NTRA - 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	3.86%	3.47%	3.37%	4.89%	3.74%	3.75%	191.14	206.00	184.11	213.86
Weekly	8.41%	8.85%	8.51%	11.20%	8.24%	8.26%	182.73	215.48	168.27	234.00
Monthly	18.10%	20.41%	17.34%	23.22%	17.16%	17.20%	167.14	235.58	140.78	279.68
Quarterly	32.38%	34.50%	34.85%	43.08%	29.72%	29.79%	147.41	267.11	109.50	359.57
Annual	—	—	—	—	59.45%	59.58%	109.50	359.57	60.43	651.56
Option Time Horizon	—	—	—	—	22.44%	22.49%	158.55	248.34	126.68	310.81

Calculation Notes: Computed at 2026-03-23T16:40 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $198.43 | Strike (K): $200.00 (first OTM call ≥ spot) | Observed Mid-Price: $17.4000 (Bid $15.8000 / Ask $19.0000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 59.45% (CRR binomial tree, American, n=20 steps) | BS IV: 59.13% (European Black-Scholes) | Yahoo IV (Annual): 59.58% → 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 0.47. Further, an Excess Kurtosis of 11.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.44%	15	0.56%	0.56%
-11.44% to -8.54%	22	0.82%	1.37%
-8.54% to -5.64%	84	3.12%	4.49%
-5.64% to -2.74%	327	12.14%	16.63%
-2.74% to 0.16%	930	34.52%	51.15%
0.16% to 3.05%	874	32.44%	83.59%
3.05% to 5.95%	326	12.10%	95.69%
5.95% to 8.85%	72	2.67%	98.37%
8.85% to 11.75%	17	0.63%	99.00%
Greater than 11.75%	27	1.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.71%	4.02%	79.18%	68.27%
±2σ	-7.57%	7.88%	95.81%	95.45%
±3σ	-11.44%	11.75%	98.44%	99.73%

Descriptive Statistics

Mean	0.156%
Standard Error	0.074%
Median	0.075%
Mode	0.000%
Standard Deviation	3.864%
Sample Variance	14.927%
Kurtosis	11.42
Skewness	0.47
Range	72.739%
Minimum	-32.785%
Maximum	39.954%
Count	2694

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.747%	1372	50.93%	1.399%
Negative	-2.560%	1308	48.55%	-1.243%
Zero	0.000%	14	0.52%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.47%
20	1 Month	4.36%
60	1 Quarter	4.01%
250	1 Year	3.83%
750	3 Years	4.05%
1,250	5 Years	4.89%
2,500	10 Years	5.13%
5,000	20 Years	5.31%
12,500	50 Years	5.31%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.34. Further, an Excess Kurtosis of 70.11 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.12%	0	0.00%	0.00%
-5.12% to -2.52%	0	0.00%	0.00%
-2.52% to 0.08%	0	0.00%	0.00%
0.08% to 2.68%	354	13.14%	13.14%
2.68% to 5.28%	1357	50.35%	63.49%
5.28% to 7.87%	638	23.67%	87.16%
7.87% to 10.47%	198	7.35%	94.51%
10.47% to 13.07%	77	2.86%	97.37%
13.07% to 15.67%	25	0.93%	98.29%
Greater than 15.67%	46	1.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.81%	8.74%	88.57%	68.27%
±2σ	-1.66%	12.21%	96.55%	95.45%
±3σ	-5.12%	15.67%	98.29%	99.73%

Descriptive Statistics

Mean	5.275%
Standard Error	0.067%
Median	4.507%
Mode	2.564%
Standard Deviation	3.466%
Sample Variance	12.013%
Kurtosis	70.11
Skewness	5.34
Range	73.171%
Minimum	1.273%
Maximum	74.444%
Count	2695

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.275%	2695	100.00%	5.275%
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 0.18. Further, an Excess Kurtosis of 5.07 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.13%	20	0.74%	0.74%
-10.13% to -7.61%	26	0.96%	1.71%
-7.61% to -5.08%	73	2.71%	4.42%
-5.08% to -2.56%	377	13.99%	18.40%
-2.56% to -0.04%	866	32.13%	50.54%
-0.04% to 2.49%	856	31.76%	82.30%
2.49% to 5.01%	324	12.02%	94.32%
5.01% to 7.54%	105	3.90%	98.22%
7.54% to 10.06%	31	1.15%	99.37%
Greater than 10.06%	17	0.63%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.40%	3.33%	76.33%	68.27%
±2σ	-6.77%	6.70%	95.14%	95.45%
±3σ	-10.13%	10.06%	98.63%	99.73%

Descriptive Statistics

Mean	-0.035%
Standard Error	0.065%
Median	-0.077%
Mode	0.000%
Standard Deviation	3.365%
Sample Variance	11.321%
Kurtosis	5.07
Skewness	0.18
Range	48.950%
Minimum	-25.445%
Maximum	23.505%
Count	2695

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.463%	1305	48.42%	1.193%
Negative	-2.419%	1368	50.76%	-1.228%
Zero	0.000%	22	0.82%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.25. Further, an Excess Kurtosis of 3.09 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.48%	3	0.54%	0.54%
-24.48% to -18.18%	6	1.07%	1.61%
-18.18% to -11.87%	24	4.29%	5.90%
-11.87% to -5.56%	60	10.73%	16.64%
-5.56% to 0.75%	190	33.99%	50.63%
0.75% to 7.05%	172	30.77%	81.40%
7.05% to 13.36%	72	12.88%	94.28%
13.36% to 19.67%	24	4.29%	98.57%
19.67% to 25.98%	3	0.54%	99.11%
Greater than 25.98%	5	0.89%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.66%	9.16%	75.13%	68.27%
±2σ	-16.07%	17.57%	95.35%	95.45%
±3σ	-24.48%	25.98%	98.57%	99.73%

Descriptive Statistics

Mean	0.746%
Standard Error	0.356%
Median	0.697%
Mode	0.000%
Standard Deviation	8.410%
Sample Variance	70.728%
Kurtosis	3.09
Skewness	0.25
Range	78.044%
Minimum	-33.303%
Maximum	44.741%
Count	559

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.356%	299	53.49%	3.400%
Negative	-5.817%	255	45.62%	-2.654%
Zero	0.000%	5	0.89%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	8.58%
12	1 Quarter	8.88%
52	1 Year	8.48%
156	3 Years	9.22%
260	5 Years	11.20%
520	10 Years	11.74%
1,040	20 Years	12.10%
2,600	50 Years	12.10%

High-to-Low 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 39.62 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.48%	0	0.00%	0.00%
-13.48% to -6.84%	0	0.00%	0.00%
-6.84% to -0.20%	0	0.00%	0.00%
-0.20% to 6.43%	78	13.93%	13.93%
6.43% to 13.07%	269	48.04%	61.96%
13.07% to 19.71%	132	23.57%	85.54%
19.71% to 26.35%	49	8.75%	94.29%
26.35% to 32.98%	18	3.21%	97.50%
32.98% to 39.62%	5	0.89%	98.39%
Greater than 39.62%	9	1.61%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.22%	21.92%	88.04%	68.27%
±2σ	-4.63%	30.77%	96.25%	95.45%
±3σ	-13.48%	39.62%	98.39%	99.73%

Descriptive Statistics

Mean	13.071%
Standard Error	0.374%
Median	11.012%
Mode	5.488%
Standard Deviation	8.850%
Sample Variance	78.328%
Kurtosis	39.62
Skewness	4.41
Range	115.967%
Minimum	2.999%
Maximum	118.966%
Count	560

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.071%	560	100.00%	13.071%
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 1.00. Further, an Excess Kurtosis of 4.92 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.59%	2	0.36%	0.36%
-25.59% to -19.21%	4	0.71%	1.07%
-19.21% to -12.82%	20	3.57%	4.64%
-12.82% to -6.44%	76	13.57%	18.21%
-6.44% to -0.06%	198	35.36%	53.57%
-0.06% to 6.32%	172	30.71%	84.29%
6.32% to 12.71%	54	9.64%	93.93%
12.71% to 19.09%	22	3.93%	97.86%
19.09% to 25.47%	6	1.07%	98.93%
Greater than 25.47%	6	1.07%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.57%	8.45%	75.36%	68.27%
±2σ	-17.08%	16.96%	95.00%	95.45%
±3σ	-25.59%	25.47%	98.57%	99.73%

Descriptive Statistics

Mean	-0.060%
Standard Error	0.360%
Median	-0.524%
Mode	0.000%
Standard Deviation	8.510%
Sample Variance	72.427%
Kurtosis	4.92
Skewness	1.00
Range	80.942%
Minimum	-30.604%
Maximum	50.338%
Count	560

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.542%	253	45.18%	2.956%
Negative	-5.555%	304	54.29%	-3.016%
Zero	0.000%	3	0.54%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.49. Further, an Excess Kurtosis of 0.35 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -50.86%	0	0.00%	0.00%
-50.86% to -37.28%	1	0.78%	0.78%
-37.28% to -23.71%	6	4.69%	5.47%
-23.71% to -10.14%	22	17.19%	22.66%
-10.14% to 3.43%	38	29.69%	52.34%
3.43% to 17.00%	34	26.56%	78.91%
17.00% to 30.57%	16	12.50%	91.41%
30.57% to 44.15%	9	7.03%	98.44%
44.15% to 57.72%	1	0.78%	99.22%
Greater than 57.72%	1	0.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.66%	21.53%	73.44%	68.27%
±2σ	-32.76%	39.62%	94.53%	95.45%
±3σ	-50.86%	57.72%	99.22%	99.73%

Descriptive Statistics

Mean	3.431%
Standard Error	1.599%
Median	3.075%
Mode	-38.129%
Standard Deviation	18.096%
Sample Variance	327.467%
Kurtosis	0.35
Skewness	0.49
Range	99.536%
Minimum	-38.129%
Maximum	61.407%
Count	128

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.795%	72	56.25%	8.885%
Negative	-12.691%	55	42.97%	-5.453%
Zero	0.000%	1	0.78%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	14.72%
12	1 Year	18.23%
36	3 Years	20.66%
60	5 Years	23.22%
120	10 Years	25.82%
240	20 Years	26.36%
600	50 Years	26.36%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.55. Further, an Excess Kurtosis of 18.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.96%	0	0.00%	0.00%
-29.96% to -14.65%	0	0.00%	0.00%
-14.65% to 0.65%	0	0.00%	0.00%
0.65% to 15.96%	24	18.60%	18.60%
15.96% to 31.27%	55	42.64%	61.24%
31.27% to 46.58%	35	27.13%	88.37%
46.58% to 61.89%	9	6.98%	95.35%
61.89% to 77.19%	4	3.10%	98.45%
77.19% to 92.50%	0	0.00%	98.45%
Greater than 92.50%	2	1.55%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.86%	51.68%	90.70%	68.27%
±2σ	-9.55%	72.09%	97.67%	95.45%
±3σ	-29.96%	92.50%	98.45%	99.73%

Descriptive Statistics

Mean	31.270%
Standard Error	1.797%
Median	27.075%
Mode	9.858%
Standard Deviation	20.411%
Sample Variance	416.623%
Kurtosis	18.75
Skewness	3.55
Range	148.954%
Minimum	9.858%
Maximum	158.812%
Count	129

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.270%	129	100.00%	31.270%
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 0.54. Further, an Excess Kurtosis of 0.56 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -52.36%	0	0.00%	0.00%
-52.36% to -39.35%	0	0.00%	0.00%
-39.35% to -26.35%	7	5.43%	5.43%
-26.35% to -13.34%	20	15.50%	20.93%
-13.34% to -0.33%	46	35.66%	56.59%
-0.33% to 12.68%	30	23.26%	79.84%
12.68% to 25.69%	15	11.63%	91.47%
25.69% to 38.69%	9	6.98%	98.45%
38.69% to 51.70%	1	0.78%	99.22%
Greater than 51.70%	1	0.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.67%	17.01%	71.32%	68.27%
±2σ	-35.02%	34.36%	96.12%	95.45%
±3σ	-52.36%	51.70%	99.22%	99.73%

Descriptive Statistics

Mean	-0.330%
Standard Error	1.527%
Median	-2.913%
Mode	15.807%
Standard Deviation	17.344%
Sample Variance	300.800%
Kurtosis	0.56
Skewness	0.54
Range	98.825%
Minimum	-37.566%
Maximum	61.259%
Count	129

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.295%	56	43.41%	6.640%
Negative	-12.316%	73	56.59%	-6.970%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.39. Further, an Excess Kurtosis of 0.72 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -86.57%	0	0.00%	0.00%
-86.57% to -62.28%	0	0.00%	0.00%
-62.28% to -37.99%	4	9.30%	9.30%
-37.99% to -13.71%	3	6.98%	16.28%
-13.71% to 10.58%	18	41.86%	58.14%
10.58% to 34.87%	7	16.28%	74.42%
34.87% to 59.16%	9	20.93%	95.35%
59.16% to 83.44%	1	2.33%	97.67%
83.44% to 107.73%	1	2.33%	100.00%
Greater than 107.73%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.80%	42.97%	74.42%	68.27%
±2σ	-54.18%	75.35%	95.35%	95.45%
±3σ	-86.57%	107.73%	100.00%	99.73%

Descriptive Statistics

Mean	10.582%
Standard Error	4.938%
Median	5.660%
Mode	-54.395%
Standard Deviation	32.383%
Sample Variance	1048.636%
Kurtosis	0.72
Skewness	0.39
Range	160.684%
Minimum	-54.395%
Maximum	106.289%
Count	43

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	28.647%	27	62.79%	17.988%
Negative	-19.902%	16	37.21%	-7.405%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	34.22%
12	3 Years	41.43%
20	5 Years	43.08%
40	10 Years	48.55%
80	20 Years	48.95%
200	50 Years	48.95%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.80. Further, an Excess Kurtosis of 3.52 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.09%	0	0.00%	0.00%
-41.09% to -15.21%	0	0.00%	0.00%
-15.21% to 10.66%	0	0.00%	0.00%
10.66% to 36.54%	8	18.18%	18.18%
36.54% to 62.41%	22	50.00%	68.18%
62.41% to 88.29%	6	13.64%	81.82%
88.29% to 114.16%	4	9.09%	90.91%
114.16% to 140.04%	2	4.55%	95.45%
140.04% to 165.91%	1	2.27%	97.73%
Greater than 165.91%	1	2.27%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	27.91%	96.91%	79.55%	68.27%
±2σ	-6.59%	131.41%	93.18%	95.45%
±3σ	-41.09%	165.91%	97.73%	99.73%

Descriptive Statistics

Mean	62.412%
Standard Error	5.201%
Median	51.723%
Mode	23.739%
Standard Deviation	34.501%
Sample Variance	1190.334%
Kurtosis	3.52
Skewness	1.80
Range	162.928%
Minimum	23.739%
Maximum	186.667%
Count	44

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	62.412%	44	100.00%	62.412%
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 1.61. Further, an Excess Kurtosis of 3.17 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -104.43%	0	0.00%	0.00%
-104.43% to -78.29%	0	0.00%	0.00%
-78.29% to -52.15%	0	0.00%	0.00%
-52.15% to -26.00%	11	25.00%	25.00%
-26.00% to 0.14%	16	36.36%	61.36%
0.14% to 26.28%	12	27.27%	88.64%
26.28% to 52.42%	0	0.00%	88.64%
52.42% to 78.56%	3	6.82%	95.45%
78.56% to 104.70%	1	2.27%	97.73%
Greater than 104.70%	1	2.27%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-34.72%	34.99%	81.82%	68.27%
±2σ	-69.57%	69.84%	95.45%	95.45%
±3σ	-104.43%	104.70%	97.73%	99.73%

Descriptive Statistics

Mean	0.136%
Standard Error	5.254%
Median	-3.919%
Mode	-51.611%
Standard Deviation	34.854%
Sample Variance	1214.805%
Kurtosis	3.17
Skewness	1.61
Range	168.702%
Minimum	-51.611%
Maximum	117.091%
Count	44

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.613%	17	38.64%	12.214%
Negative	-19.683%	27	61.36%	-12.078%
Zero	0.000%	0	0.00%	0.000%

NTRA Deep Analytical Report