SONO - 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	3.43%	2.61%	2.85%	3.96%	3.88%	3.84%	13.02	14.08	12.53	14.63
Weekly	7.47%	7.00%	7.46%	9.50%	8.54%	8.45%	12.43	14.75	11.41	16.06
Monthly	16.00%	14.77%	15.77%	20.63%	17.78%	17.59%	11.33	16.17	9.49	19.32
Quarterly	29.18%	32.79%	25.67%	37.99%	30.80%	30.47%	9.95	18.42	7.31	25.07
Annual	—	—	—	—	61.60%	60.94%	7.31	25.07	3.95	46.41
Option Time Horizon	—	—	—	—	23.25%	23.00%	10.73	17.08	8.51	21.56

Calculation Notes: Computed at 2026-03-23T16:36 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $13.54 | Strike (K): $15.00 (first OTM call ≥ spot) | Observed Mid-Price: $0.7250 (Bid $0.5500 / Ask $0.9000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 61.60% (CRR binomial tree, American, n=20 steps) | BS IV: 60.87% (European Black-Scholes) | Yahoo IV (Annual): 60.94% → 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 0.23. Further, an Excess Kurtosis of 9.80 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.24%	11	0.57%	0.57%
-10.24% to -7.67%	11	0.57%	1.15%
-7.67% to -5.10%	65	3.39%	4.54%
-5.10% to -2.53%	229	11.94%	16.48%
-2.53% to 0.04%	655	34.15%	50.63%
0.04% to 2.61%	632	32.95%	83.58%
2.61% to 5.18%	231	12.04%	95.62%
5.18% to 7.75%	52	2.71%	98.33%
7.75% to 10.32%	14	0.73%	99.06%
Greater than 10.32%	18	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.39%	3.47%	78.88%	68.27%
±2σ	-6.82%	6.89%	95.67%	95.45%
±3σ	-10.24%	10.32%	98.49%	99.73%

Descriptive Statistics

Mean	0.039%
Standard Error	0.078%
Median	0.000%
Mode	0.000%
Standard Deviation	3.428%
Sample Variance	11.749%
Kurtosis	9.80
Skewness	0.23
Range	54.787%
Minimum	-24.945%
Maximum	29.842%
Count	1918

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.416%	947	49.37%	1.193%
Negative	-2.314%	957	49.90%	-1.155%
Zero	0.000%	14	0.73%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.97%
20	1 Month	4.42%
60	1 Quarter	4.65%
250	1 Year	4.29%
750	3 Years	3.79%
1,250	5 Years	3.96%
2,500	10 Years	4.48%
5,000	20 Years	4.48%
12,500	50 Years	4.48%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.99. Further, an Excess Kurtosis of 17.77 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.49%	0	0.00%	0.00%
-3.49% to -1.53%	0	0.00%	0.00%
-1.53% to 0.42%	0	0.00%	0.00%
0.42% to 2.37%	301	15.69%	15.69%
2.37% to 4.33%	921	47.99%	63.68%
4.33% to 6.28%	413	21.52%	85.20%
6.28% to 8.24%	151	7.87%	93.07%
8.24% to 10.19%	71	3.70%	96.77%
10.19% to 12.15%	30	1.56%	98.33%
Greater than 12.15%	32	1.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.72%	6.93%	84.21%	68.27%
±2σ	-0.88%	9.54%	95.52%	95.45%
±3σ	-3.49%	12.15%	98.33%	99.73%

Descriptive Statistics

Mean	4.329%
Standard Error	0.059%
Median	3.656%
Mode	3.223%
Standard Deviation	2.606%
Sample Variance	6.792%
Kurtosis	17.77
Skewness	2.99
Range	34.556%
Minimum	0.841%
Maximum	35.397%
Count	1919

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.329%	1919	100.00%	4.329%
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 -0.12. Further, an Excess Kurtosis of 3.71 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.44%	17	0.89%	0.89%
-8.44% to -6.30%	15	0.78%	1.67%
-6.30% to -4.16%	61	3.18%	4.85%
-4.16% to -2.03%	254	13.24%	18.08%
-2.03% to 0.11%	637	33.19%	51.28%
0.11% to 2.24%	590	30.75%	82.02%
2.24% to 4.38%	232	12.09%	94.11%
4.38% to 6.52%	77	4.01%	98.12%
6.52% to 8.65%	18	0.94%	99.06%
Greater than 8.65%	18	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.74%	2.96%	75.46%	68.27%
±2σ	-5.59%	5.80%	94.95%	95.45%
±3σ	-8.44%	8.65%	98.18%	99.73%

Descriptive Statistics

Mean	0.108%
Standard Error	0.065%
Median	0.053%
Mode	0.000%
Standard Deviation	2.848%
Sample Variance	8.113%
Kurtosis	3.71
Skewness	-0.12
Range	32.889%
Minimum	-19.638%
Maximum	13.251%
Count	1919

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.172%	963	50.18%	1.090%
Negative	-2.024%	931	48.52%	-0.982%
Zero	0.000%	25	1.30%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.09. Further, an Excess Kurtosis of 2.72 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -22.22%	4	1.01%	1.01%
-22.22% to -16.61%	1	0.25%	1.26%
-16.61% to -11.01%	17	4.27%	5.53%
-11.01% to -5.41%	45	11.31%	16.83%
-5.41% to 0.19%	140	35.18%	52.01%
0.19% to 5.79%	120	30.15%	82.16%
5.79% to 11.39%	47	11.81%	93.97%
11.39% to 17.00%	15	3.77%	97.74%
17.00% to 22.60%	7	1.76%	99.50%
Greater than 22.60%	2	0.50%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.28%	7.66%	77.14%	68.27%
±2σ	-14.75%	15.13%	93.97%	95.45%
±3σ	-22.22%	22.60%	98.49%	99.73%

Descriptive Statistics

Mean	0.191%
Standard Error	0.374%
Median	-0.180%
Mode	-7.465%
Standard Deviation	7.469%
Sample Variance	55.788%
Kurtosis	2.72
Skewness	0.09
Range	63.630%
Minimum	-28.663%
Maximum	34.967%
Count	398

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.770%	194	48.74%	2.813%
Negative	-5.165%	202	50.75%	-2.621%
Zero	0.000%	2	0.50%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	10.27%
12	1 Quarter	10.76%
52	1 Year	10.47%
156	3 Years	9.22%
260	5 Years	9.50%
520	10 Years	10.56%
1,040	20 Years	10.56%
2,600	50 Years	10.56%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.16. Further, an Excess Kurtosis of 6.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.73%	0	0.00%	0.00%
-9.73% to -4.48%	0	0.00%	0.00%
-4.48% to 0.77%	0	0.00%	0.00%
0.77% to 6.02%	70	17.54%	17.54%
6.02% to 11.27%	190	47.62%	65.16%
11.27% to 16.52%	74	18.55%	83.71%
16.52% to 21.78%	35	8.77%	92.48%
21.78% to 27.03%	15	3.76%	96.24%
27.03% to 32.28%	6	1.50%	97.74%
Greater than 32.28%	9	2.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.27%	18.27%	82.46%	68.27%
±2σ	-2.73%	25.28%	95.74%	95.45%
±3σ	-9.73%	32.28%	97.74%	99.73%

Descriptive Statistics

Mean	11.274%
Standard Error	0.350%
Median	8.868%
Mode	3.096%
Standard Deviation	7.001%
Sample Variance	49.008%
Kurtosis	6.75
Skewness	2.16
Range	49.091%
Minimum	3.069%
Maximum	52.160%
Count	399

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.274%	399	100.00%	11.274%
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 0.93. Further, an Excess Kurtosis of 4.78 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -22.10%	1	0.25%	0.25%
-22.10% to -16.51%	3	0.75%	1.00%
-16.51% to -10.91%	17	4.26%	5.26%
-10.91% to -5.32%	49	12.28%	17.54%
-5.32% to 0.27%	138	34.59%	52.13%
0.27% to 5.87%	124	31.08%	83.21%
5.87% to 11.46%	43	10.78%	93.98%
11.46% to 17.05%	15	3.76%	97.74%
17.05% to 22.65%	5	1.25%	99.00%
Greater than 22.65%	4	1.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.19%	7.73%	77.44%	68.27%
±2σ	-14.64%	15.19%	95.49%	95.45%
±3σ	-22.10%	22.65%	98.75%	99.73%

Descriptive Statistics

Mean	0.272%
Standard Error	0.373%
Median	0.077%
Mode	-9.770%
Standard Deviation	7.458%
Sample Variance	55.618%
Kurtosis	4.78
Skewness	0.93
Range	64.549%
Minimum	-23.184%
Maximum	41.365%
Count	399

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.527%	202	50.63%	2.798%
Negative	-5.116%	197	49.37%	-2.526%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.75. Further, an Excess Kurtosis of 1.18 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -47.17%	0	0.00%	0.00%
-47.17% to -35.17%	0	0.00%	0.00%
-35.17% to -23.18%	3	3.30%	3.30%
-23.18% to -11.18%	18	19.78%	23.08%
-11.18% to 0.82%	29	31.87%	54.95%
0.82% to 12.82%	27	29.67%	84.62%
12.82% to 24.81%	8	8.79%	93.41%
24.81% to 36.81%	3	3.30%	96.70%
36.81% to 48.81%	1	1.10%	97.80%
Greater than 48.81%	2	2.20%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.18%	16.82%	71.43%	68.27%
±2σ	-31.18%	32.81%	92.31%	95.45%
±3σ	-47.17%	48.81%	97.80%	99.73%

Descriptive Statistics

Mean	0.818%
Standard Error	1.677%
Median	-0.387%
Mode	-31.976%
Standard Deviation	15.997%
Sample Variance	255.892%
Kurtosis	1.18
Skewness	0.75
Range	84.305%
Minimum	-31.976%
Maximum	52.329%
Count	91

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.518%	44	48.35%	6.536%
Negative	-11.312%	46	50.55%	-5.718%
Zero	0.000%	1	1.10%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	21.63%
12	1 Year	21.58%
36	3 Years	20.40%
60	5 Years	20.63%
120	10 Years	23.38%
240	20 Years	23.38%
600	50 Years	23.38%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.33. Further, an Excess Kurtosis of 1.48 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.06%	0	0.00%	0.00%
-17.06% to -5.98%	0	0.00%	0.00%
-5.98% to 5.09%	0	0.00%	0.00%
5.09% to 16.17%	24	26.09%	26.09%
16.17% to 27.25%	31	33.70%	59.78%
27.25% to 38.33%	21	22.83%	82.61%
38.33% to 49.41%	5	5.43%	88.04%
49.41% to 60.49%	7	7.61%	95.65%
60.49% to 71.56%	3	3.26%	98.91%
Greater than 71.56%	1	1.09%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.48%	42.02%	77.17%	68.27%
±2σ	-2.29%	56.79%	93.48%	95.45%
±3σ	-17.06%	71.56%	98.91%	99.73%

Descriptive Statistics

Mean	27.251%
Standard Error	1.540%
Median	23.024%
Mode	9.520%
Standard Deviation	14.771%
Sample Variance	218.193%
Kurtosis	1.48
Skewness	1.33
Range	70.723%
Minimum	9.520%
Maximum	80.243%
Count	92

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	27.251%	92	100.00%	27.251%
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 0.30. Further, an Excess Kurtosis of 0.30 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -45.84%	0	0.00%	0.00%
-45.84% to -34.01%	1	1.09%	1.09%
-34.01% to -22.19%	5	5.43%	6.52%
-22.19% to -10.36%	12	13.04%	19.57%
-10.36% to 1.46%	29	31.52%	51.09%
1.46% to 13.29%	26	28.26%	79.35%
13.29% to 25.11%	14	15.22%	94.57%
25.11% to 36.94%	2	2.17%	96.74%
36.94% to 48.76%	3	3.26%	100.00%
Greater than 48.76%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.30%	17.23%	70.65%	68.27%
±2σ	-30.07%	33.00%	94.57%	95.45%
±3σ	-45.84%	48.76%	100.00%	99.73%

Descriptive Statistics

Mean	1.462%
Standard Error	1.644%
Median	0.795%
Mode	-34.038%
Standard Deviation	15.767%
Sample Variance	248.588%
Kurtosis	0.30
Skewness	0.30
Range	79.324%
Minimum	-34.038%
Maximum	45.286%
Count	92

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.635%	47	51.09%	6.966%
Negative	-11.252%	45	48.91%	-5.504%
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 1.03. Further, an Excess Kurtosis of 0.51 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -83.77%	0	0.00%	0.00%
-83.77% to -61.89%	0	0.00%	0.00%
-61.89% to -40.00%	0	0.00%	0.00%
-40.00% to -18.11%	7	23.33%	23.33%
-18.11% to 3.77%	11	36.67%	60.00%
3.77% to 25.66%	7	23.33%	83.33%
25.66% to 47.54%	1	3.33%	86.67%
47.54% to 69.43%	3	10.00%	96.67%
69.43% to 91.31%	1	3.33%	100.00%
Greater than 91.31%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.41%	32.95%	70.00%	68.27%
±2σ	-54.59%	62.13%	96.67%	95.45%
±3σ	-83.77%	91.31%	100.00%	99.73%

Descriptive Statistics

Mean	3.771%
Standard Error	5.328%
Median	-4.343%
Mode	-37.106%
Standard Deviation	29.181%
Sample Variance	851.544%
Kurtosis	0.51
Skewness	1.03
Range	116.216%
Minimum	-37.106%
Maximum	79.110%
Count	30

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.874%	12	40.00%	12.750%
Negative	-14.964%	18	60.00%	-8.978%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	44.11%
12	3 Years	41.47%
20	5 Years	37.99%
40	10 Years	44.33%
80	20 Years	44.33%
200	50 Years	44.33%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.12. Further, an Excess Kurtosis of 1.03 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -40.27%	0	0.00%	0.00%
-40.27% to -15.68%	0	0.00%	0.00%
-15.68% to 8.92%	0	0.00%	0.00%
8.92% to 33.51%	8	25.81%	25.81%
33.51% to 58.10%	10	32.26%	58.06%
58.10% to 82.69%	7	22.58%	80.65%
82.69% to 107.28%	4	12.90%	93.55%
107.28% to 131.87%	1	3.23%	96.77%
131.87% to 156.46%	0	0.00%	96.77%
Greater than 156.46%	1	3.23%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	25.31%	90.88%	70.97%	68.27%
±2σ	-7.48%	123.67%	96.77%	95.45%
±3σ	-40.26%	156.46%	96.77%	99.73%

Descriptive Statistics

Mean	58.096%
Standard Error	5.889%
Median	42.581%
Mode	21.192%
Standard Deviation	32.787%
Sample Variance	1074.990%
Kurtosis	1.03
Skewness	1.12
Range	135.343%
Minimum	21.192%
Maximum	156.535%
Count	31

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	58.096%	31	100.00%	58.096%
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 -0.05. Further, an Excess Kurtosis of -0.42 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -73.09%	0	0.00%	0.00%
-73.09% to -53.84%	0	0.00%	0.00%
-53.84% to -34.59%	3	9.68%	9.68%
-34.59% to -15.34%	3	9.68%	19.35%
-15.34% to 3.91%	9	29.03%	48.39%
3.91% to 23.16%	9	29.03%	77.42%
23.16% to 42.41%	5	16.13%	93.55%
42.41% to 61.66%	2	6.45%	100.00%
61.66% to 80.91%	0	0.00%	100.00%
Greater than 80.91%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.75%	29.58%	67.74%	68.27%
±2σ	-47.42%	55.25%	96.77%	95.45%
±3σ	-73.09%	80.91%	100.00%	99.73%

Descriptive Statistics

Mean	3.913%
Standard Error	4.610%
Median	6.494%
Mode	-43.901%
Standard Deviation	25.667%
Sample Variance	658.814%
Kurtosis	-0.42
Skewness	-0.05
Range	101.879%
Minimum	-43.901%
Maximum	57.978%
Count	31

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.159%	19	61.29%	12.355%
Negative	-21.811%	12	38.71%	-8.443%
Zero	0.000%	0	0.00%	0.000%

SONO Deep Analytical Report