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	4.11%	3.76%	3.35%	4.48%	3.78%	3.80%	76.05	82.03	73.22	85.19
Weekly	9.30%	12.41%	10.04%	10.73%	8.33%	8.37%	72.67	85.84	66.86	93.30
Monthly	17.21%	32.18%	17.84%	22.52%	17.34%	17.43%	66.41	93.93	55.83	111.72
Quarterly	33.40%	67.15%	43.61%	37.03%	30.03%	30.19%	58.49	106.65	43.32	144.01
Annual	—	—	—	—	60.07%	60.38%	43.32	144.01	23.76	262.57
Option Time Horizon	—	—	—	—	22.67%	22.79%	62.96	99.08	50.19	124.29

Calculation Notes: Computed at 2026-03-23T16:34 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $78.98 | Strike (K): $80.00 (first OTM call ≥ spot) | Observed Mid-Price: $6.8500 (Bid $6.4000 / Ask $7.3000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 60.07% (CRR binomial tree, American, n=20 steps) | BS IV: 59.97% (European Black-Scholes) | Yahoo IV (Annual): 60.38% → 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 20.36 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.21%	43	0.85%	0.85%
-12.21% to -9.13%	38	0.75%	1.60%
-9.13% to -6.04%	105	2.08%	3.68%
-6.04% to -2.96%	468	9.25%	12.92%
-2.96% to 0.12%	1898	37.51%	50.43%
0.12% to 3.20%	1841	36.38%	86.82%
3.20% to 6.29%	461	9.11%	95.93%
6.29% to 9.37%	118	2.33%	98.26%
9.37% to 12.45%	40	0.79%	99.05%
Greater than 12.45%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.99%	4.23%	83.22%	68.27%
±2σ	-8.10%	8.34%	95.73%	95.45%
±3σ	-12.21%	12.45%	98.20%	99.73%

Descriptive Statistics

Mean	0.121%
Standard Error	0.058%
Median	0.092%
Mode	0.000%
Standard Deviation	4.111%
Sample Variance	16.897%
Kurtosis	20.36
Skewness	-0.53
Range	86.658%
Minimum	-44.737%
Maximum	41.921%
Count	5060

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.608%	2588	51.15%	1.334%
Negative	-2.535%	2420	47.83%	-1.212%
Zero	0.000%	52	1.03%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.51%
20	1 Month	3.73%
60	1 Quarter	4.03%
250	1 Year	4.11%
750	3 Years	3.94%
1,250	5 Years	4.48%
2,500	10 Years	4.47%
5,000	20 Years	4.89%
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 4.49. Further, an Excess Kurtosis of 36.82 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.56%	0	0.00%	0.00%
-6.56% to -3.74%	0	0.00%	0.00%
-3.74% to -0.92%	0	0.00%	0.00%
-0.92% to 1.91%	348	6.88%	6.88%
1.91% to 4.73%	3048	60.23%	67.10%
4.73% to 7.55%	1056	20.87%	87.97%
7.55% to 10.37%	310	6.13%	94.09%
10.37% to 13.19%	129	2.55%	96.64%
13.19% to 16.01%	66	1.30%	97.95%
Greater than 16.01%	104	2.05%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.97%	8.49%	90.77%	68.27%
±2σ	-2.80%	12.25%	96.15%	95.45%
±3σ	-6.56%	16.01%	97.96%	99.73%

Descriptive Statistics

Mean	4.728%
Standard Error	0.053%
Median	3.704%
Mode	1.966%
Standard Deviation	3.762%
Sample Variance	14.155%
Kurtosis	36.82
Skewness	4.49
Range	60.673%
Minimum	0.756%
Maximum	61.429%
Count	5061

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.728%	5061	100.00%	4.728%
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 11.41 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.98%	36	0.71%	0.71%
-9.98% to -7.47%	43	0.85%	1.56%
-7.47% to -4.95%	145	2.87%	4.43%
-4.95% to -2.44%	579	11.44%	15.87%
-2.44% to 0.08%	1762	34.82%	50.68%
0.08% to 2.60%	1747	34.52%	85.20%
2.60% to 5.11%	515	10.18%	95.38%
5.11% to 7.63%	122	2.41%	97.79%
7.63% to 10.15%	64	1.26%	99.05%
Greater than 10.15%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.27%	3.44%	79.89%	68.27%
±2σ	-6.63%	6.79%	95.18%	95.45%
±3σ	-9.98%	10.15%	98.34%	99.73%

Descriptive Statistics

Mean	0.081%
Standard Error	0.047%
Median	0.037%
Mode	0.000%
Standard Deviation	3.355%
Sample Variance	11.257%
Kurtosis	11.41
Skewness	0.72
Range	63.518%
Minimum	-27.077%
Maximum	36.441%
Count	5061

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.309%	2541	50.21%	1.159%
Negative	-2.214%	2466	48.73%	-1.079%
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 -27.29%	6	0.57%	0.57%
-27.29% to -20.31%	21	2.00%	2.57%
-20.31% to -13.34%	24	2.29%	4.86%
-13.34% to -6.37%	90	8.58%	13.44%
-6.37% to 0.60%	385	36.70%	50.14%
0.60% to 7.58%	365	34.80%	84.94%
7.58% to 14.55%	117	11.15%	96.09%
14.55% to 21.52%	25	2.38%	98.47%
21.52% to 28.50%	9	0.86%	99.33%
Greater than 28.50%	7	0.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.69%	9.90%	81.98%	68.27%
±2σ	-17.99%	19.20%	94.09%	95.45%
±3σ	-27.29%	28.50%	98.76%	99.73%

Descriptive Statistics

Mean	0.604%
Standard Error	0.287%
Median	0.550%
Mode	0.000%
Standard Deviation	9.297%
Sample Variance	86.436%
Kurtosis	13.09
Skewness	0.64
Range	140.988%
Minimum	-52.893%
Maximum	88.095%
Count	1049

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.237%	557	53.10%	3.312%
Negative	-5.845%	486	46.33%	-2.708%
Zero	0.000%	6	0.57%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.30%
12	1 Quarter	8.74%
52	1 Year	9.53%
156	3 Years	9.27%
260	5 Years	10.73%
520	10 Years	10.55%
1,040	20 Years	11.65%
2,600	50 Years	11.66%

High-to-Low 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 70.87 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.56%	0	0.00%	0.00%
-24.56% to -15.25%	0	0.00%	0.00%
-15.25% to -5.94%	0	0.00%	0.00%
-5.94% to 3.37%	9	0.86%	0.86%
3.37% to 12.68%	713	67.90%	68.76%
12.68% to 21.99%	207	19.71%	88.48%
21.99% to 31.30%	70	6.67%	95.14%
31.30% to 40.60%	23	2.19%	97.33%
40.60% to 49.91%	14	1.33%	98.67%
Greater than 49.91%	14	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.27%	25.09%	91.62%	68.27%
±2σ	-12.15%	37.50%	96.86%	95.45%
±3σ	-24.56%	49.91%	98.67%	99.73%

Descriptive Statistics

Mean	12.678%
Standard Error	0.383%
Median	9.178%
Mode	4.767%
Standard Deviation	12.412%
Sample Variance	154.064%
Kurtosis	70.87
Skewness	6.43
Range	199.951%
Minimum	2.581%
Maximum	202.532%
Count	1050

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.678%	1050	100.00%	12.678%
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 3.32. Further, an Excess Kurtosis of 31.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.77%	3	0.29%	0.29%
-29.77% to -22.24%	5	0.48%	0.76%
-22.24% to -14.71%	23	2.19%	2.95%
-14.71% to -7.17%	101	9.62%	12.57%
-7.17% to 0.36%	445	42.38%	54.95%
0.36% to 7.89%	354	33.71%	88.67%
7.89% to 15.43%	72	6.86%	95.52%
15.43% to 22.96%	17	1.62%	97.14%
22.96% to 30.49%	14	1.33%	98.48%
Greater than 30.49%	16	1.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.68%	10.41%	85.33%	68.27%
±2σ	-19.73%	20.45%	95.81%	95.45%
±3σ	-29.77%	30.49%	98.19%	99.73%

Descriptive Statistics

Mean	0.361%
Standard Error	0.310%
Median	-0.456%
Mode	0.000%
Standard Deviation	10.044%
Sample Variance	100.885%
Kurtosis	31.10
Skewness	3.32
Range	154.847%
Minimum	-44.620%
Maximum	110.227%
Count	1050

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.807%	493	46.95%	3.196%
Negative	-5.393%	552	52.57%	-2.835%
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 0.82. Further, an Excess Kurtosis of 4.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -49.48%	0	0.00%	0.00%
-49.48% to -36.57%	4	1.66%	1.66%
-36.57% to -23.66%	9	3.73%	5.39%
-23.66% to -10.75%	26	10.79%	16.18%
-10.75% to 2.16%	88	36.51%	52.70%
2.16% to 15.07%	74	30.71%	83.40%
15.07% to 27.98%	28	11.62%	95.02%
27.98% to 40.89%	6	2.49%	97.51%
40.89% to 53.80%	4	1.66%	99.17%
Greater than 53.80%	2	0.83%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.05%	19.37%	78.84%	68.27%
±2σ	-32.27%	36.59%	95.02%	95.45%
±3σ	-49.48%	53.80%	99.17%	99.73%

Descriptive Statistics

Mean	2.160%
Standard Error	1.109%
Median	1.040%
Mode	-49.402%
Standard Deviation	17.214%
Sample Variance	296.326%
Kurtosis	4.88
Skewness	0.82
Range	138.478%
Minimum	-49.402%
Maximum	89.076%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.303%	129	53.53%	7.121%
Negative	-10.770%	111	46.06%	-4.960%
Zero	0.000%	1	0.41%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	16.74%
12	1 Year	18.30%
36	3 Years	19.77%
60	5 Years	22.52%
120	10 Years	22.89%
240	20 Years	25.19%
600	50 Years	25.21%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.94. Further, an Excess Kurtosis of 48.66 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -65.90%	0	0.00%	0.00%
-65.90% to -41.76%	0	0.00%	0.00%
-41.76% to -17.63%	0	0.00%	0.00%
-17.63% to 6.51%	2	0.83%	0.83%
6.51% to 30.64%	173	71.49%	72.31%
30.64% to 54.77%	45	18.60%	90.91%
54.77% to 78.91%	12	4.96%	95.87%
78.91% to 103.04%	3	1.24%	97.11%
103.04% to 127.18%	4	1.65%	98.76%
Greater than 127.18%	3	1.24%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.54%	62.82%	94.63%	68.27%
±2σ	-33.72%	95.00%	96.28%	95.45%
±3σ	-65.90%	127.18%	98.76%	99.73%

Descriptive Statistics

Mean	30.641%
Standard Error	2.069%
Median	23.390%
Mode	6.209%
Standard Deviation	32.179%
Sample Variance	1035.487%
Kurtosis	48.66
Skewness	5.94
Range	345.690%
Minimum	6.209%
Maximum	351.899%
Count	242

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.641%	242	100.00%	30.641%
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 1.68. Further, an Excess Kurtosis of 6.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -52.72%	0	0.00%	0.00%
-52.72% to -39.35%	2	0.83%	0.83%
-39.35% to -25.97%	6	2.48%	3.31%
-25.97% to -12.60%	33	13.64%	16.94%
-12.60% to 0.78%	96	39.67%	56.61%
0.78% to 14.16%	72	29.75%	86.36%
14.16% to 27.53%	17	7.02%	93.39%
27.53% to 40.91%	9	3.72%	97.11%
40.91% to 54.29%	3	1.24%	98.35%
Greater than 54.29%	4	1.65%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.05%	18.62%	80.17%	68.27%
±2σ	-34.89%	36.45%	95.04%	95.45%
±3σ	-52.72%	54.29%	98.35%	99.73%

Descriptive Statistics

Mean	0.781%
Standard Error	1.146%
Median	-0.768%
Mode	5.081%
Standard Deviation	17.835%
Sample Variance	318.099%
Kurtosis	6.93
Skewness	1.68
Range	150.866%
Minimum	-45.354%
Maximum	105.512%
Count	242

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.985%	111	45.87%	6.415%
Negative	-10.487%	130	53.72%	-5.634%
Zero	0.000%	1	0.41%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.31. Further, an Excess Kurtosis of 0.67 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -92.53%	0	0.00%	0.00%
-92.53% to -67.47%	1	1.25%	1.25%
-67.47% to -42.42%	5	6.25%	7.50%
-42.42% to -17.37%	12	15.00%	22.50%
-17.37% to 7.68%	21	26.25%	48.75%
7.68% to 32.73%	25	31.25%	80.00%
32.73% to 57.79%	12	15.00%	95.00%
57.79% to 82.84%	2	2.50%	97.50%
82.84% to 107.89%	1	1.25%	98.75%
Greater than 107.89%	1	1.25%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.72%	41.09%	71.25%	68.27%
±2σ	-59.12%	74.49%	95.00%	95.45%
±3σ	-92.53%	107.89%	98.75%	99.73%

Descriptive Statistics

Mean	7.683%
Standard Error	3.735%
Median	7.825%
Mode	-70.652%
Standard Deviation	33.403%
Sample Variance	1115.761%
Kurtosis	0.67
Skewness	0.31
Range	182.964%
Minimum	-70.652%
Maximum	112.312%
Count	80

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	29.114%	47	58.75%	17.104%
Negative	-22.841%	33	41.25%	-9.422%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	29.12%
12	3 Years	31.15%
20	5 Years	37.03%
40	10 Years	41.40%
80	20 Years	46.63%
200	50 Years	46.63%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.04. Further, an Excess Kurtosis of 9.95 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -134.28%	0	0.00%	0.00%
-134.28% to -83.92%	0	0.00%	0.00%
-83.92% to -33.55%	0	0.00%	0.00%
-33.55% to 16.81%	2	2.47%	2.47%
16.81% to 67.17%	57	70.37%	72.84%
67.17% to 117.53%	13	16.05%	88.89%
117.53% to 167.89%	4	4.94%	93.83%
167.89% to 218.25%	1	1.23%	95.06%
218.25% to 268.61%	1	1.23%	96.30%
Greater than 268.61%	3	3.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.02%	134.31%	91.36%	68.27%
±2σ	-67.13%	201.46%	95.06%	95.45%
±3σ	-134.28%	268.61%	96.30%	99.73%

Descriptive Statistics

Mean	67.166%
Standard Error	7.461%
Median	45.049%
Mode	14.488%
Standard Deviation	67.147%
Sample Variance	4508.686%
Kurtosis	9.95
Skewness	3.04
Range	359.083%
Minimum	14.488%
Maximum	373.571%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	67.166%	81	100.00%	67.166%
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 2.84. 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 -126.68%	0	0.00%	0.00%
-126.68% to -93.97%	0	0.00%	0.00%
-93.97% to -61.27%	0	0.00%	0.00%
-61.27% to -28.56%	9	11.11%	11.11%
-28.56% to 4.15%	43	53.09%	64.20%
4.15% to 36.86%	17	20.99%	85.19%
36.86% to 69.56%	7	8.64%	93.83%
69.56% to 102.27%	2	2.47%	96.30%
102.27% to 134.98%	2	2.47%	98.77%
Greater than 134.98%	1	1.23%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-39.46%	47.76%	83.95%	68.27%
±2σ	-83.07%	91.37%	96.30%	95.45%
±3σ	-126.68%	134.98%	98.77%	99.73%

Descriptive Statistics

Mean	4.150%
Standard Error	4.846%
Median	-6.304%
Mode	-53.195%
Standard Deviation	43.610%
Sample Variance	1901.874%
Kurtosis	12.11
Skewness	2.84
Range	300.893%
Minimum	-53.195%
Maximum	247.698%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	37.547%	34	41.98%	15.760%
Negative	-20.010%	47	58.02%	-11.611%
Zero	0.000%	0	0.00%	0.000%

CROX Deep Analytical Report