TWLO - 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	3.80%	2.56%	3.05%	4.67%	2.84%	3.20%	109.97	116.40	106.89	119.76
Weekly	8.82%	8.49%	9.01%	11.19%	6.26%	7.04%	106.28	120.45	99.83	128.23
Monthly	17.32%	17.57%	17.94%	23.37%	13.03%	14.65%	99.32	128.88	87.19	146.82
Quarterly	36.29%	29.50%	29.76%	41.48%	22.57%	25.37%	90.28	141.78	72.04	177.68
Annual	—	—	—	—	45.13%	50.74%	72.04	177.68	45.88	279.03
Option Time Horizon	—	—	—	—	17.36%	19.52%	95.11	134.59	79.95	160.11

Calculation Notes: Computed at 2026-02-22T14:25 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $113.14 | Strike (K): $115.00 (first OTM call ≥ spot) | Observed Mid-Price: $7.3000 (Bid $6.5500 / Ask $8.0500) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 45.13% (CRR binomial tree, American, n=20 steps) | BS IV: 45.31% (European Black-Scholes) | Yahoo IV (Annual): 50.74% → 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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.61. Further, an Excess Kurtosis of 14.17 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.28%	10	0.41%	0.41%
-11.28% to -8.43%	21	0.86%	1.28%
-8.43% to -5.58%	98	4.04%	5.31%
-5.58% to -2.73%	272	11.20%	16.52%
-2.73% to 0.13%	808	33.28%	49.79%
0.13% to 2.98%	846	34.84%	84.64%
2.98% to 5.83%	259	10.67%	95.30%
5.83% to 8.69%	70	2.88%	98.19%
8.69% to 11.54%	23	0.95%	99.14%
Greater than 11.54%	21	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.68%	3.93%	78.75%	68.27%
±2σ	-7.48%	7.74%	95.76%	95.45%
±3σ	-11.28%	11.54%	98.72%	99.73%

Descriptive Statistics

Mean	0.128%
Standard Error	0.077%
Median	0.142%
Mode	0.000%
Standard Deviation	3.804%
Sample Variance	14.469%
Kurtosis	14.17
Skewness	0.61
Range	74.224%
Minimum	-34.608%
Maximum	39.616%
Count	2428

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.587%	1263	52.02%	1.346%
Negative	-2.548%	1160	47.78%	-1.217%
Zero	0.000%	5	0.21%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	6.58%
20	1 Month	6.01%
60	1 Quarter	4.34%
250	1 Year	4.35%
750	3 Years	3.83%
1,250	5 Years	4.67%
2,500	10 Years	4.81%
5,000	20 Years	4.81%
12,500	50 Years	4.81%

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 6.33 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.08%	0	0.00%	0.00%
-3.08% to -1.16%	0	0.00%	0.00%
-1.16% to 0.76%	0	0.00%	0.00%
0.76% to 2.68%	520	21.41%	21.41%
2.68% to 4.59%	976	40.18%	61.59%
4.59% to 6.51%	523	21.53%	83.12%
6.51% to 8.43%	223	9.18%	92.30%
8.43% to 10.35%	94	3.87%	96.17%
10.35% to 12.27%	54	2.22%	98.39%
Greater than 12.27%	39	1.61%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.04%	7.15%	78.39%	68.27%
±2σ	-0.52%	9.71%	95.14%	95.45%
±3σ	-3.08%	12.27%	98.39%	99.73%

Descriptive Statistics

Mean	4.595%
Standard Error	0.052%
Median	3.946%
Mode	2.033%
Standard Deviation	2.557%
Sample Variance	6.536%
Kurtosis	6.33
Skewness	1.92
Range	24.307%
Minimum	0.841%
Maximum	25.148%
Count	2429

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.595%	2429	100.00%	4.595%
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 0.07. Further, an Excess Kurtosis of 2.24 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.10%	16	0.66%	0.66%
-9.10% to -6.82%	25	1.03%	1.69%
-6.82% to -4.53%	79	3.25%	4.94%
-4.53% to -2.24%	334	13.75%	18.69%
-2.24% to 0.04%	794	32.69%	51.38%
0.04% to 2.33%	743	30.59%	81.97%
2.33% to 4.62%	275	11.32%	93.29%
4.62% to 6.90%	105	4.32%	97.61%
6.90% to 9.19%	41	1.69%	99.30%
Greater than 9.19%	17	0.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.00%	3.09%	74.31%	68.27%
±2σ	-6.05%	6.14%	94.65%	95.45%
±3σ	-9.10%	9.19%	98.64%	99.73%

Descriptive Statistics

Mean	0.044%
Standard Error	0.062%
Median	-0.036%
Mode	0.000%
Standard Deviation	3.049%
Sample Variance	9.295%
Kurtosis	2.24
Skewness	0.07
Range	32.407%
Minimum	-16.672%
Maximum	15.735%
Count	2429

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.310%	1198	49.32%	1.139%
Negative	-2.175%	1224	50.39%	-1.096%
Zero	0.000%	7	0.29%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.68. Further, an Excess Kurtosis of 6.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.81%	3	0.60%	0.60%
-25.81% to -19.19%	4	0.79%	1.39%
-19.19% to -12.57%	25	4.96%	6.35%
-12.57% to -5.95%	55	10.91%	17.26%
-5.95% to 0.67%	169	33.53%	50.79%
0.67% to 7.28%	161	31.94%	82.74%
7.28% to 13.90%	61	12.10%	94.84%
13.90% to 20.52%	17	3.37%	98.21%
20.52% to 27.14%	6	1.19%	99.40%
Greater than 27.14%	3	0.60%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.16%	9.49%	76.19%	68.27%
±2σ	-16.98%	18.31%	95.63%	95.45%
±3σ	-25.81%	27.14%	98.81%	99.73%

Descriptive Statistics

Mean	0.665%
Standard Error	0.393%
Median	0.518%
Mode	-7.307%
Standard Deviation	8.825%
Sample Variance	77.877%
Kurtosis	6.93
Skewness	0.68
Range	103.354%
Minimum	-38.577%
Maximum	64.777%
Count	504

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.201%	276	54.76%	3.396%
Negative	-6.036%	228	45.24%	-2.731%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	12.32%
12	1 Quarter	10.10%
52	1 Year	10.53%
156	3 Years	9.34%
260	5 Years	11.19%
520	10 Years	11.59%
1,040	20 Years	11.59%
2,600	50 Years	11.59%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.45. Further, an Excess Kurtosis of 21.79 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.05%	0	0.00%	0.00%
-13.05% to -6.68%	0	0.00%	0.00%
-6.68% to -0.31%	0	0.00%	0.00%
-0.31% to 6.06%	73	14.46%	14.46%
6.06% to 12.43%	252	49.90%	64.36%
12.43% to 18.80%	101	20.00%	84.36%
18.80% to 25.16%	50	9.90%	94.26%
25.16% to 31.53%	18	3.56%	97.82%
31.53% to 37.90%	3	0.59%	98.42%
Greater than 37.90%	8	1.58%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.93%	20.92%	86.14%	68.27%
±2σ	-4.56%	29.41%	96.63%	95.45%
±3σ	-13.05%	37.90%	98.42%	99.73%

Descriptive Statistics

Mean	12.426%
Standard Error	0.378%
Median	10.183%
Mode	4.509%
Standard Deviation	8.492%
Sample Variance	72.108%
Kurtosis	21.79
Skewness	3.45
Range	90.470%
Minimum	2.506%
Maximum	92.976%
Count	505

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.426%	505	100.00%	12.426%
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 1.40. Further, an Excess Kurtosis of 8.59 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.75%	1	0.20%	0.20%
-26.75% to -19.99%	2	0.40%	0.59%
-19.99% to -13.23%	16	3.17%	3.76%
-13.23% to -6.47%	65	12.87%	16.63%
-6.47% to 0.29%	198	39.21%	55.84%
0.29% to 7.05%	141	27.92%	83.76%
7.05% to 13.81%	48	9.50%	93.27%
13.81% to 20.57%	24	4.75%	98.02%
20.57% to 27.33%	5	0.99%	99.01%
Greater than 27.33%	5	0.99%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.72%	9.30%	78.22%	68.27%
±2σ	-17.74%	18.32%	95.05%	95.45%
±3σ	-26.75%	27.33%	98.81%	99.73%

Descriptive Statistics

Mean	0.289%
Standard Error	0.401%
Median	-0.379%
Mode	-3.875%
Standard Deviation	9.014%
Sample Variance	81.261%
Kurtosis	8.59
Skewness	1.40
Range	106.854%
Minimum	-38.224%
Maximum	68.630%
Count	505

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.752%	240	47.52%	3.209%
Negative	-5.585%	264	52.28%	-2.920%
Zero	0.000%	1	0.20%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.63. Further, an Excess Kurtosis of 2.22 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -49.56%	0	0.00%	0.00%
-49.56% to -36.56%	1	0.86%	0.86%
-36.56% to -23.57%	3	2.59%	3.45%
-23.57% to -10.58%	20	17.24%	20.69%
-10.58% to 2.41%	37	31.90%	52.59%
2.41% to 15.40%	33	28.45%	81.03%
15.40% to 28.40%	14	12.07%	93.10%
28.40% to 41.39%	7	6.03%	99.14%
41.39% to 54.38%	0	0.00%	99.14%
Greater than 54.38%	1	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.91%	19.74%	67.24%	68.27%
±2σ	-32.23%	37.06%	96.55%	95.45%
±3σ	-49.56%	54.38%	99.14%	99.73%

Descriptive Statistics

Mean	2.413%
Standard Error	1.608%
Median	0.779%
Mode	-46.986%
Standard Deviation	17.323%
Sample Variance	300.089%
Kurtosis	2.22
Skewness	0.63
Range	122.943%
Minimum	-46.986%
Maximum	75.957%
Count	116

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.889%	60	51.72%	7.701%
Negative	-10.955%	56	48.28%	-5.289%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	17.81%
12	1 Year	21.15%
36	3 Years	20.60%
60	5 Years	23.37%
120	10 Years	24.86%
240	20 Years	24.86%
600	50 Years	24.86%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.89. Further, an Excess Kurtosis of 4.35 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.24%	0	0.00%	0.00%
-23.24% to -10.06%	0	0.00%	0.00%
-10.06% to 3.11%	0	0.00%	0.00%
3.11% to 16.29%	23	19.66%	19.66%
16.29% to 29.47%	52	44.44%	64.10%
29.47% to 42.65%	24	20.51%	84.62%
42.65% to 55.83%	10	8.55%	93.16%
55.83% to 69.01%	3	2.56%	95.73%
69.01% to 82.18%	2	1.71%	97.44%
Greater than 82.18%	3	2.56%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	11.90%	47.04%	82.05%	68.27%
±2σ	-5.67%	64.61%	94.87%	95.45%
±3σ	-23.24%	82.18%	97.44%	99.73%

Descriptive Statistics

Mean	29.471%
Standard Error	1.624%
Median	24.701%
Mode	9.172%
Standard Deviation	17.571%
Sample Variance	308.742%
Kurtosis	4.35
Skewness	1.89
Range	88.443%
Minimum	9.172%
Maximum	97.615%
Count	117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	29.471%	117	100.00%	29.471%
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 1.07. Further, an Excess Kurtosis of 3.81 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -53.38%	0	0.00%	0.00%
-53.38% to -39.92%	1	0.85%	0.85%
-39.92% to -26.47%	2	1.71%	2.56%
-26.47% to -13.01%	21	17.95%	20.51%
-13.01% to 0.45%	39	33.33%	53.85%
0.45% to 13.91%	34	29.06%	82.91%
13.91% to 27.36%	14	11.97%	94.87%
27.36% to 40.82%	3	2.56%	97.44%
40.82% to 54.28%	1	0.85%	98.29%
Greater than 54.28%	2	1.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.49%	18.39%	70.94%	68.27%
±2σ	-35.44%	36.34%	96.58%	95.45%
±3σ	-53.38%	54.28%	98.29%	99.73%

Descriptive Statistics

Mean	0.449%
Standard Error	1.659%
Median	-0.592%
Mode	-44.641%
Standard Deviation	17.944%
Sample Variance	321.991%
Kurtosis	3.81
Skewness	1.07
Range	127.818%
Minimum	-44.641%
Maximum	83.177%
Count	117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.076%	56	47.86%	6.737%
Negative	-12.060%	61	52.14%	-6.288%
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 1.71. Further, an Excess Kurtosis of 4.61 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -101.26%	0	0.00%	0.00%
-101.26% to -74.04%	0	0.00%	0.00%
-74.04% to -46.82%	0	0.00%	0.00%
-46.82% to -19.61%	7	17.95%	17.95%
-19.61% to 7.61%	17	43.59%	61.54%
7.61% to 34.83%	8	20.51%	82.05%
34.83% to 62.05%	5	12.82%	94.87%
62.05% to 89.26%	1	2.56%	97.44%
89.26% to 116.48%	0	0.00%	97.44%
Greater than 116.48%	1	2.56%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.68%	43.90%	79.49%	68.27%
±2σ	-64.97%	80.19%	94.87%	95.45%
±3σ	-101.26%	116.48%	97.44%	99.73%

Descriptive Statistics

Mean	7.612%
Standard Error	5.811%
Median	0.559%
Mode	-45.750%
Standard Deviation	36.290%
Sample Variance	1316.973%
Kurtosis	4.61
Skewness	1.71
Range	192.785%
Minimum	-45.750%
Maximum	147.035%
Count	39

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	32.632%	20	51.28%	16.734%
Negative	-18.725%	19	48.72%	-9.122%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	28.57%
12	3 Years	39.55%
20	5 Years	41.48%
40	10 Years	46.02%
80	20 Years	46.02%
200	50 Years	46.02%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.32. Further, an Excess Kurtosis of 2.40 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.69%	0	0.00%	0.00%
-29.69% to -7.57%	0	0.00%	0.00%
-7.57% to 14.56%	0	0.00%	0.00%
14.56% to 36.68%	9	22.50%	22.50%
36.68% to 58.81%	18	45.00%	67.50%
58.81% to 80.93%	3	7.50%	75.00%
80.93% to 103.06%	7	17.50%	92.50%
103.06% to 125.18%	2	5.00%	97.50%
125.18% to 147.31%	0	0.00%	97.50%
Greater than 147.31%	1	2.50%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	29.31%	88.31%	70.00%	68.27%
±2σ	-0.19%	117.81%	97.50%	95.45%
±3σ	-29.69%	147.31%	97.50%	99.73%

Descriptive Statistics

Mean	58.809%
Standard Error	4.664%
Median	55.043%
Mode	17.814%
Standard Deviation	29.500%
Sample Variance	870.235%
Kurtosis	2.40
Skewness	1.32
Range	143.319%
Minimum	17.814%
Maximum	161.133%
Count	40

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	58.809%	40	100.00%	58.809%
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 0.36. Further, an Excess Kurtosis of 0.41 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -88.29%	0	0.00%	0.00%
-88.29% to -65.97%	0	0.00%	0.00%
-65.97% to -43.65%	2	5.00%	5.00%
-43.65% to -21.33%	10	25.00%	30.00%
-21.33% to 0.98%	7	17.50%	47.50%
0.98% to 23.30%	13	32.50%	80.00%
23.30% to 45.62%	6	15.00%	95.00%
45.62% to 67.94%	1	2.50%	97.50%
67.94% to 90.26%	1	2.50%	100.00%
Greater than 90.26%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.77%	30.74%	75.00%	68.27%
±2σ	-58.53%	60.50%	95.00%	95.45%
±3σ	-88.29%	90.26%	100.00%	99.73%

Descriptive Statistics

Mean	0.984%
Standard Error	4.705%
Median	1.142%
Mode	-60.569%
Standard Deviation	29.758%
Sample Variance	885.523%
Kurtosis	0.41
Skewness	0.36
Range	146.788%
Minimum	-60.569%
Maximum	86.219%
Count	40

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.612%	21	52.50%	12.396%
Negative	-24.025%	19	47.50%	-11.412%
Zero	0.000%	0	0.00%	0.000%

TWLO Deep Analytical Report