YELP - 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	3.29%	2.65%	2.65%	3.01%	2.60%	4.71%	20.71	21.81	20.18	22.38
Weekly	6.86%	7.49%	7.01%	7.03%	5.71%	10.37%	20.07	22.50	18.96	23.82
Monthly	13.97%	17.04%	14.86%	14.91%	11.89%	21.59%	18.87	23.93	16.75	26.96
Quarterly	23.43%	29.33%	19.78%	24.20%	20.60%	37.40%	17.29	26.11	14.07	32.08
Annual	—	—	—	—	41.20%	74.80%	14.07	32.08	9.32	48.44
Option Time Horizon	—	—	—	—	19.53%	35.46%	17.48	25.83	14.38	31.40

Calculation Notes: Computed at 2026-02-22T14:21 UTC | Reference Contract: Call option expiring 2026-05-15 (82 days) | Spot Price (S): $21.25 | Strike (K): $22.00 (first OTM call ≥ spot) | Observed Mid-Price: $1.4250 (Bid $0.1500 / Ask $2.7000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 41.20% (CRR binomial tree, American, n=20 steps) | BS IV: 41.66% (European Black-Scholes) | Yahoo IV (Annual): 74.80% → 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 0.31. Further, an Excess Kurtosis of 14.83 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.82%	28	0.80%	0.80%
-9.82% to -7.35%	22	0.63%	1.42%
-7.35% to -4.88%	80	2.28%	3.70%
-4.88% to -2.42%	343	9.77%	13.47%
-2.42% to 0.05%	1317	37.50%	50.97%
0.05% to 2.52%	1224	34.85%	85.82%
2.52% to 4.98%	366	10.42%	96.24%
4.98% to 7.45%	78	2.22%	98.46%
7.45% to 9.92%	22	0.63%	99.09%
Greater than 9.92%	32	0.91%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.24%	3.34%	82.29%	68.27%
±2σ	-6.53%	6.63%	95.93%	95.45%
±3σ	-9.82%	9.92%	98.29%	99.73%

Descriptive Statistics

Mean	0.050%
Standard Error	0.056%
Median	0.000%
Mode	0.000%
Standard Deviation	3.289%
Sample Variance	10.819%
Kurtosis	14.83
Skewness	0.31
Range	54.268%
Minimum	-26.598%
Maximum	27.670%
Count	3512

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.132%	1750	49.83%	1.062%
Negative	-2.046%	1737	49.46%	-1.012%
Zero	0.000%	25	0.71%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	6.72%
20	1 Month	4.59%
60	1 Quarter	3.12%
250	1 Year	2.82%
750	3 Years	2.62%
1,250	5 Years	3.01%
2,500	10 Years	3.46%
5,000	20 Years	4.02%
12,500	50 Years	4.02%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.18. Further, an Excess Kurtosis of 17.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -4.10%	0	0.00%	0.00%
-4.10% to -2.11%	0	0.00%	0.00%
-2.11% to -0.12%	0	0.00%	0.00%
-0.12% to 1.87%	517	14.72%	14.72%
1.87% to 3.86%	1732	49.30%	64.02%
3.86% to 5.85%	759	21.61%	85.62%
5.85% to 7.84%	282	8.03%	93.65%
7.84% to 9.83%	106	3.02%	96.67%
9.83% to 11.82%	56	1.59%	98.26%
Greater than 11.82%	61	1.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.21%	6.51%	87.73%	68.27%
±2σ	-1.45%	9.16%	96.19%	95.45%
±3σ	-4.10%	11.82%	98.26%	99.73%

Descriptive Statistics

Mean	3.858%
Standard Error	0.045%
Median	3.159%
Mode	1.601%
Standard Deviation	2.653%
Sample Variance	7.037%
Kurtosis	17.75
Skewness	3.18
Range	30.704%
Minimum	0.638%
Maximum	31.342%
Count	3513

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.858%	3513	100.00%	3.858%
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 7.50 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -7.93%	21	0.60%	0.60%
-7.93% to -5.94%	33	0.94%	1.54%
-5.94% to -3.95%	116	3.30%	4.84%
-3.95% to -1.95%	407	11.59%	16.42%
-1.95% to 0.04%	1215	34.59%	51.01%
0.04% to 2.03%	1145	32.59%	83.60%
2.03% to 4.02%	404	11.50%	95.10%
4.02% to 6.01%	96	2.73%	97.84%
6.01% to 8.00%	46	1.31%	99.15%
Greater than 8.00%	30	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.62%	2.69%	78.62%	68.27%
±2σ	-5.27%	5.35%	94.90%	95.45%
±3σ	-7.93%	8.00%	98.55%	99.73%

Descriptive Statistics

Mean	0.037%
Standard Error	0.045%
Median	0.000%
Mode	0.000%
Standard Deviation	2.655%
Sample Variance	7.050%
Kurtosis	7.50
Skewness	0.42
Range	41.662%
Minimum	-21.726%
Maximum	19.936%
Count	3513

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.883%	1740	49.53%	0.933%
Negative	-1.796%	1753	49.90%	-0.896%
Zero	0.000%	20	0.57%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.35. Further, an Excess Kurtosis of 4.95 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.34%	8	1.10%	1.10%
-20.34% to -15.19%	8	1.10%	2.19%
-15.19% to -10.05%	26	3.57%	5.76%
-10.05% to -4.91%	68	9.33%	15.09%
-4.91% to 0.24%	256	35.12%	50.21%
0.24% to 5.38%	248	34.02%	84.22%
5.38% to 10.52%	81	11.11%	95.34%
10.52% to 15.66%	20	2.74%	98.08%
15.66% to 20.81%	3	0.41%	98.49%
Greater than 20.81%	11	1.51%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.62%	7.09%	80.38%	68.27%
±2σ	-13.48%	13.95%	94.24%	95.45%
±3σ	-20.34%	20.81%	97.39%	99.73%

Descriptive Statistics

Mean	0.236%
Standard Error	0.254%
Median	0.212%
Mode	-2.554%
Standard Deviation	6.857%
Sample Variance	47.015%
Kurtosis	4.95
Skewness	0.35
Range	76.714%
Minimum	-33.030%
Maximum	43.684%
Count	729

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.685%	384	52.67%	2.468%
Negative	-4.716%	345	47.33%	-2.232%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	12.44%
12	1 Quarter	7.84%
52	1 Year	6.96%
156	3 Years	6.28%
260	5 Years	7.03%
520	10 Years	8.19%
1,040	20 Years	9.44%
2,600	50 Years	9.44%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.61. Further, an Excess Kurtosis of 8.99 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.37%	0	0.00%	0.00%
-12.37% to -6.76%	0	0.00%	0.00%
-6.76% to -1.14%	0	0.00%	0.00%
-1.14% to 4.48%	95	13.01%	13.01%
4.48% to 10.09%	388	53.15%	66.16%
10.09% to 15.71%	150	20.55%	86.71%
15.71% to 21.33%	40	5.48%	92.19%
21.33% to 26.94%	23	3.15%	95.34%
26.94% to 32.56%	14	1.92%	97.26%
Greater than 32.56%	20	2.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.60%	17.58%	88.90%	68.27%
±2σ	-4.88%	25.07%	95.21%	95.45%
±3σ	-12.37%	32.56%	97.26%	99.73%

Descriptive Statistics

Mean	10.093%
Standard Error	0.277%
Median	7.996%
Mode	6.085%
Standard Deviation	7.489%
Sample Variance	56.092%
Kurtosis	8.99
Skewness	2.61
Range	57.134%
Minimum	2.236%
Maximum	59.370%
Count	730

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.093%	730	100.00%	10.093%
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 1.22. Further, an Excess Kurtosis of 7.78 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.72%	3	0.41%	0.41%
-20.72% to -15.46%	8	1.10%	1.51%
-15.46% to -10.21%	16	2.19%	3.70%
-10.21% to -4.95%	88	12.05%	15.75%
-4.95% to 0.31%	280	38.36%	54.11%
0.31% to 5.56%	239	32.74%	86.85%
5.56% to 10.82%	54	7.40%	94.25%
10.82% to 16.07%	21	2.88%	97.12%
16.07% to 21.33%	10	1.37%	98.49%
Greater than 21.33%	11	1.51%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.70%	7.31%	80.27%	68.27%
±2σ	-13.71%	14.32%	94.38%	95.45%
±3σ	-20.72%	21.33%	98.08%	99.73%

Descriptive Statistics

Mean	0.306%
Standard Error	0.259%
Median	-0.090%
Mode	-4.102%
Standard Deviation	7.008%
Sample Variance	49.110%
Kurtosis	7.78
Skewness	1.22
Range	83.977%
Minimum	-29.282%
Maximum	54.695%
Count	730

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.185%	353	48.36%	2.507%
Negative	-4.284%	375	51.37%	-2.201%
Zero	0.000%	2	0.27%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.42. Further, an Excess Kurtosis of 2.28 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.09%	1	0.60%	0.60%
-41.09% to -30.61%	1	0.60%	1.20%
-30.61% to -20.13%	9	5.39%	6.59%
-20.13% to -9.65%	22	13.17%	19.76%
-9.65% to 0.82%	55	32.93%	52.69%
0.82% to 11.30%	48	28.74%	81.44%
11.30% to 21.78%	18	10.78%	92.22%
21.78% to 32.26%	10	5.99%	98.20%
32.26% to 42.74%	2	1.20%	99.40%
Greater than 42.74%	1	0.60%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.15%	14.80%	76.05%	68.27%
±2σ	-27.12%	28.77%	95.81%	95.45%
±3σ	-41.09%	42.74%	98.80%	99.73%

Descriptive Statistics

Mean	0.825%
Standard Error	1.081%
Median	0.433%
Mode	-42.341%
Standard Deviation	13.972%
Sample Variance	195.217%
Kurtosis	2.28
Skewness	0.42
Range	104.720%
Minimum	-42.341%
Maximum	62.379%
Count	167

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.892%	85	50.90%	5.544%
Negative	-9.611%	82	49.10%	-4.719%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	18.19%
12	1 Year	14.90%
36	3 Years	13.71%
60	5 Years	14.91%
120	10 Years	17.70%
240	20 Years	20.45%
600	50 Years	20.45%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.23. Further, an Excess Kurtosis of 17.94 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.20%	0	0.00%	0.00%
-27.20% to -14.42%	0	0.00%	0.00%
-14.42% to -1.64%	0	0.00%	0.00%
-1.64% to 11.14%	24	14.29%	14.29%
11.14% to 23.92%	81	48.21%	62.50%
23.92% to 36.71%	38	22.62%	85.12%
36.71% to 49.49%	13	7.74%	92.86%
49.49% to 62.27%	6	3.57%	96.43%
62.27% to 75.05%	3	1.79%	98.21%
Greater than 75.05%	3	1.79%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	6.88%	40.97%	88.69%	68.27%
±2σ	-10.16%	58.01%	96.43%	95.45%
±3σ	-27.20%	75.05%	98.21%	99.73%

Descriptive Statistics

Mean	23.924%
Standard Error	1.315%
Median	18.907%
Mode	13.912%
Standard Deviation	17.043%
Sample Variance	290.478%
Kurtosis	17.94
Skewness	3.23
Range	144.831%
Minimum	4.665%
Maximum	149.496%
Count	168

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.924%	168	100.00%	23.924%
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 1.29. Further, an Excess Kurtosis of 4.66 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -43.40%	0	0.00%	0.00%
-43.40% to -32.25%	1	0.60%	0.60%
-32.25% to -21.11%	5	2.98%	3.57%
-21.11% to -9.96%	24	14.29%	17.86%
-9.96% to 1.19%	64	38.10%	55.95%
1.19% to 12.34%	47	27.98%	83.93%
12.34% to 23.49%	14	8.33%	92.26%
23.49% to 34.63%	9	5.36%	97.62%
34.63% to 45.78%	2	1.19%	98.81%
Greater than 45.78%	2	1.19%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.67%	16.05%	77.98%	68.27%
±2σ	-28.54%	30.92%	96.43%	95.45%
±3σ	-43.40%	45.78%	98.81%	99.73%

Descriptive Statistics

Mean	1.190%
Standard Error	1.147%
Median	-0.642%
Mode	-38.103%
Standard Deviation	14.864%
Sample Variance	220.937%
Kurtosis	4.66
Skewness	1.29
Range	112.424%
Minimum	-38.103%
Maximum	74.321%
Count	168

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.495%	78	46.43%	5.801%
Negative	-8.608%	90	53.57%	-4.611%
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 1.22. 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 -68.08%	0	0.00%	0.00%
-68.08% to -50.51%	0	0.00%	0.00%
-50.51% to -32.94%	1	1.79%	1.79%
-32.94% to -15.37%	9	16.07%	17.86%
-15.37% to 2.20%	26	46.43%	64.29%
2.20% to 19.77%	9	16.07%	80.36%
19.77% to 37.34%	5	8.93%	89.29%
37.34% to 54.91%	3	5.36%	94.64%
54.91% to 72.48%	3	5.36%	100.00%
Greater than 72.48%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.22%	25.63%	76.79%	68.27%
±2σ	-44.65%	49.05%	91.07%	95.45%
±3σ	-68.08%	72.48%	100.00%	99.73%

Descriptive Statistics

Mean	2.201%
Standard Error	3.130%
Median	-5.642%
Mode	-32.978%
Standard Deviation	23.426%
Sample Variance	548.789%
Kurtosis	1.05
Skewness	1.22
Range	98.662%
Minimum	-32.978%
Maximum	65.684%
Count	56

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.817%	22	39.29%	9.750%
Negative	-12.433%	34	60.71%	-7.549%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	22.15%
12	3 Years	23.72%
20	5 Years	24.20%
40	10 Years	30.73%
80	20 Years	35.91%
200	50 Years	35.91%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.39. Further, an Excess Kurtosis of 8.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.54%	0	0.00%	0.00%
-42.54% to -20.54%	0	0.00%	0.00%
-20.54% to 1.46%	0	0.00%	0.00%
1.46% to 23.46%	10	17.54%	17.54%
23.46% to 45.46%	24	42.11%	59.65%
45.46% to 67.45%	13	22.81%	82.46%
67.45% to 89.45%	6	10.53%	92.98%
89.45% to 111.45%	2	3.51%	96.49%
111.45% to 133.45%	1	1.75%	98.25%
Greater than 133.45%	1	1.75%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	16.12%	74.79%	87.72%	68.27%
±2σ	-13.21%	104.12%	96.49%	95.45%
±3σ	-42.54%	133.45%	98.25%	99.73%

Descriptive Statistics

Mean	45.456%
Standard Error	3.885%
Median	35.818%
Mode	12.807%
Standard Deviation	29.332%
Sample Variance	860.380%
Kurtosis	8.61
Skewness	2.39
Range	173.151%
Minimum	12.807%
Maximum	185.958%
Count	57

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	45.456%	57	100.00%	45.456%
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 -0.13. Further, an Excess Kurtosis of -0.02 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -56.65%	0	0.00%	0.00%
-56.65% to -41.82%	0	0.00%	0.00%
-41.82% to -26.99%	5	8.77%	8.77%
-26.99% to -12.16%	9	15.79%	24.56%
-12.16% to 2.67%	10	17.54%	42.11%
2.67% to 17.51%	21	36.84%	78.95%
17.51% to 32.34%	9	15.79%	94.74%
32.34% to 47.17%	2	3.51%	98.25%
47.17% to 62.00%	1	1.75%	100.00%
Greater than 62.00%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.10%	22.45%	71.93%	68.27%
±2σ	-36.88%	42.23%	92.98%	95.45%
±3σ	-56.65%	62.00%	100.00%	99.73%

Descriptive Statistics

Mean	2.675%
Standard Error	2.619%
Median	5.949%
Mode	-39.337%
Standard Deviation	19.776%
Sample Variance	391.101%
Kurtosis	-0.02
Skewness	-0.13
Range	88.239%
Minimum	-39.337%
Maximum	48.902%
Count	57

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.130%	35	61.40%	9.290%
Negative	-17.140%	22	38.60%	-6.615%
Zero	0.000%	0	0.00%	0.000%

YELP Deep Analytical Report