329%
Standard Deviation	7.644%
Sample Variance	58.431%
Kurtosis	6.23
Skewness	-0.44
Range	83.073%
Minimum	-47.882%
Maximum	35.191%
Count	600

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.466%	322	53.67%	2.933%
Negative	-5.017%	278	46.33%	-2.325%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	17.99%
12	1 Quarter	13.78%
52	1 Year	10.50%
156	3 Years	9.51%
260	5 Years	10.60%
520	10 Years	10.94%
1,040	20 Years	10.82%
2,600	50 Years	10.82%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 6.95. Further, an Excess Kurtosis of 69.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -22.37%	0	0.00%	0.00%
-22.37% to -13.81%	0	0.00%	0.00%
-13.81% to -5.25%	0	0.00%	0.00%
-5.25% to 3.31%	3	0.50%	0.50%
3.31% to 11.87%	400	66.56%	67.05%
11.87% to 20.43%	147	24.46%	91.51%
20.43% to 28.99%	30	4.99%	96.51%
28.99% to 37.56%	10	1.66%	98.17%
37.56% to 46.12%	3	0.50%	98.67%
Greater than 46.12%	8	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.46%	23.29%	94.18%	68.27%
±2σ	-10.96%	34.70%	97.50%	95.45%
±3σ	-22.37%	46.12%	98.67%	99.73%

Descriptive Statistics

Mean	11.872%
Standard Error	0.466%
Median	9.340%
Mode	6.211%
Standard Deviation	11.415%
Sample Variance	130.305%
Kurtosis	69.49
Skewness	6.95
Range	157.674%
Minimum	2.127%
Maximum	159.801%
Count	601

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.872%	601	100.00%	11.872%
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 4.11. Further, an Excess Kurtosis of 39.04 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.48%	0	0.00%	0.00%
-26.48% to -19.78%	2	0.33%	0.33%
-19.78% to -13.08%	12	2.00%	2.33%
-13.08% to -6.39%	73	12.15%	14.48%
-6.39% to 0.31%	251	41.76%	56.24%
0.31% to 7.01%	196	32.61%	88.85%
7.01% to 13.71%	41	6.82%	95.67%
13.71% to 20.41%	15	2.50%	98.17%
20.41% to 27.11%	3	0.50%	98.67%
Greater than 27.11%	8	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.62%	9.24%	83.53%	68.27%
±2σ	-17.55%	18.17%	97.00%	95.45%
±3σ	-26.48%	27.11%	98.67%	99.73%

Descriptive Statistics

Mean	0.312%
Standard Error	0.364%
Median	-0.326%
Mode	0.000%
Standard Deviation	8.931%
Sample Variance	79.756%
Kurtosis	39.04
Skewness	4.11
Range	128.731%
Minimum	-24.667%
Maximum	104.064%
Count	601

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.025%	281	46.76%	2.817%
Negative	-4.764%	316	52.58%	-2.505%
Zero	0.000%	4	0.67%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.15. Further, an Excess Kurtosis of 3.40 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -48.32%	1	0.72%	0.72%
-48.32% to -35.52%	0	0.00%	0.72%
-35.52% to -22.71%	6	4.35%	5.07%
-22.71% to -9.90%	17	12.32%	17.39%
-9.90% to 2.90%	51	36.96%	54.35%
2.90% to 15.71%	38	27.54%	81.88%
15.71% to 28.52%	17	12.32%	94.20%
28.52% to 41.32%	4	2.90%	97.10%
41.32% to 54.13%	3	2.17%	99.28%
Greater than 54.13%	1	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.17%	19.98%	75.36%	68.27%
±2σ	-31.25%	37.05%	95.65%	95.45%
±3σ	-48.32%	54.13%	98.55%	99.73%

Descriptive Statistics

Mean	2.903%
Standard Error	1.454%
Median	0.668%
Mode	-71.303%
Standard Deviation	17.075%
Sample Variance	291.569%
Kurtosis	3.40
Skewness	0.15
Range	136.695%
Minimum	-71.303%
Maximum	65.392%
Count	138

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.896%	76	55.07%	7.653%
Negative	-10.573%	62	44.93%	-4.750%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	30.83%
12	1 Year	22.52%
36	3 Years	19.92%
60	5 Years	22.27%
120	10 Years	23.99%
240	20 Years	23.51%
600	50 Years	23.51%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 10.39. Further, an Excess Kurtosis of 115.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -156.86%	0	0.00%	0.00%
-156.86% to -109.79%	0	0.00%	0.00%
-109.79% to -62.73%	0	0.00%	0.00%
-62.73% to -15.66%	0	0.00%	0.00%
-15.66% to 31.40%	104	74.82%	74.82%
31.40% to 78.47%	32	23.02%	97.84%
78.47% to 125.53%	1	0.72%	98.56%
125.53% to 172.60%	1	0.72%	99.28%
172.60% to 219.66%	0	0.00%	99.28%
Greater than 219.66%	1	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.35%	94.16%	97.84%	68.27%
±2σ	-94.10%	156.91%	99.28%	95.45%
±3σ	-156.86%	219.66%	99.28%	99.73%

Descriptive Statistics

Mean	31.403%
Standard Error	5.323%
Median	21.555%
Mode	8.039%
Standard Deviation	62.754%
Sample Variance	3938.007%
Kurtosis	115.93
Skewness	10.39
Range	725.848%
Minimum	8.039%
Maximum	733.887%
Count	139

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.403%	139	100.00%	31.403%
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 6.42. Further, an Excess Kurtosis of 60.94 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -76.68%	0	0.00%	0.00%
-76.68% to -57.30%	0	0.00%	0.00%
-57.30% to -37.92%	1	0.72%	0.72%
-37.92% to -18.54%	10	7.19%	7.91%
-18.54% to 0.84%	66	47.48%	55.40%
0.84% to 20.22%	48	34.53%	89.93%
20.22% to 39.60%	13	9.35%	99.28%
39.60% to 58.98%	0	0.00%	99.28%
58.98% to 78.36%	0	0.00%	99.28%
Greater than 78.36%	1	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.00%	26.68%	89.93%	68.27%
±2σ	-50.84%	52.52%	99.28%	95.45%
±3σ	-76.68%	78.36%	99.28%	99.73%

Descriptive Statistics

Mean	0.841%
Standard Error	2.192%
Median	-1.235%
Mode	-0.823%
Standard Deviation	25.839%
Sample Variance	667.674%
Kurtosis	60.94
Skewness	6.42
Range	292.652%
Minimum	-44.388%
Maximum	248.264%
Count	139

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.519%	63	45.32%	7.034%
Negative	-11.326%	76	54.68%	-6.193%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.01. Further, an Excess Kurtosis of 0.56 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -72.09%	0	0.00%	0.00%
-72.09% to -52.13%	1	2.17%	2.17%
-52.13% to -32.17%	1	2.17%	4.35%
-32.17% to -12.21%	9	19.57%	23.91%
-12.21% to 7.74%	13	28.26%	52.17%
7.74% to 27.70%	11	23.91%	76.09%
27.70% to 47.66%	9	19.57%	95.65%
47.66% to 67.62%	1	2.17%	97.83%
67.62% to 87.58%	1	2.17%	100.00%
Greater than 87.58%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-18.87%	34.36%	73.91%	68.27%
±2σ	-45.48%	60.97%	95.65%	95.45%
±3σ	-72.09%	87.58%	100.00%	99.73%

Descriptive Statistics

Mean	7.744%
Standard Error	3.924%
Median	6.283%
Mode	-64.135%
Standard Deviation	26.611%
Sample Variance	708.154%
Kurtosis	0.56
Skewness	-0.01
Range	141.941%
Minimum	-64.135%
Maximum	77.806%
Count	46

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.244%	28	60.87%	14.757%
Negative	-17.921%	18	39.13%	-7.012%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	39.73%
12	3 Years	36.11%
20	5 Years	37.54%
40	10 Years	43.01%
80	20 Years	42.66%
200	50 Years	42.66%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 6.49. Further, an Excess Kurtosis of 43.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -387.31%	0	0.00%	0.00%
-387.31% to -272.92%	0	0.00%	0.00%
-272.92% to -158.52%	0	0.00%	0.00%
-158.52% to -44.13%	0	0.00%	0.00%
-44.13% to 70.27%	39	82.98%	82.98%
70.27% to 184.66%	7	14.89%	97.87%
184.66% to 299.06%	0	0.00%	97.87%
299.06% to 413.45%	0	0.00%	97.87%
413.45% to 527.85%	0	0.00%	97.87%
Greater than 527.85%	1	2.13%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-82.26%	222.80%	97.87%	68.27%
±2σ	-234.78%	375.32%	97.87%	95.45%
±3σ	-387.31%	527.85%	97.87%	99.73%

Descriptive Statistics

Mean	70.269%
Standard Error	22.248%
Median	38.933%
Mode	15.789%
Standard Deviation	152.526%
Sample Variance	23264.146%
Kurtosis	43.46
Skewness	6.49
Range	1059.294%
Minimum	15.789%
Maximum	1075.083%
Count	47

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	70.269%	47	100.00%	70.269%
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 3.19. Further, an Excess Kurtosis of 15.40 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -103.72%	0	0.00%	0.00%
-103.72% to -77.68%	0	0.00%	0.00%
-77.68% to -51.63%	0	0.00%	0.00%
-51.63% to -25.58%	6	12.77%	12.77%
-25.58% to 0.47%	22	46.81%	59.57%
0.47% to 26.52%	13	27.66%	87.23%
26.52% to 52.57%	5	10.64%	97.87%
52.57% to 78.61%	0	0.00%	97.87%
78.61% to 104.66%	0	0.00%	97.87%
Greater than 104.66%	1	2.13%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-34.26%	35.20%	85.11%	68.27%
±2σ	-68.99%	69.93%	97.87%	95.45%
±3σ	-103.72%	104.66%	97.87%	99.73%

Descriptive Statistics

Mean	0.469%
Standard Error	5.066%
Median	-5.489%
Mode	-44.255%
Standard Deviation	34.731%
Sample Variance	1206.262%
Kurtosis	15.40
Skewness	3.19
Range	225.458%
Minimum	-44.255%
Maximum	181.203%
Count	47

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	27.811%	19	40.43%	11.243%
Negative	-18.084%	28	59.57%	-10.773%
Zero	0.000%	0	0.00%	0.000%

CZR Deep Analytical Report