CTRM - 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	6.47%	11.64%	5.54%	6.21%	5.37%	5.46%	2.22	2.47	2.10	2.61
Weekly	12.69%	31.95%	13.95%	13.93%	11.82%	12.03%	2.08	2.63	1.85	2.96
Monthly	33.58%	76.41%	26.24%	29.23%	24.60%	25.03%	1.83	2.99	1.43	3.83
Quarterly	88.56%	153.70%	67.34%	51.32%	42.61%	43.36%	1.53	3.58	1.00	5.49
Annual	—	—	—	—	85.22%	86.72%	1.00	5.49	0.43	12.87
Option Time Horizon	—	—	—	—	32.78%	33.36%	1.69	3.25	1.21	4.51

Calculation Notes: Computed at 2026-02-22T14:22 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $2.34 | Strike (K): $2.50 (first OTM call ≥ spot) | Observed Mid-Price: $0.2500 (Bid $0.1000 / Ask $0.4000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 85.22% (CRR binomial tree, American, n=20 steps) | BS IV: 86.24% (European Black-Scholes) | Yahoo IV (Annual): 86.72% → 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 Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.39. Further, an Excess Kurtosis of 61.18 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.54%	7	0.40%	0.40%
-19.54% to -14.68%	10	0.57%	0.97%
-14.68% to -9.83%	32	1.83%	2.80%
-9.83% to -4.97%	144	8.23%	11.03%
-4.97% to -0.12%	755	43.17%	54.20%
-0.12% to 4.73%	623	35.62%	89.82%
4.73% to 9.59%	113	6.46%	96.28%
9.59% to 14.44%	33	1.89%	98.17%
14.44% to 19.30%	15	0.86%	99.03%
Greater than 19.30%	17	0.97%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.59%	6.35%	85.82%	68.27%
±2σ	-13.06%	12.82%	96.28%	95.45%
±3σ	-19.54%	19.30%	98.63%	99.73%

Descriptive Statistics

Mean	-0.120%
Standard Error	0.155%
Median	-0.578%
Mode	0.000%
Standard Deviation	6.472%
Sample Variance	41.882%
Kurtosis	61.18
Skewness	4.39
Range	139.365%
Minimum	-32.222%
Maximum	107.143%
Count	1749

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.347%	719	41.11%	1.787%
Negative	-3.519%	948	54.20%	-1.907%
Zero	0.000%	82	4.69%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.76%
20	1 Month	6.33%
60	1 Quarter	5.63%
250	1 Year	4.82%
750	3 Years	5.64%
1,250	5 Years	6.21%
2,500	10 Years	8.23%
5,000	20 Years	8.23%
12,500	50 Years	8.23%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 12.21. Further, an Excess Kurtosis of 234.45 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.54%	0	0.00%	0.00%
-26.54% to -17.82%	0	0.00%	0.00%
-17.82% to -9.09%	0	0.00%	0.00%
-9.09% to -0.36%	0	0.00%	0.00%
-0.36% to 8.37%	1237	70.69%	70.69%
8.37% to 17.10%	387	22.11%	92.80%
17.10% to 25.83%	80	4.57%	97.37%
25.83% to 34.56%	17	0.97%	98.34%
34.56% to 43.28%	7	0.40%	98.74%
Greater than 43.28%	22	1.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.27%	20.01%	95.14%	68.27%
±2σ	-14.91%	31.65%	98.11%	95.45%
±3σ	-26.54%	43.28%	98.74%	99.73%

Descriptive Statistics

Mean	8.370%
Standard Error	0.278%
Median	5.978%
Mode	6.667%
Standard Deviation	11.638%
Sample Variance	135.454%
Kurtosis	234.45
Skewness	12.21
Range	287.270%
Minimum	0.465%
Maximum	287.735%
Count	1750

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.370%	1750	100.00%	8.370%
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 1.13. Further, an Excess Kurtosis of 26.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.18%	9	0.51%	0.51%
-16.18% to -12.03%	16	0.91%	1.43%
-12.03% to -7.87%	48	2.74%	4.17%
-7.87% to -3.71%	147	8.40%	12.57%
-3.71% to 0.45%	656	37.49%	50.06%
0.45% to 4.60%	653	37.31%	87.37%
4.60% to 8.76%	168	9.60%	96.97%
8.76% to 12.92%	32	1.83%	98.80%
12.92% to 17.08%	9	0.51%	99.31%
Greater than 17.08%	12	0.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.10%	5.99%	83.54%	68.27%
±2σ	-10.64%	11.53%	96.63%	95.45%
±3σ	-16.18%	17.08%	98.80%	99.73%

Descriptive Statistics

Mean	0.447%
Standard Error	0.132%
Median	0.445%
Mode	0.000%
Standard Deviation	5.543%
Sample Variance	30.720%
Kurtosis	26.42
Skewness	1.13
Range	106.965%
Minimum	-52.739%
Maximum	54.226%
Count	1750

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.556%	933	53.31%	1.896%
Negative	-3.567%	711	40.63%	-1.449%
Zero	0.000%	106	6.06%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.71. Further, an Excess Kurtosis of 11.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -38.78%	2	0.55%	0.55%
-38.78% to -29.26%	2	0.55%	1.10%
-29.26% to -19.74%	7	1.92%	3.01%
-19.74% to -10.22%	44	12.05%	15.07%
-10.22% to -0.70%	136	37.26%	52.33%
-0.70% to 8.82%	125	34.25%	86.58%
8.82% to 18.34%	33	9.04%	95.62%
18.34% to 27.86%	7	1.92%	97.53%
27.86% to 37.38%	2	0.55%	98.08%
Greater than 37.38%	7	1.92%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.40%	11.99%	81.92%	68.27%
±2σ	-26.09%	24.68%	95.62%	95.45%
±3σ	-38.78%	37.38%	97.53%	99.73%

Descriptive Statistics

Mean	-0.703%
Standard Error	0.664%
Median	-1.210%
Mode	0.000%
Standard Deviation	12.693%
Sample Variance	161.112%
Kurtosis	11.49
Skewness	1.71
Range	136.352%
Minimum	-50.000%
Maximum	86.352%
Count	365

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.803%	154	42.19%	3.714%
Negative	-8.061%	200	54.80%	-4.417%
Zero	0.000%	11	3.01%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	17.13%
12	1 Quarter	14.30%
52	1 Year	10.61%
156	3 Years	12.55%
260	5 Years	13.93%
520	10 Years	19.92%
1,040	20 Years	19.92%
2,600	50 Years	19.92%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 6.06. Further, an Excess Kurtosis of 45.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -72.97%	0	0.00%	0.00%
-72.97% to -49.01%	0	0.00%	0.00%
-49.01% to -25.05%	0	0.00%	0.00%
-25.05% to -1.08%	0	0.00%	0.00%
-1.08% to 22.88%	265	72.40%	72.40%
22.88% to 46.84%	75	20.49%	92.90%
46.84% to 70.81%	11	3.01%	95.90%
70.81% to 94.77%	4	1.09%	96.99%
94.77% to 118.73%	5	1.37%	98.36%
Greater than 118.73%	6	1.64%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.07%	54.83%	93.72%	68.27%
±2σ	-41.02%	86.78%	96.72%	95.45%
±3σ	-72.97%	118.73%	98.36%	99.73%

Descriptive Statistics

Mean	22.881%
Standard Error	1.670%
Median	14.843%
Mode	14.286%
Standard Deviation	31.951%
Sample Variance	1020.881%
Kurtosis	45.93
Skewness	6.06
Range	304.519%
Minimum	2.970%
Maximum	307.489%
Count	366

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.881%	366	100.00%	22.881%
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 2.09. Further, an Excess Kurtosis of 12.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -39.31%	2	0.55%	0.55%
-39.31% to -28.85%	3	0.82%	1.37%
-28.85% to -18.39%	4	1.09%	2.46%
-18.39% to -7.92%	45	12.30%	14.75%
-7.92% to 2.54%	150	40.98%	55.74%
2.54% to 13.01%	115	31.42%	87.16%
13.01% to 23.47%	29	7.92%	95.08%
23.47% to 33.93%	8	2.19%	97.27%
33.93% to 44.40%	5	1.37%	98.63%
Greater than 44.40%	5	1.37%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.41%	16.49%	83.06%	68.27%
±2σ	-25.36%	30.44%	95.36%	95.45%
±3σ	-39.31%	44.40%	98.09%	99.73%

Descriptive Statistics

Mean	2.542%
Standard Error	0.729%
Median	1.280%
Mode	0.000%
Standard Deviation	13.951%
Sample Variance	194.629%
Kurtosis	12.85
Skewness	2.09
Range	145.235%
Minimum	-44.789%
Maximum	100.446%
Count	366

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.158%	202	55.19%	5.606%
Negative	-7.284%	154	42.08%	-3.065%
Zero	0.000%	10	2.73%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.43. Further, an Excess Kurtosis of 29.96 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -103.22%	0	0.00%	0.00%
-103.22% to -78.04%	0	0.00%	0.00%
-78.04% to -52.86%	0	0.00%	0.00%
-52.86% to -27.68%	10	11.90%	11.90%
-27.68% to -2.50%	36	42.86%	54.76%
-2.50% to 22.68%	30	35.71%	90.48%
22.68% to 47.86%	5	5.95%	96.43%
47.86% to 73.04%	2	2.38%	98.81%
73.04% to 98.22%	0	0.00%	98.81%
Greater than 98.22%	1	1.19%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-36.08%	31.08%	84.52%	68.27%
±2σ	-69.65%	64.65%	98.81%	95.45%
±3σ	-103.23%	98.23%	98.81%	99.73%

Descriptive Statistics

Mean	-2.500%
Standard Error	3.663%
Median	-5.033%
Mode	-49.000%
Standard Deviation	33.575%
Sample Variance	1127.306%
Kurtosis	29.96
Skewness	4.43
Range	284.135%
Minimum	-49.000%
Maximum	235.135%
Count	84

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	25.072%	28	33.33%	8.357%
Negative	-16.582%	55	65.48%	-10.857%
Zero	0.000%	1	1.19%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	25.60%
12	1 Year	18.81%
36	3 Years	25.67%
60	5 Years	29.23%
120	10 Years	46.65%
240	20 Years	46.65%
600	50 Years	46.65%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.76. Further, an Excess Kurtosis of 15.18 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -168.88%	0	0.00%	0.00%
-168.88% to -111.57%	0	0.00%	0.00%
-111.57% to -54.27%	0	0.00%	0.00%
-54.27% to 3.04%	0	0.00%	0.00%
3.04% to 60.34%	61	71.76%	71.76%
60.34% to 117.65%	17	20.00%	91.76%
117.65% to 174.95%	3	3.53%	95.29%
174.95% to 232.26%	1	1.18%	96.47%
232.26% to 289.56%	0	0.00%	96.47%
Greater than 289.56%	3	3.53%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.06%	136.75%	92.94%	68.27%
±2σ	-92.47%	213.16%	95.29%	95.45%
±3σ	-168.88%	289.56%	96.47%	99.73%

Descriptive Statistics

Mean	60.344%
Standard Error	8.288%
Median	40.594%
Mode	31.579%
Standard Deviation	76.407%
Sample Variance	5838.033%
Kurtosis	15.18
Skewness	3.76
Range	430.723%
Minimum	8.696%
Maximum	439.419%
Count	85

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	60.344%	85	100.00%	60.344%
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 0.44. Further, an Excess Kurtosis of 1.01 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -68.71%	1	1.18%	1.18%
-68.71% to -49.03%	0	0.00%	1.18%
-49.03% to -29.36%	2	2.35%	3.53%
-29.36% to -9.68%	11	12.94%	16.47%
-9.68% to 10.00%	35	41.18%	57.65%
10.00% to 29.68%	18	21.18%	78.82%
29.68% to 49.36%	10	11.76%	90.59%
49.36% to 69.04%	7	8.24%	98.82%
69.04% to 88.72%	0	0.00%	98.82%
Greater than 88.72%	1	1.18%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.24%	36.24%	76.47%	68.27%
±2σ	-42.48%	62.48%	92.94%	95.45%
±3σ	-68.71%	88.72%	97.65%	99.73%

Descriptive Statistics

Mean	10.003%
Standard Error	2.846%
Median	4.563%
Mode	-69.355%
Standard Deviation	26.239%
Sample Variance	688.500%
Kurtosis	1.01
Skewness	0.44
Range	160.531%
Minimum	-69.355%
Maximum	91.176%
Count	85

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.165%	53	62.35%	15.068%
Negative	-13.454%	32	37.65%	-5.065%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.01. Further, an Excess Kurtosis of 18.69 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -266.62%	0	0.00%	0.00%
-266.62% to -200.20%	0	0.00%	0.00%
-200.20% to -133.78%	0	0.00%	0.00%
-133.78% to -67.37%	1	3.57%	3.57%
-67.37% to -0.95%	17	60.71%	64.29%
-0.95% to 65.47%	8	28.57%	92.86%
65.47% to 131.88%	1	3.57%	96.43%
131.88% to 198.30%	0	0.00%	96.43%
198.30% to 264.72%	0	0.00%	96.43%
Greater than 264.72%	1	3.57%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-89.50%	87.61%	96.43%	68.27%
±2σ	-178.06%	176.16%	96.43%	95.45%
±3σ	-266.62%	264.72%	96.43%	99.73%

Descriptive Statistics

Mean	-0.949%
Standard Error	16.736%
Median	-21.482%
Mode	-70.088%
Standard Deviation	88.556%
Sample Variance	7842.180%
Kurtosis	18.69
Skewness	4.01
Range	482.485%
Minimum	-70.088%
Maximum	412.397%
Count	28

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	80.294%	8	28.57%	22.941%
Negative	-33.446%	20	71.43%	-23.890%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	24.26%
12	3 Years	43.82%
20	5 Years	51.32%
40	10 Years	88.07%
80	20 Years	88.07%
200	50 Years	88.07%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.91. Further, an Excess Kurtosis of 3.64 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -310.45%	0	0.00%	0.00%
-310.45% to -195.18%	0	0.00%	0.00%
-195.18% to -79.91%	0	0.00%	0.00%
-79.91% to 35.37%	3	10.34%	10.34%
35.37% to 150.65%	18	62.07%	72.41%
150.65% to 265.92%	3	10.34%	82.76%
265.92% to 381.19%	3	10.34%	93.10%
381.19% to 496.47%	1	3.45%	96.55%
496.47% to 611.75%	0	0.00%	96.55%
Greater than 611.75%	1	3.45%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.05%	304.35%	82.76%	68.27%
±2σ	-156.75%	458.04%	96.55%	95.45%
±3σ	-310.45%	611.75%	96.55%	99.73%

Descriptive Statistics

Mean	150.645%
Standard Error	28.541%
Median	81.558%
Mode	17.476%
Standard Deviation	153.700%
Sample Variance	23623.636%
Kurtosis	3.64
Skewness	1.91
Range	651.755%
Minimum	17.476%
Maximum	669.231%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	150.645%	29	100.00%	150.645%
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 0.96. Further, an Excess Kurtosis of 1.17 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -163.62%	0	0.00%	0.00%
-163.62% to -113.12%	0	0.00%	0.00%
-113.12% to -62.62%	1	3.45%	3.45%
-62.62% to -12.12%	5	17.24%	20.69%
-12.12% to 38.39%	11	37.93%	58.62%
38.39% to 88.89%	6	20.69%	79.31%
88.89% to 139.39%	3	10.34%	89.66%
139.39% to 189.90%	2	6.90%	96.55%
189.90% to 240.40%	1	3.45%	100.00%
Greater than 240.40%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.95%	105.72%	75.86%	68.27%
±2σ	-96.29%	173.06%	93.10%	95.45%
±3σ	-163.62%	240.40%	100.00%	99.73%

Descriptive Statistics

Mean	38.387%
Standard Error	12.504%
Median	30.081%
Mode	-80.645%
Standard Deviation	67.337%
Sample Variance	4534.316%
Kurtosis	1.17
Skewness	0.96
Range	303.684%
Minimum	-80.645%
Maximum	223.039%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	67.413%	20	68.97%	46.492%
Negative	-26.113%	9	31.03%	-8.104%
Zero	0.000%	0	0.00%	0.000%

CTRM Deep Analytical Report