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	3.68%	3.54%	3.17%	4.61%	3.41%	3.42%	20.52	21.97	19.83	22.73
Weekly	7.63%	11.41%	8.93%	10.53%	7.50%	7.53%	19.70	22.88	18.27	24.67
Monthly	17.06%	62.97%	25.87%	21.94%	15.62%	15.67%	18.16	24.82	15.53	29.02
Quarterly	26.42%	152.66%	34.55%	35.50%	27.06%	27.15%	16.20	27.83	12.36	36.47
Annual	—	—	—	—	54.12%	54.30%	12.36	36.47	7.19	62.66
Option Time Horizon	—	—	—	—	17.69%	17.75%	17.79	25.34	14.90	30.24

Calculation Notes: Computed at 2026-02-22T14:20 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $21.23 | Strike (K): $24.50 (first OTM call ≥ spot) | Observed Mid-Price: $0.4950 (Bid $0.4100 / Ask $0.5800) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 54.12% (CRR binomial tree, American, n=20 steps) | BS IV: 54.18% (European Black-Scholes) | Yahoo IV (Annual): 54.30% → 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 0.38. Further, an Excess Kurtosis of 20.72 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.93%	18	0.63%	0.63%
-10.93% to -8.17%	18	0.63%	1.26%
-8.17% to -5.40%	73	2.55%	3.80%
-5.40% to -2.64%	316	11.02%	14.82%
-2.64% to 0.12%	1044	36.40%	51.22%
0.12% to 2.89%	976	34.03%	85.25%
2.89% to 5.65%	296	10.32%	95.57%
5.65% to 8.41%	75	2.62%	98.19%
8.41% to 11.18%	33	1.15%	99.34%
Greater than 11.18%	19	0.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.56%	3.81%	80.58%	68.27%
±2σ	-7.25%	7.49%	95.96%	95.45%
±3σ	-10.93%	11.17%	98.71%	99.73%

Descriptive Statistics

Mean	0.123%
Standard Error	0.069%
Median	0.032%
Mode	0.000%
Standard Deviation	3.684%
Sample Variance	13.572%
Kurtosis	20.72
Skewness	0.38
Range	81.662%
Minimum	-37.505%
Maximum	44.157%
Count	2868

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.503%	1440	50.21%	1.257%
Negative	-2.351%	1382	48.19%	-1.133%
Zero	0.000%	46	1.60%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	7.48%
20	1 Month	5.77%
60	1 Quarter	4.48%
250	1 Year	4.58%
750	3 Years	4.11%
1,250	5 Years	4.61%
2,500	10 Years	4.74%
5,000	20 Years	4.77%
12,500	50 Years	4.77%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.61. Further, an Excess Kurtosis of 56.45 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.97%	0	0.00%	0.00%
-5.97% to -3.31%	0	0.00%	0.00%
-3.31% to -0.65%	0	0.00%	0.00%
-0.65% to 2.00%	212	7.39%	7.39%
2.00% to 4.66%	1650	57.51%	64.90%
4.66% to 7.32%	696	24.26%	89.16%
7.32% to 9.98%	181	6.31%	95.47%
9.98% to 12.64%	60	2.09%	97.56%
12.64% to 15.29%	21	0.73%	98.29%
Greater than 15.29%	49	1.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.12%	8.21%	91.50%	68.27%
±2σ	-2.42%	11.75%	96.65%	95.45%
±3σ	-5.97%	15.29%	98.29%	99.73%

Descriptive Statistics

Mean	4.663%
Standard Error	0.066%
Median	3.849%
Mode	3.226%
Standard Deviation	3.544%
Sample Variance	12.557%
Kurtosis	56.45
Skewness	5.61
Range	59.830%
Minimum	0.498%
Maximum	60.328%
Count	2869

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.663%	2869	100.00%	4.663%
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 2.03. Further, an Excess Kurtosis of 28.17 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.42%	14	0.49%	0.49%
-9.42% to -7.05%	24	0.84%	1.32%
-7.05% to -4.67%	83	2.89%	4.22%
-4.67% to -2.30%	355	12.37%	16.59%
-2.30% to 0.08%	1018	35.48%	52.07%
0.08% to 2.45%	912	31.79%	83.86%
2.45% to 4.83%	322	11.22%	95.09%
4.83% to 7.21%	94	3.28%	98.36%
7.21% to 9.58%	25	0.87%	99.23%
Greater than 9.58%	22	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.09%	3.25%	78.25%	68.27%
±2σ	-6.25%	6.41%	95.99%	95.45%
±3σ	-9.42%	9.58%	98.75%	99.73%

Descriptive Statistics

Mean	0.079%
Standard Error	0.059%
Median	0.000%
Mode	0.000%
Standard Deviation	3.167%
Sample Variance	10.033%
Kurtosis	28.17
Skewness	2.03
Range	65.283%
Minimum	-18.603%
Maximum	46.680%
Count	2869

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.287%	1393	48.55%	1.110%
Negative	-2.074%	1427	49.74%	-1.032%
Zero	0.000%	49	1.71%	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 6.27 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 -16.60%	3	0.50%	1.68%
-16.60% to -10.88%	15	2.52%	4.19%
-10.88% to -5.15%	64	10.74%	14.93%
-5.15% to 0.57%	219	36.74%	51.68%
0.57% to 6.29%	183	30.70%	82.38%
6.29% to 12.02%	78	13.09%	95.47%
12.02% to 17.74%	16	2.68%	98.15%
17.74% to 23.47%	6	1.01%	99.16%
Greater than 23.47%	5	0.84%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.06%	8.20%	78.19%	68.27%
±2σ	-14.69%	15.83%	95.30%	95.45%
±3σ	-22.32%	23.47%	97.99%	99.73%

Descriptive Statistics

Mean	0.571%
Standard Error	0.313%
Median	0.400%
Mode	-4.329%
Standard Deviation	7.632%
Sample Variance	58.246%
Kurtosis	6.27
Skewness	-0.46
Range	83.073%
Minimum	-47.882%
Maximum	35.191%
Count	596

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.435%	319	53.52%	2.909%
Negative	-5.029%	277	46.48%	-2.337%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	15.37%
12	1 Quarter	9.82%
52	1 Year	10.05%
156	3 Years	9.39%
260	5 Years	10.53%
520	10 Years	10.86%
1,040	20 Years	10.78%
2,600	50 Years	10.78%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 7.00. Further, an Excess Kurtosis of 70.15 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -22.41%	0	0.00%	0.00%
-22.41% to -13.85%	0	0.00%	0.00%
-13.85% to -5.29%	0	0.00%	0.00%
-5.29% to 3.27%	3	0.50%	0.50%
3.27% to 11.83%	395	66.16%	66.67%
11.83% to 20.39%	150	25.13%	91.79%
20.39% to 28.95%	29	4.86%	96.65%
28.95% to 37.51%	9	1.51%	98.16%
37.51% to 46.06%	3	0.50%	98.66%
Greater than 46.06%	8	1.34%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.42%	23.24%	94.14%	68.27%
±2σ	-11.00%	34.65%	97.49%	95.45%
±3σ	-22.41%	46.06%	98.66%	99.73%

Descriptive Statistics

Mean	11.828%
Standard Error	0.467%
Median	9.330%
Mode	6.211%
Standard Deviation	11.412%
Sample Variance	130.237%
Kurtosis	70.15
Skewness	7.00
Range	157.674%
Minimum	2.127%
Maximum	159.801%
Count	597

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.828%	597	100.00%	11.828%
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 4.13. Further, an Excess Kurtosis of 39.22 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.44%	0	0.00%	0.00%
-26.44% to -19.74%	2	0.34%	0.34%
-19.74% to -13.04%	13	2.18%	2.51%
-13.04% to -6.34%	72	12.06%	14.57%
-6.34% to 0.35%	248	41.54%	56.11%
0.35% to 7.05%	195	32.66%	88.78%
7.05% to 13.75%	41	6.87%	95.64%
13.75% to 20.45%	15	2.51%	98.16%
20.45% to 27.15%	3	0.50%	98.66%
Greater than 27.15%	8	1.34%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.58%	9.28%	83.25%	68.27%
±2σ	-17.51%	18.22%	97.15%	95.45%
±3σ	-26.44%	27.15%	98.66%	99.73%

Descriptive Statistics

Mean	0.354%
Standard Error	0.366%
Median	-0.298%
Mode	0.000%
Standard Deviation	8.931%
Sample Variance	79.758%
Kurtosis	39.22
Skewness	4.13
Range	128.731%
Minimum	-24.667%
Maximum	104.064%
Count	597

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.040%	280	46.90%	2.833%
Negative	-4.727%	313	52.43%	-2.478%
Zero	0.000%	4	0.67%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.17. Further, an Excess Kurtosis of 3.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -48.45%	1	0.73%	0.73%
-48.45% to -35.65%	0	0.00%	0.73%
-35.65% to -22.86%	6	4.38%	5.11%
-22.86% to -10.06%	17	12.41%	17.52%
-10.06% to 2.74%	52	37.96%	55.47%
2.74% to 15.54%	37	27.01%	82.48%
15.54% to 28.34%	16	11.68%	94.16%
28.34% to 41.13%	4	2.92%	97.08%
41.13% to 53.93%	3	2.19%	99.27%
Greater than 53.93%	1	0.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.32%	19.80%	76.64%	68.27%
±2σ	-31.39%	36.87%	95.62%	95.45%
±3σ	-48.45%	53.93%	98.54%	99.73%

Descriptive Statistics

Mean	2.740%
Standard Error	1.458%
Median	0.621%
Mode	-71.303%
Standard Deviation	17.064%
Sample Variance	291.169%
Kurtosis	3.46
Skewness	0.17
Range	136.695%
Minimum	-71.303%
Maximum	65.392%
Count	137

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.746%	75	54.74%	7.525%
Negative	-10.573%	62	45.26%	-4.785%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	18.97%
12	1 Year	21.15%
36	3 Years	19.43%
60	5 Years	21.94%
120	10 Years	23.78%
240	20 Years	23.34%
600	50 Years	23.34%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 10.36. Further, an Excess Kurtosis of 115.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -157.65%	0	0.00%	0.00%
-157.65% to -110.42%	0	0.00%	0.00%
-110.42% to -63.19%	0	0.00%	0.00%
-63.19% to -15.96%	0	0.00%	0.00%
-15.96% to 31.27%	105	76.09%	76.09%
31.27% to 78.50%	30	21.74%	97.83%
78.50% to 125.72%	1	0.72%	98.55%
125.72% to 172.95%	1	0.72%	99.28%
172.95% to 220.18%	0	0.00%	99.28%
Greater than 220.18%	1	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.71%	94.24%	97.83%	68.27%
±2σ	-94.68%	157.21%	99.28%	95.45%
±3σ	-157.65%	220.18%	99.28%	99.73%

Descriptive Statistics

Mean	31.265%
Standard Error	5.361%
Median	21.450%
Mode	8.039%
Standard Deviation	62.973%
Sample Variance	3965.572%
Kurtosis	115.27
Skewness	10.36
Range	725.848%
Minimum	8.039%
Maximum	733.887%
Count	138

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.265%	138	100.00%	31.265%
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 6.43. Further, an Excess Kurtosis of 60.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -76.60%	0	0.00%	0.00%
-76.60% to -57.19%	0	0.00%	0.00%
-57.19% to -37.79%	1	0.72%	0.72%
-37.79% to -18.39%	10	7.25%	7.97%
-18.39% to 1.02%	66	47.83%	55.80%
1.02% to 20.42%	47	34.06%	89.86%
20.42% to 39.82%	13	9.42%	99.28%
39.82% to 59.23%	0	0.00%	99.28%
59.23% to 78.63%	0	0.00%	99.28%
Greater than 78.63%	1	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.85%	26.89%	89.86%	68.27%
±2σ	-50.73%	52.76%	99.28%	95.45%
±3σ	-76.60%	78.63%	99.28%	99.73%

Descriptive Statistics

Mean	1.016%
Standard Error	2.202%
Median	-1.080%
Mode	-0.823%
Standard Deviation	25.871%
Sample Variance	669.314%
Kurtosis	60.93
Skewness	6.43
Range	292.652%
Minimum	-44.388%
Maximum	248.264%
Count	138

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.519%	63	45.65%	7.085%
Negative	-11.165%	75	54.35%	-6.068%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.05. 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 -72.11%	0	0.00%	0.00%
-72.11% to -52.29%	1	2.17%	2.17%
-52.29% to -32.47%	0	0.00%	2.17%
-32.47% to -12.65%	10	21.74%	23.91%
-12.65% to 7.17%	14	30.43%	54.35%
7.17% to 26.98%	10	21.74%	76.09%
26.98% to 46.80%	9	19.57%	95.65%
46.80% to 66.62%	1	2.17%	97.83%
66.62% to 86.44%	1	2.17%	100.00%
Greater than 86.44%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.26%	33.59%	73.91%	68.27%
±2σ	-45.68%	60.02%	95.65%	95.45%
±3σ	-72.10%	86.44%	100.00%	99.73%

Descriptive Statistics

Mean	7.167%
Standard Error	3.896%
Median	5.850%
Mode	-64.135%
Standard Deviation	26.424%
Sample Variance	698.247%
Kurtosis	0.66
Skewness	0.05
Range	141.941%
Minimum	-64.135%
Maximum	77.806%
Count	46

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.295%	28	60.87%	14.180%
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	29.55%
12	3 Years	32.71%
20	5 Years	35.50%
40	10 Years	41.99%
80	20 Years	41.77%
200	50 Years	41.77%

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 43.48 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -388.70%	0	0.00%	0.00%
-388.70% to -274.21%	0	0.00%	0.00%
-274.21% to -159.71%	0	0.00%	0.00%
-159.71% to -45.22%	0	0.00%	0.00%
-45.22% to 69.28%	40	85.11%	85.11%
69.28% to 183.77%	6	12.77%	97.87%
183.77% to 298.27%	0	0.00%	97.87%
298.27% to 412.76%	0	0.00%	97.87%
412.76% to 527.26%	0	0.00%	97.87%
Greater than 527.26%	1	2.13%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-83.38%	221.94%	97.87%	68.27%
±2σ	-236.04%	374.60%	97.87%	95.45%
±3σ	-388.70%	527.26%	97.87%	99.73%

Descriptive Statistics

Mean	69.277%
Standard Error	22.268%
Median	38.889%
Mode	15.789%
Standard Deviation	152.660%
Sample Variance	23305.094%
Kurtosis	43.48
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	69.277%	47	100.00%	69.277%
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 3.20. Further, an Excess Kurtosis of 15.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -102.77%	0	0.00%	0.00%
-102.77% to -76.86%	0	0.00%	0.00%
-76.86% to -50.94%	0	0.00%	0.00%
-50.94% to -25.02%	6	12.77%	12.77%
-25.02% to 0.89%	23	48.94%	61.70%
0.89% to 26.81%	12	25.53%	87.23%
26.81% to 52.72%	5	10.64%	97.87%
52.72% to 78.64%	0	0.00%	97.87%
78.64% to 104.55%	0	0.00%	97.87%
Greater than 104.55%	1	2.13%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-33.66%	35.45%	85.11%	68.27%
±2σ	-68.22%	70.00%	97.87%	95.45%
±3σ	-102.77%	104.55%	97.87%	99.73%

Descriptive Statistics

Mean	0.891%
Standard Error	5.040%
Median	-5.017%
Mode	-44.255%
Standard Deviation	34.554%
Sample Variance	1193.970%
Kurtosis	15.61
Skewness	3.20
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	-17.376%	28	59.57%	-10.352%
Zero	0.000%	0	0.00%	0.000%

CZR Deep Analytical Report