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	6.44%	11.58%	5.52%	6.10%	4.15%	4.18%	2.09	2.27	2.01	2.37
Weekly	12.62%	31.81%	13.88%	13.69%	9.14%	9.21%	1.99	2.39	1.82	2.62
Monthly	33.39%	76.06%	26.08%	28.89%	19.02%	19.17%	1.80	2.64	1.49	3.19
Quarterly	88.61%	153.40%	67.00%	51.68%	32.94%	33.20%	1.57	3.03	1.13	4.21
Annual	—	—	—	—	65.88%	66.41%	1.13	4.21	0.58	8.14
Option Time Horizon	—	—	—	—	23.64%	23.83%	1.72	2.76	1.36	3.50

Calculation Notes: Computed at 2026-03-01T12:16 UTC | Reference Contract: Call option expiring 2026-04-17 (47 days) | Spot Price (S): $2.18 | Strike (K): $2.50 (first OTM call ≥ spot) | Observed Mid-Price: $0.1000 (Bid $0.0000 / Ask $0.2000) | Risk-Free Rate (r): 3.74% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 65.88% (CRR binomial tree, American, n=20 steps) | BS IV: 66.29% (European Black-Scholes) | Yahoo IV (Annual): 66.41% → 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 4.41. Further, an Excess Kurtosis of 61.80 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.44%	7	0.40%	0.40%
-19.44% to -14.61%	11	0.62%	1.02%
-14.61% to -9.78%	31	1.75%	2.77%
-9.78% to -4.96%	148	8.36%	11.13%
-4.96% to -0.13%	764	43.16%	54.29%
-0.13% to 4.70%	630	35.59%	89.89%
4.70% to 9.53%	113	6.38%	96.27%
9.53% to 14.35%	34	1.92%	98.19%
14.35% to 19.18%	14	0.79%	98.98%
Greater than 19.18%	18	1.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.57%	6.31%	85.99%	68.27%
±2σ	-13.00%	12.75%	96.27%	95.45%
±3σ	-19.44%	19.18%	98.59%	99.73%

Descriptive Statistics

Mean	-0.129%
Standard Error	0.153%
Median	-0.577%
Mode	0.000%
Standard Deviation	6.437%
Sample Variance	41.438%
Kurtosis	61.80
Skewness	4.41
Range	139.365%
Minimum	-32.222%
Maximum	107.143%
Count	1770

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.315%	726	41.02%	1.770%
Negative	-3.498%	961	54.29%	-1.899%
Zero	0.000%	83	4.69%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	4.18%
20	1 Month	4.72%
60	1 Quarter	5.27%
250	1 Year	4.73%
750	3 Years	5.46%
1,250	5 Years	6.10%
2,500	10 Years	8.19%
5,000	20 Years	8.19%
12,500	50 Years	8.19%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 12.27. Further, an Excess Kurtosis of 236.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.40%	0	0.00%	0.00%
-26.40% to -17.72%	0	0.00%	0.00%
-17.72% to -9.04%	0	0.00%	0.00%
-9.04% to -0.35%	0	0.00%	0.00%
-0.35% to 8.33%	1253	70.75%	70.75%
8.33% to 17.01%	389	21.96%	92.72%
17.01% to 25.69%	81	4.57%	97.29%
25.69% to 34.38%	19	1.07%	98.36%
34.38% to 43.06%	7	0.40%	98.76%
Greater than 43.06%	22	1.24%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.25%	19.91%	94.97%	68.27%
±2σ	-14.83%	31.48%	98.08%	95.45%
±3σ	-26.40%	43.06%	98.76%	99.73%

Descriptive Statistics

Mean	8.328%
Standard Error	0.275%
Median	5.952%
Mode	6.667%
Standard Deviation	11.577%
Sample Variance	134.019%
Kurtosis	236.85
Skewness	12.27
Range	287.270%
Minimum	0.465%
Maximum	287.735%
Count	1771

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.328%	1771	100.00%	8.328%
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 26.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.09%	9	0.51%	0.51%
-16.09% to -11.95%	16	0.90%	1.41%
-11.95% to -7.82%	49	2.77%	4.18%
-7.82% to -3.68%	147	8.30%	12.48%
-3.68% to 0.46%	665	37.55%	50.03%
0.46% to 4.59%	662	37.38%	87.41%
4.59% to 8.73%	170	9.60%	97.01%
8.73% to 12.87%	32	1.81%	98.81%
12.87% to 17.00%	9	0.51%	99.32%
Greater than 17.00%	12	0.68%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.06%	5.97%	83.46%	68.27%
±2σ	-10.57%	11.49%	96.61%	95.45%
±3σ	-16.09%	17.01%	98.81%	99.73%

Descriptive Statistics

Mean	0.457%
Standard Error	0.131%
Median	0.451%
Mode	0.000%
Standard Deviation	5.516%
Sample Variance	30.426%
Kurtosis	26.63
Skewness	1.13
Range	106.965%
Minimum	-52.739%
Maximum	54.226%
Count	1771

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.545%	946	53.42%	1.894%
Negative	-3.550%	717	40.49%	-1.437%
Zero	0.000%	108	6.10%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.73. Further, an Excess Kurtosis of 11.67 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -38.62%	2	0.54%	0.54%
-38.62% to -29.15%	2	0.54%	1.08%
-29.15% to -19.68%	7	1.90%	2.98%
-19.68% to -10.21%	45	12.20%	15.18%
-10.21% to -0.75%	137	37.13%	52.30%
-0.75% to 8.72%	127	34.42%	86.72%
8.72% to 18.19%	33	8.94%	95.66%
18.19% to 27.66%	7	1.90%	97.56%
27.66% to 37.12%	2	0.54%	98.10%
Greater than 37.12%	7	1.90%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.37%	11.88%	82.11%	68.27%
±2σ	-26.00%	24.50%	95.66%	95.45%
±3σ	-38.62%	37.12%	97.56%	99.73%

Descriptive Statistics

Mean	-0.747%
Standard Error	0.657%
Median	-1.220%
Mode	0.000%
Standard Deviation	12.624%
Sample Variance	159.377%
Kurtosis	11.67
Skewness	1.73
Range	136.352%
Minimum	-50.000%
Maximum	86.352%
Count	369

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.713%	155	42.01%	3.660%
Negative	-8.010%	203	55.01%	-4.407%
Zero	0.000%	11	2.98%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.49%
12	1 Quarter	11.77%
52	1 Year	10.34%
156	3 Years	12.16%
260	5 Years	13.69%
520	10 Years	19.81%
1,040	20 Years	19.81%
2,600	50 Years	19.81%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 6.08. Further, an Excess Kurtosis of 46.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -72.67%	0	0.00%	0.00%
-72.67% to -48.82%	0	0.00%	0.00%
-48.82% to -24.96%	0	0.00%	0.00%
-24.96% to -1.11%	0	0.00%	0.00%
-1.11% to 22.74%	269	72.70%	72.70%
22.74% to 46.60%	74	20.00%	92.70%
46.60% to 70.45%	12	3.24%	95.95%
70.45% to 94.31%	4	1.08%	97.03%
94.31% to 118.16%	5	1.35%	98.38%
Greater than 118.16%	6	1.62%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.06%	54.55%	93.78%	68.27%
±2σ	-40.87%	86.36%	96.76%	95.45%
±3σ	-72.67%	118.16%	98.38%	99.73%

Descriptive Statistics

Mean	22.744%
Standard Error	1.653%
Median	14.690%
Mode	14.286%
Standard Deviation	31.806%
Sample Variance	1011.602%
Kurtosis	46.39
Skewness	6.08
Range	304.519%
Minimum	2.970%
Maximum	307.489%
Count	370

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.744%	370	100.00%	22.744%
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 2.10. Further, an Excess Kurtosis of 12.99 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -39.06%	2	0.54%	0.54%
-39.06% to -28.65%	3	0.81%	1.35%
-28.65% to -18.24%	4	1.08%	2.43%
-18.24% to -7.84%	44	11.89%	14.32%
-7.84% to 2.57%	153	41.35%	55.68%
2.57% to 12.98%	116	31.35%	87.03%
12.98% to 23.39%	30	8.11%	95.14%
23.39% to 33.80%	8	2.16%	97.30%
33.80% to 44.21%	5	1.35%	98.65%
Greater than 44.21%	5	1.35%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.30%	16.45%	82.97%	68.27%
±2σ	-25.18%	30.33%	95.41%	95.45%
±3σ	-39.06%	44.21%	98.11%	99.73%

Descriptive Statistics

Mean	2.573%
Standard Error	0.721%
Median	1.303%
Mode	0.000%
Standard Deviation	13.878%
Sample Variance	192.604%
Kurtosis	12.99
Skewness	2.10
Range	145.235%
Minimum	-44.789%
Maximum	100.446%
Count	370

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.107%	205	55.41%	5.600%
Negative	-7.225%	155	41.89%	-3.027%
Zero	0.000%	10	2.70%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.46. Further, an Excess Kurtosis of 30.37 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -102.86%	0	0.00%	0.00%
-102.86% to -77.81%	0	0.00%	0.00%
-77.81% to -52.77%	0	0.00%	0.00%
-52.77% to -27.73%	10	11.76%	11.76%
-27.73% to -2.68%	38	44.71%	56.47%
-2.68% to 22.36%	29	34.12%	90.59%
22.36% to 47.40%	5	5.88%	96.47%
47.40% to 72.45%	2	2.35%	98.82%
72.45% to 97.49%	0	0.00%	98.82%
Greater than 97.49%	1	1.18%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-36.07%	30.71%	84.71%	68.27%
±2σ	-69.46%	64.10%	98.82%	95.45%
±3σ	-102.86%	97.49%	98.82%	99.73%

Descriptive Statistics

Mean	-2.682%
Standard Error	3.622%
Median	-5.785%
Mode	-49.000%
Standard Deviation	33.391%
Sample Variance	1114.976%
Kurtosis	30.37
Skewness	4.46
Range	284.135%
Minimum	-49.000%
Maximum	235.135%
Count	85

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	25.072%	28	32.94%	8.259%
Negative	-16.608%	56	65.88%	-10.942%
Zero	0.000%	1	1.18%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	22.56%
12	1 Year	18.41%
36	3 Years	24.47%
60	5 Years	28.89%
120	10 Years	46.33%
240	20 Years	46.33%
600	50 Years	46.33%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.78. Further, an Excess Kurtosis of 15.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -168.27%	0	0.00%	0.00%
-168.27% to -111.22%	0	0.00%	0.00%
-111.22% to -54.18%	0	0.00%	0.00%
-54.18% to 2.87%	0	0.00%	0.00%
2.87% to 59.91%	62	72.09%	72.09%
59.91% to 116.96%	17	19.77%	91.86%
116.96% to 174.00%	3	3.49%	95.35%
174.00% to 231.05%	1	1.16%	96.51%
231.05% to 288.10%	0	0.00%	96.51%
Greater than 288.10%	3	3.49%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.15%	135.97%	93.02%	68.27%
±2σ	-92.21%	212.04%	95.35%	95.45%
±3σ	-168.27%	288.10%	96.51%	99.73%

Descriptive Statistics

Mean	59.913%
Standard Error	8.202%
Median	40.373%
Mode	31.579%
Standard Deviation	76.061%
Sample Variance	5785.261%
Kurtosis	15.38
Skewness	3.78
Range	430.723%
Minimum	8.696%
Maximum	439.419%
Count	86

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	59.913%	86	100.00%	59.913%
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 0.42. Further, an Excess Kurtosis of 1.05 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -68.03%	1	1.16%	1.16%
-68.03% to -48.47%	0	0.00%	1.16%
-48.47% to -28.91%	2	2.33%	3.49%
-28.91% to -9.35%	11	12.79%	16.28%
-9.35% to 10.21%	35	40.70%	56.98%
10.21% to 29.77%	19	22.09%	79.07%
29.77% to 49.33%	10	11.63%	90.70%
49.33% to 68.89%	7	8.14%	98.84%
68.89% to 88.45%	0	0.00%	98.84%
Greater than 88.45%	1	1.16%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.87%	36.29%	75.58%	68.27%
±2σ	-41.95%	62.37%	93.02%	95.45%
±3σ	-68.03%	88.45%	97.67%	99.73%

Descriptive Statistics

Mean	10.213%
Standard Error	2.812%
Median	5.072%
Mode	-69.355%
Standard Deviation	26.080%
Sample Variance	680.180%
Kurtosis	1.05
Skewness	0.42
Range	160.531%
Minimum	-69.355%
Maximum	91.176%
Count	86

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.742%	55	63.95%	15.184%
Negative	-13.791%	31	36.05%	-4.971%
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 4.03. Further, an Excess Kurtosis of 18.76 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -267.40%	0	0.00%	0.00%
-267.40% to -200.94%	0	0.00%	0.00%
-200.94% to -134.48%	0	0.00%	0.00%
-134.48% to -68.02%	1	3.57%	3.57%
-68.02% to -1.56%	18	64.29%	67.86%
-1.56% to 64.89%	7	25.00%	92.86%
64.89% to 131.35%	1	3.57%	96.43%
131.35% to 197.81%	0	0.00%	96.43%
197.81% to 264.27%	0	0.00%	96.43%
Greater than 264.27%	1	3.57%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-90.18%	87.05%	96.43%	68.27%
±2σ	-178.79%	175.66%	96.43%	95.45%
±3σ	-267.40%	264.27%	96.43%	99.73%

Descriptive Statistics

Mean	-1.564%
Standard Error	16.746%
Median	-21.482%
Mode	-70.088%
Standard Deviation	88.612%
Sample Variance	7852.120%
Kurtosis	18.76
Skewness	4.03
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	-34.308%	20	71.43%	-24.506%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	26.06%
12	3 Years	44.42%
20	5 Years	51.68%
40	10 Years	88.33%
80	20 Years	88.33%
200	50 Years	88.33%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.92. Further, an Excess Kurtosis of 3.67 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -309.20%	0	0.00%	0.00%
-309.20% to -194.16%	0	0.00%	0.00%
-194.16% to -79.11%	0	0.00%	0.00%
-79.11% to 35.94%	3	10.34%	10.34%
35.94% to 150.99%	18	62.07%	72.41%
150.99% to 266.04%	3	10.34%	82.76%
266.04% to 381.09%	3	10.34%	93.10%
381.09% to 496.14%	1	3.45%	96.55%
496.14% to 611.19%	0	0.00%	96.55%
Greater than 611.19%	1	3.45%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.41%	304.39%	82.76%	68.27%
±2σ	-155.81%	457.79%	96.55%	95.45%
±3σ	-309.20%	611.19%	96.55%	99.73%

Descriptive Statistics

Mean	150.993%
Standard Error	28.485%
Median	81.558%
Mode	18.848%
Standard Deviation	153.399%
Sample Variance	23531.134%
Kurtosis	3.67
Skewness	1.92
Range	650.383%
Minimum	18.848%
Maximum	669.231%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	150.993%	29	100.00%	150.993%
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 0.95. Further, an Excess Kurtosis of 1.21 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -161.90%	0	0.00%	0.00%
-161.90% to -111.65%	0	0.00%	0.00%
-111.65% to -61.41%	1	3.45%	3.45%
-61.41% to -11.16%	5	17.24%	20.69%
-11.16% to 39.09%	11	37.93%	58.62%
39.09% to 89.33%	6	20.69%	79.31%
89.33% to 139.58%	3	10.34%	89.66%
139.58% to 189.83%	2	6.90%	96.55%
189.83% to 240.07%	1	3.45%	100.00%
Greater than 240.07%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-27.91%	106.08%	75.86%	68.27%
±2σ	-94.90%	173.08%	93.10%	95.45%
±3σ	-161.90%	240.07%	100.00%	99.73%

Descriptive Statistics

Mean	39.087%
Standard Error	12.441%
Median	30.081%
Mode	-80.645%
Standard Deviation	66.996%
Sample Variance	4488.522%
Kurtosis	1.21
Skewness	0.95
Range	303.684%
Minimum	-80.645%
Maximum	223.039%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	65.027%	21	72.41%	47.089%
Negative	-29.003%	8	27.59%	-8.001%
Zero	0.000%	0	0.00%	0.000%

CTRM Deep Analytical Report