CROX - 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	4.12%	3.77%	3.36%	4.51%	2.33%	2.49%	97.73	102.40	95.48	104.81
Weekly	9.30%	12.43%	10.05%	10.80%	5.13%	5.48%	95.04	105.31	90.28	110.85
Monthly	17.25%	32.24%	17.88%	22.57%	10.68%	11.41%	89.90	111.32	80.79	123.87
Quarterly	33.39%	67.23%	43.67%	36.59%	18.50%	19.77%	83.14	120.37	69.10	144.84
Annual	—	—	—	—	37.01%	39.54%	69.10	144.84	47.72	209.70
Option Time Horizon	—	—	—	—	14.23%	15.21%	86.77	115.34	75.26	132.99

Calculation Notes: Computed at 2026-02-22T14:19 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $100.04 | Strike (K): $105.00 (first OTM call ≥ spot) | Observed Mid-Price: $3.9000 (Bid $3.7000 / Ask $4.1000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 37.01% (CRR binomial tree, American, n=20 steps) | BS IV: 37.21% (European Black-Scholes) | Yahoo IV (Annual): 39.54% → 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 -0.53. Further, an Excess Kurtosis of 20.34 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.22%	43	0.85%	0.85%
-12.22% to -9.13%	38	0.75%	1.61%
-9.13% to -6.05%	105	2.08%	3.69%
-6.05% to -2.96%	464	9.21%	12.90%
-2.96% to 0.13%	1892	37.55%	50.45%
0.13% to 3.21%	1833	36.38%	86.82%
3.21% to 6.30%	459	9.11%	95.93%
6.30% to 9.39%	117	2.32%	98.25%
9.39% to 12.47%	40	0.79%	99.05%
Greater than 12.47%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.99%	4.24%	83.25%	68.27%
±2σ	-8.11%	8.36%	95.71%	95.45%
±3σ	-12.22%	12.47%	98.19%	99.73%

Descriptive Statistics

Mean	0.126%
Standard Error	0.058%
Median	0.103%
Mode	0.000%
Standard Deviation	4.116%
Sample Variance	16.941%
Kurtosis	20.34
Skewness	-0.53
Range	86.658%
Minimum	-44.737%
Maximum	41.921%
Count	5039

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.609%	2582	51.24%	1.337%
Negative	-2.537%	2405	47.73%	-1.211%
Zero	0.000%	52	1.03%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.47%
20	1 Month	4.59%
60	1 Quarter	3.84%
250	1 Year	4.17%
750	3 Years	3.96%
1,250	5 Years	4.51%
2,500	10 Years	4.47%
5,000	20 Years	4.90%
12,500	50 Years	4.90%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.48. Further, an Excess Kurtosis of 36.68 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.57%	0	0.00%	0.00%
-6.57% to -3.75%	0	0.00%	0.00%
-3.75% to -0.92%	0	0.00%	0.00%
-0.92% to 1.91%	348	6.90%	6.90%
1.91% to 4.73%	3030	60.12%	67.02%
4.73% to 7.56%	1054	20.91%	87.94%
7.56% to 10.39%	311	6.17%	94.11%
10.39% to 13.21%	129	2.56%	96.67%
13.21% to 16.04%	65	1.29%	97.96%
Greater than 16.04%	103	2.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.96%	8.50%	90.77%	68.27%
±2σ	-2.81%	12.27%	96.13%	95.45%
±3σ	-6.57%	16.04%	97.96%	99.73%

Descriptive Statistics

Mean	4.733%
Standard Error	0.053%
Median	3.704%
Mode	1.966%
Standard Deviation	3.769%
Sample Variance	14.205%
Kurtosis	36.68
Skewness	4.48
Range	60.673%
Minimum	0.756%
Maximum	61.429%
Count	5040

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.733%	5040	100.00%	4.733%
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 0.72. Further, an Excess Kurtosis of 11.40 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.00%	36	0.71%	0.71%
-10.00% to -7.48%	43	0.85%	1.57%
-7.48% to -4.96%	144	2.86%	4.42%
-4.96% to -2.44%	575	11.41%	15.83%
-2.44% to 0.08%	1756	34.84%	50.67%
0.08% to 2.60%	1741	34.54%	85.22%
2.60% to 5.12%	511	10.14%	95.36%
5.12% to 7.64%	122	2.42%	97.78%
7.64% to 10.15%	64	1.27%	99.05%
Greater than 10.15%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.28%	3.44%	79.92%	68.27%
±2σ	-6.64%	6.80%	95.16%	95.45%
±3σ	-10.00%	10.15%	98.33%	99.73%

Descriptive Statistics

Mean	0.078%
Standard Error	0.047%
Median	0.027%
Mode	0.000%
Standard Deviation	3.359%
Sample Variance	11.285%
Kurtosis	11.40
Skewness	0.72
Range	63.518%
Minimum	-27.077%
Maximum	36.441%
Count	5040

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.311%	2527	50.14%	1.159%
Negative	-2.216%	2459	48.79%	-1.081%
Zero	0.000%	54	1.07%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.64. Further, an Excess Kurtosis of 13.09 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.28%	6	0.57%	0.57%
-27.28% to -20.31%	21	2.01%	2.58%
-20.31% to -13.33%	24	2.30%	4.88%
-13.33% to -6.35%	88	8.42%	13.30%
-6.35% to 0.63%	386	36.94%	50.24%
0.63% to 7.61%	364	34.83%	85.07%
7.61% to 14.58%	115	11.00%	96.08%
14.58% to 21.56%	25	2.39%	98.47%
21.56% to 28.54%	9	0.86%	99.33%
Greater than 28.54%	7	0.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.68%	9.93%	82.11%	68.27%
±2σ	-17.98%	19.24%	94.07%	95.45%
±3σ	-27.28%	28.54%	98.76%	99.73%

Descriptive Statistics

Mean	0.628%
Standard Error	0.288%
Median	0.620%
Mode	0.000%
Standard Deviation	9.304%
Sample Variance	86.566%
Kurtosis	13.09
Skewness	0.64
Range	140.988%
Minimum	-52.893%
Maximum	88.095%
Count	1045

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.254%	556	53.21%	3.328%
Negative	-5.841%	483	46.22%	-2.700%
Zero	0.000%	6	0.57%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	10.61%
12	1 Quarter	8.08%
52	1 Year	9.48%
156	3 Years	9.30%
260	5 Years	10.80%
520	10 Years	10.56%
1,040	20 Years	11.67%
2,600	50 Years	11.67%

High-to-Low 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 70.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.61%	0	0.00%	0.00%
-24.61% to -15.29%	0	0.00%	0.00%
-15.29% to -5.96%	0	0.00%	0.00%
-5.96% to 3.36%	9	0.86%	0.86%
3.36% to 12.69%	711	67.97%	68.83%
12.69% to 22.01%	205	19.60%	88.43%
22.01% to 31.34%	70	6.69%	95.12%
31.34% to 40.66%	23	2.20%	97.32%
40.66% to 49.99%	14	1.34%	98.66%
Greater than 49.99%	14	1.34%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.25%	25.12%	91.68%	68.27%
±2σ	-12.18%	37.56%	96.85%	95.45%
±3σ	-24.61%	49.99%	98.66%	99.73%

Descriptive Statistics

Mean	12.688%
Standard Error	0.384%
Median	9.178%
Mode	4.767%
Standard Deviation	12.434%
Sample Variance	154.609%
Kurtosis	70.61
Skewness	6.42
Range	199.951%
Minimum	2.581%
Maximum	202.532%
Count	1046

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.688%	1046	100.00%	12.688%
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 3.33. Further, an Excess Kurtosis of 31.13 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.82%	3	0.29%	0.29%
-29.82% to -22.28%	5	0.48%	0.76%
-22.28% to -14.74%	23	2.20%	2.96%
-14.74% to -7.20%	100	9.56%	12.52%
-7.20% to 0.34%	443	42.35%	54.88%
0.34% to 7.88%	355	33.94%	88.81%
7.88% to 15.42%	70	6.69%	95.51%
15.42% to 22.96%	17	1.63%	97.13%
22.96% to 30.50%	14	1.34%	98.47%
Greater than 30.50%	16	1.53%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.71%	10.39%	85.56%	68.27%
±2σ	-19.77%	20.45%	95.79%	95.45%
±3σ	-29.82%	30.50%	98.18%	99.73%

Descriptive Statistics

Mean	0.341%
Standard Error	0.311%
Median	-0.474%
Mode	0.000%
Standard Deviation	10.053%
Sample Variance	101.069%
Kurtosis	31.13
Skewness	3.33
Range	154.847%
Minimum	-44.620%
Maximum	110.227%
Count	1046

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.816%	489	46.75%	3.186%
Negative	-5.393%	552	52.77%	-2.846%
Zero	0.000%	5	0.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.81. Further, an Excess Kurtosis of 4.82 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -49.49%	0	0.00%	0.00%
-49.49% to -36.55%	4	1.67%	1.67%
-36.55% to -23.61%	9	3.75%	5.42%
-23.61% to -10.67%	25	10.42%	15.83%
-10.67% to 2.27%	90	37.50%	53.33%
2.27% to 15.21%	72	30.00%	83.33%
15.21% to 28.15%	28	11.67%	95.00%
28.15% to 41.09%	6	2.50%	97.50%
41.09% to 54.03%	4	1.67%	99.17%
Greater than 54.03%	2	0.83%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.98%	19.52%	78.75%	68.27%
±2σ	-32.24%	36.77%	95.00%	95.45%
±3σ	-49.49%	54.03%	99.17%	99.73%

Descriptive Statistics

Mean	2.269%
Standard Error	1.114%
Median	1.058%
Mode	-49.402%
Standard Deviation	17.253%
Sample Variance	297.672%
Kurtosis	4.82
Skewness	0.81
Range	138.478%
Minimum	-49.402%
Maximum	89.076%
Count	240

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.389%	129	53.75%	7.197%
Negative	-10.751%	110	45.83%	-4.928%
Zero	0.000%	1	0.42%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	14.06%
12	1 Year	18.35%
36	3 Years	19.76%
60	5 Years	22.57%
120	10 Years	22.92%
240	20 Years	25.24%
600	50 Years	25.24%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.93. Further, an Excess Kurtosis of 48.48 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -66.02%	0	0.00%	0.00%
-66.02% to -41.84%	0	0.00%	0.00%
-41.84% to -17.66%	0	0.00%	0.00%
-17.66% to 6.51%	2	0.83%	0.83%
6.51% to 30.69%	172	71.37%	72.20%
30.69% to 54.87%	45	18.67%	90.87%
54.87% to 79.05%	12	4.98%	95.85%
79.05% to 103.22%	3	1.24%	97.10%
103.22% to 127.40%	4	1.66%	98.76%
Greater than 127.40%	3	1.24%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.54%	62.93%	94.61%	68.27%
±2σ	-33.78%	95.16%	96.27%	95.45%
±3σ	-66.02%	127.40%	98.76%	99.73%

Descriptive Statistics

Mean	30.692%
Standard Error	2.077%
Median	23.457%
Mode	6.209%
Standard Deviation	32.236%
Sample Variance	1039.177%
Kurtosis	48.48
Skewness	5.93
Range	345.690%
Minimum	6.209%
Maximum	351.899%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.692%	241	100.00%	30.692%
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 1.69. Further, an Excess Kurtosis of 6.89 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -52.95%	0	0.00%	0.00%
-52.95% to -39.54%	2	0.83%	0.83%
-39.54% to -26.12%	5	2.07%	2.90%
-26.12% to -12.71%	34	14.11%	17.01%
-12.71% to 0.70%	96	39.83%	56.85%
0.70% to 14.11%	71	29.46%	86.31%
14.11% to 27.53%	17	7.05%	93.36%
27.53% to 40.94%	9	3.73%	97.10%
40.94% to 54.35%	3	1.24%	98.34%
Greater than 54.35%	4	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.18%	18.59%	80.08%	68.27%
±2σ	-35.07%	36.47%	95.02%	95.45%
±3σ	-52.95%	54.35%	98.34%	99.73%

Descriptive Statistics

Mean	0.701%
Standard Error	1.152%
Median	-0.805%
Mode	5.081%
Standard Deviation	17.884%
Sample Variance	319.849%
Kurtosis	6.89
Skewness	1.69
Range	150.866%
Minimum	-45.354%
Maximum	105.512%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.009%	110	45.64%	6.394%
Negative	-10.554%	130	53.94%	-5.693%
Zero	0.000%	1	0.41%	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 0.66 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -92.18%	0	0.00%	0.00%
-92.18% to -67.14%	1	1.25%	1.25%
-67.14% to -42.09%	5	6.25%	7.50%
-42.09% to -17.05%	12	15.00%	22.50%
-17.05% to 8.00%	22	27.50%	50.00%
8.00% to 33.04%	25	31.25%	81.25%
33.04% to 58.08%	11	13.75%	95.00%
58.08% to 83.13%	2	2.50%	97.50%
83.13% to 108.17%	1	1.25%	98.75%
Greater than 108.17%	1	1.25%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.40%	41.39%	71.25%	68.27%
±2σ	-58.79%	74.78%	95.00%	95.45%
±3σ	-92.18%	108.17%	98.75%	99.73%

Descriptive Statistics

Mean	7.996%
Standard Error	3.733%
Median	7.942%
Mode	-70.652%
Standard Deviation	33.392%
Sample Variance	1115.021%
Kurtosis	0.66
Skewness	0.29
Range	182.964%
Minimum	-70.652%
Maximum	112.312%
Count	80

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	28.908%	48	60.00%	17.345%
Negative	-23.371%	32	40.00%	-9.348%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	26.89%
12	3 Years	30.40%
20	5 Years	36.59%
40	10 Years	41.17%
80	20 Years	46.52%
200	50 Years	46.52%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.03. Further, an Excess Kurtosis of 9.91 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -134.69%	0	0.00%	0.00%
-134.69% to -84.26%	0	0.00%	0.00%
-84.26% to -33.84%	0	0.00%	0.00%
-33.84% to 16.59%	1	1.23%	1.23%
16.59% to 67.02%	58	71.60%	72.84%
67.02% to 117.44%	13	16.05%	88.89%
117.44% to 167.87%	4	4.94%	93.83%
167.87% to 218.29%	1	1.23%	95.06%
218.29% to 268.72%	1	1.23%	96.30%
Greater than 268.72%	3	3.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-0.22%	134.25%	91.36%	68.27%
±2σ	-67.45%	201.48%	95.06%	95.45%
±3σ	-134.69%	268.72%	96.30%	99.73%

Descriptive Statistics

Mean	67.015%
Standard Error	7.470%
Median	45.049%
Mode	14.488%
Standard Deviation	67.234%
Sample Variance	4520.381%
Kurtosis	9.91
Skewness	3.03
Range	359.083%
Minimum	14.488%
Maximum	373.571%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	67.015%	81	100.00%	67.015%
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 2.85. Further, an Excess Kurtosis of 12.11 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -127.13%	0	0.00%	0.00%
-127.13% to -94.38%	0	0.00%	0.00%
-94.38% to -61.63%	0	0.00%	0.00%
-61.63% to -28.88%	9	11.11%	11.11%
-28.88% to 3.87%	44	54.32%	65.43%
3.87% to 36.62%	16	19.75%	85.19%
36.62% to 69.37%	7	8.64%	93.83%
69.37% to 102.12%	2	2.47%	96.30%
102.12% to 134.88%	2	2.47%	98.77%
Greater than 134.88%	1	1.23%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-39.79%	47.54%	82.72%	68.27%
±2σ	-83.46%	91.21%	96.30%	95.45%
±3σ	-127.13%	134.88%	98.77%	99.73%

Descriptive Statistics

Mean	3.874%
Standard Error	4.852%
Median	-6.742%
Mode	-53.195%
Standard Deviation	43.667%
Sample Variance	1906.836%
Kurtosis	12.11
Skewness	2.85
Range	300.893%
Minimum	-53.195%
Maximum	247.698%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	38.494%	33	40.74%	15.683%
Negative	-19.928%	48	59.26%	-11.809%
Zero	0.000%	0	0.00%	0.000%

CROX Deep Analytical Report