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	3.79%	2.55%	3.04%	4.62%	3.68%	3.72%	122.40	131.74	117.98	136.67
Weekly	8.81%	8.47%	9.00%	11.07%	8.10%	8.19%	117.11	137.69	108.00	149.30
Monthly	17.23%	17.58%	17.86%	23.08%	16.85%	17.05%	107.29	150.29	90.65	177.87
Quarterly	36.22%	29.30%	29.75%	41.75%	29.19%	29.54%	94.84	170.02	70.83	227.65
Annual	—	—	—	—	58.38%	59.07%	70.83	227.65	39.51	408.12
Option Time Horizon	—	—	—	—	22.03%	22.30%	101.87	158.28	81.72	197.30

Calculation Notes: Computed at 2026-03-23T16:42 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $126.98 | Strike (K): $130.00 (first OTM call ≥ spot) | Observed Mid-Price: $10.1750 (Bid $9.3500 / Ask $11.0000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 58.38% (CRR binomial tree, American, n=20 steps) | BS IV: 58.69% (European Black-Scholes) | Yahoo IV (Annual): 59.07% → 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 14.23 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.24%	10	0.41%	0.41%
-11.24% to -8.40%	21	0.86%	1.27%
-8.40% to -5.56%	98	4.00%	5.27%
-5.56% to -2.71%	275	11.23%	16.50%
-2.71% to 0.13%	816	33.32%	49.82%
0.13% to 2.98%	854	34.87%	84.69%
2.98% to 5.82%	260	10.62%	95.30%
5.82% to 8.66%	70	2.86%	98.16%
8.66% to 11.51%	24	0.98%	99.14%
Greater than 11.51%	21	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.66%	3.92%	78.77%	68.27%
±2σ	-7.45%	7.72%	95.75%	95.45%
±3σ	-11.24%	11.51%	98.73%	99.73%

Descriptive Statistics

Mean	0.132%
Standard Error	0.077%
Median	0.145%
Mode	0.000%
Standard Deviation	3.792%
Sample Variance	14.382%
Kurtosis	14.23
Skewness	0.61
Range	74.224%
Minimum	-34.608%
Maximum	39.616%
Count	2449

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.581%	1275	52.06%	1.344%
Negative	-2.538%	1169	47.73%	-1.211%
Zero	0.000%	5	0.20%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.42%
20	1 Month	4.13%
60	1 Quarter	4.68%
250	1 Year	4.28%
750	3 Years	3.81%
1,250	5 Years	4.62%
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 1.93. Further, an Excess Kurtosis of 6.37 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.06%	0	0.00%	0.00%
-3.06% to -1.15%	0	0.00%	0.00%
-1.15% to 0.77%	0	0.00%	0.00%
0.77% to 2.68%	520	21.22%	21.22%
2.68% to 4.59%	990	40.41%	61.63%
4.59% to 6.50%	529	21.59%	83.22%
6.50% to 8.41%	223	9.10%	92.33%
8.41% to 10.32%	94	3.84%	96.16%
10.32% to 12.24%	55	2.24%	98.41%
Greater than 12.24%	39	1.59%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.04%	7.14%	78.45%	68.27%
±2σ	-0.51%	9.69%	95.18%	95.45%
±3σ	-3.06%	12.24%	98.41%	99.73%

Descriptive Statistics

Mean	4.590%
Standard Error	0.052%
Median	3.941%
Mode	2.033%
Standard Deviation	2.549%
Sample Variance	6.500%
Kurtosis	6.37
Skewness	1.93
Range	24.307%
Minimum	0.841%
Maximum	25.148%
Count	2450

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.590%	2450	100.00%	4.590%
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 2.25 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.09%	16	0.65%	0.65%
-9.09% to -6.80%	26	1.06%	1.71%
-6.80% to -4.52%	80	3.27%	4.98%
-4.52% to -2.24%	336	13.71%	18.69%
-2.24% to 0.04%	796	32.49%	51.18%
0.04% to 2.32%	753	30.73%	81.92%
2.32% to 4.60%	278	11.35%	93.27%
4.60% to 6.88%	107	4.37%	97.63%
6.88% to 9.16%	40	1.63%	99.27%
Greater than 9.16%	18	0.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.00%	3.08%	74.33%	68.27%
±2σ	-6.04%	6.12%	94.69%	95.45%
±3σ	-9.09%	9.16%	98.61%	99.73%

Descriptive Statistics

Mean	0.037%
Standard Error	0.061%
Median	-0.043%
Mode	0.000%
Standard Deviation	3.041%
Sample Variance	9.245%
Kurtosis	2.25
Skewness	0.07
Range	32.407%
Minimum	-16.672%
Maximum	15.735%
Count	2450

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.303%	1205	49.18%	1.133%
Negative	-2.169%	1238	50.53%	-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 6.91 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.75%	3	0.59%	0.59%
-25.75% to -19.14%	4	0.79%	1.38%
-19.14% to -12.54%	25	4.92%	6.30%
-12.54% to -5.93%	55	10.83%	17.13%
-5.93% to 0.68%	172	33.86%	50.98%
0.68% to 7.29%	161	31.69%	82.68%
7.29% to 13.90%	61	12.01%	94.69%
13.90% to 20.51%	18	3.54%	98.23%
20.51% to 27.12%	6	1.18%	99.41%
Greater than 27.12%	3	0.59%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.13%	9.50%	76.18%	68.27%
±2σ	-16.94%	18.31%	95.67%	95.45%
±3σ	-25.75%	27.12%	98.82%	99.73%

Descriptive Statistics

Mean	0.684%
Standard Error	0.391%
Median	0.518%
Mode	-7.307%
Standard Deviation	8.813%
Sample Variance	77.668%
Kurtosis	6.91
Skewness	0.68
Range	103.354%
Minimum	-38.577%
Maximum	64.777%
Count	508

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.216%	278	54.72%	3.402%
Negative	-6.002%	230	45.28%	-2.717%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.34%
12	1 Quarter	11.27%
52	1 Year	10.36%
156	3 Years	9.24%
260	5 Years	11.07%
520	10 Years	11.57%
1,040	20 Years	11.57%
2,600	50 Years	11.57%

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

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.01%	0	0.00%	0.00%
-13.01% to -6.66%	0	0.00%	0.00%
-6.66% to -0.30%	0	0.00%	0.00%
-0.30% to 6.05%	74	14.54%	14.54%
6.05% to 12.40%	254	49.90%	64.44%
12.40% to 18.76%	102	20.04%	84.48%
18.76% to 25.11%	49	9.63%	94.11%
25.11% to 31.46%	19	3.73%	97.84%
31.46% to 37.82%	3	0.59%	98.43%
Greater than 37.82%	8	1.57%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.93%	20.88%	86.05%	68.27%
±2σ	-4.54%	29.35%	96.66%	95.45%
±3σ	-13.01%	37.82%	98.43%	99.73%

Descriptive Statistics

Mean	12.403%
Standard Error	0.376%
Median	10.163%
Mode	4.509%
Standard Deviation	8.472%
Sample Variance	71.779%
Kurtosis	21.86
Skewness	3.45
Range	90.470%
Minimum	2.506%
Maximum	92.976%
Count	509

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.403%	509	100.00%	12.403%
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 8.60 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.73%	1	0.20%	0.20%
-26.73% to -19.99%	2	0.39%	0.59%
-19.99% to -13.24%	16	3.14%	3.73%
-13.24% to -6.49%	64	12.57%	16.31%
-6.49% to 0.26%	199	39.10%	55.40%
0.26% to 7.01%	145	28.49%	83.89%
7.01% to 13.76%	48	9.43%	93.32%
13.76% to 20.51%	24	4.72%	98.04%
20.51% to 27.26%	5	0.98%	99.02%
Greater than 27.26%	5	0.98%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.74%	9.26%	78.19%	68.27%
±2σ	-17.74%	18.26%	94.89%	95.45%
±3σ	-26.73%	27.26%	98.82%	99.73%

Descriptive Statistics

Mean	0.262%
Standard Error	0.399%
Median	-0.379%
Mode	-3.875%
Standard Deviation	8.999%
Sample Variance	80.983%
Kurtosis	8.60
Skewness	1.40
Range	106.854%
Minimum	-38.224%
Maximum	68.630%
Count	509

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.708%	242	47.54%	3.189%
Negative	-5.602%	266	52.26%	-2.928%
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 0.62. Further, an Excess Kurtosis of 2.27 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -49.21%	0	0.00%	0.00%
-49.21% to -36.28%	1	0.85%	0.85%
-36.28% to -23.36%	3	2.56%	3.42%
-23.36% to -10.44%	20	17.09%	20.51%
-10.44% to 2.49%	37	31.62%	52.14%
2.49% to 15.41%	34	29.06%	81.20%
15.41% to 28.34%	14	11.97%	93.16%
28.34% to 41.26%	7	5.98%	99.15%
41.26% to 54.19%	0	0.00%	99.15%
Greater than 54.19%	1	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.74%	19.72%	67.52%	68.27%
±2σ	-31.97%	36.95%	95.73%	95.45%
±3σ	-49.21%	54.18%	99.15%	99.73%

Descriptive Statistics

Mean	2.489%
Standard Error	1.593%
Median	1.181%
Mode	-46.986%
Standard Deviation	17.232%
Sample Variance	296.958%
Kurtosis	2.27
Skewness	0.62
Range	122.943%
Minimum	-46.986%
Maximum	75.957%
Count	117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.493%	62	52.99%	7.680%
Negative	-11.043%	55	47.01%	-5.191%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	15.16%
12	1 Year	20.08%
36	3 Years	20.21%
60	5 Years	23.08%
120	10 Years	24.73%
240	20 Years	24.73%
600	50 Years	24.73%

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 -23.44%	0	0.00%	0.00%
-23.44% to -10.26%	0	0.00%	0.00%
-10.26% to 2.93%	0	0.00%	0.00%
2.93% to 16.12%	23	19.49%	19.49%
16.12% to 29.31%	52	44.07%	63.56%
29.31% to 42.50%	25	21.19%	84.75%
42.50% to 55.69%	10	8.47%	93.22%
55.69% to 68.87%	3	2.54%	95.76%
68.87% to 82.06%	2	1.69%	97.46%
Greater than 82.06%	3	2.54%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	11.73%	46.89%	82.20%	68.27%
±2σ	-5.86%	64.48%	94.92%	95.45%
±3σ	-23.44%	82.06%	97.46%	99.73%

Descriptive Statistics

Mean	29.309%
Standard Error	1.619%
Median	24.683%
Mode	9.172%
Standard Deviation	17.584%
Sample Variance	309.214%
Kurtosis	4.35
Skewness	1.89
Range	88.443%
Minimum	9.172%
Maximum	97.615%
Count	118

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	29.309%	118	100.00%	29.309%
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 1.09. Further, an Excess Kurtosis of 3.90 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -53.22%	0	0.00%	0.00%
-53.22% to -39.83%	1	0.85%	0.85%
-39.83% to -26.43%	2	1.69%	2.54%
-26.43% to -13.04%	21	17.80%	20.34%
-13.04% to 0.36%	40	33.90%	54.24%
0.36% to 13.76%	33	27.97%	82.20%
13.76% to 27.15%	15	12.71%	94.92%
27.15% to 40.55%	3	2.54%	97.46%
40.55% to 53.94%	1	0.85%	98.31%
Greater than 53.94%	2	1.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.50%	18.22%	71.19%	68.27%
±2σ	-35.36%	36.08%	95.76%	95.45%
±3σ	-53.22%	53.94%	98.31%	99.73%

Descriptive Statistics

Mean	0.360%
Standard Error	1.644%
Median	-1.113%
Mode	-44.641%
Standard Deviation	17.861%
Sample Variance	318.999%
Kurtosis	3.90
Skewness	1.09
Range	127.818%
Minimum	-44.641%
Maximum	83.177%
Count	118

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.208%	55	46.61%	6.622%
Negative	-11.729%	63	53.39%	-6.262%
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 1.69. 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 -100.77%	0	0.00%	0.00%
-100.77% to -73.60%	0	0.00%	0.00%
-73.60% to -46.43%	0	0.00%	0.00%
-46.43% to -19.27%	8	20.51%	20.51%
-19.27% to 7.90%	16	41.03%	61.54%
7.90% to 35.07%	8	20.51%	82.05%
35.07% to 62.24%	5	12.82%	94.87%
62.24% to 89.40%	1	2.56%	97.44%
89.40% to 116.57%	0	0.00%	97.44%
Greater than 116.57%	1	2.56%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.32%	44.12%	79.49%	68.27%
±2σ	-64.54%	80.35%	94.87%	95.45%
±3σ	-100.77%	116.57%	97.44%	99.73%

Descriptive Statistics

Mean	7.902%
Standard Error	5.800%
Median	1.436%
Mode	-45.750%
Standard Deviation	36.223%
Sample Variance	1312.105%
Kurtosis	4.61
Skewness	1.69
Range	192.785%
Minimum	-45.750%
Maximum	147.035%
Count	39

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.328%	21	53.85%	16.869%
Negative	-19.428%	18	46.15%	-8.967%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	29.91%
12	3 Years	40.00%
20	5 Years	41.75%
40	10 Years	46.15%
80	20 Years	46.15%
200	50 Years	46.15%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.36. Further, an Excess Kurtosis of 2.47 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -28.93%	0	0.00%	0.00%
-28.93% to -6.95%	0	0.00%	0.00%
-6.95% to 15.02%	0	0.00%	0.00%
15.02% to 36.99%	10	25.00%	25.00%
36.99% to 58.96%	18	45.00%	70.00%
58.96% to 80.93%	2	5.00%	75.00%
80.93% to 102.91%	7	17.50%	92.50%
102.91% to 124.88%	2	5.00%	97.50%
124.88% to 146.85%	0	0.00%	97.50%
Greater than 146.85%	1	2.50%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	29.67%	88.26%	67.50%	68.27%
±2σ	0.37%	117.55%	97.50%	95.45%
±3σ	-28.93%	146.85%	97.50%	99.73%

Descriptive Statistics

Mean	58.962%
Standard Error	4.632%
Median	55.043%
Mode	21.710%
Standard Deviation	29.296%
Sample Variance	858.255%
Kurtosis	2.47
Skewness	1.36
Range	139.423%
Minimum	21.710%
Maximum	161.133%
Count	40

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	58.962%	40	100.00%	58.962%
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 0.39. 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 -88.57%	0	0.00%	0.00%
-88.57% to -66.25%	0	0.00%	0.00%
-66.25% to -43.94%	2	5.00%	5.00%
-43.94% to -21.62%	10	25.00%	30.00%
-21.62% to 0.70%	8	20.00%	50.00%
0.70% to 23.01%	12	30.00%	80.00%
23.01% to 45.33%	6	15.00%	95.00%
45.33% to 67.65%	1	2.50%	97.50%
67.65% to 89.96%	1	2.50%	100.00%
Greater than 89.96%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-29.06%	30.45%	75.00%	68.27%
±2σ	-58.81%	60.21%	95.00%	95.45%
±3σ	-88.57%	89.96%	100.00%	99.73%

Descriptive Statistics

Mean	0.697%
Standard Error	4.705%
Median	0.181%
Mode	-60.569%
Standard Deviation	29.755%
Sample Variance	885.376%
Kurtosis	0.42
Skewness	0.39
Range	146.788%
Minimum	-60.569%
Maximum	86.219%
Count	40

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.451%	20	50.00%	12.226%
Negative	-23.056%	20	50.00%	-11.528%
Zero	0.000%	0	0.00%	0.000%

TWLO Deep Analytical Report