UAL - 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	4.01%	6.00%	3.52%	4.01%	3.22%	3.26%	109.45	116.73	105.97	120.56
Weekly	8.74%	15.76%	8.44%	9.67%	7.09%	7.18%	105.29	121.34	98.08	130.26
Monthly	16.18%	39.44%	17.62%	21.30%	14.77%	14.94%	97.51	131.02	84.12	151.87
Quarterly	28.11%	82.53%	35.21%	38.16%	25.58%	25.88%	87.52	145.98	67.76	188.53
Annual	—	—	—	—	51.16%	51.76%	67.76	188.53	40.63	314.47
Option Time Horizon	—	—	—	—	16.72%	16.92%	95.62	133.61	80.90	157.93

Calculation Notes: Computed at 2026-02-22T14:24 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $113.03 | Strike (K): $114.00 (first OTM call ≥ spot) | Observed Mid-Price: $7.2750 (Bid $6.6000 / Ask $7.9500) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 51.16% (CRR binomial tree, American, n=20 steps) | BS IV: 51.04% (European Black-Scholes) | Yahoo IV (Annual): 51.76% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Historical Dividend Payouts

Ex-Dividend Date	Amount
2008-01-07	$2.1500

Close-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 28.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.92%	39	0.77%	0.77%
-11.92% to -8.91%	47	0.93%	1.71%
-8.91% to -5.91%	107	2.12%	3.83%
-5.91% to -2.90%	480	9.52%	13.35%
-2.90% to 0.10%	1916	38.01%	51.36%
0.10% to 3.11%	1789	35.49%	86.85%
3.11% to 6.12%	455	9.03%	95.87%
6.12% to 9.12%	121	2.40%	98.27%
9.12% to 12.13%	32	0.63%	98.91%
Greater than 12.13%	55	1.09%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.90%	4.11%	83.02%	68.27%
±2σ	-7.91%	8.12%	95.26%	95.45%
±3σ	-11.92%	12.13%	98.14%	99.73%

Descriptive Statistics

Mean	0.104%
Standard Error	0.056%
Median	0.028%
Mode	0.000%
Standard Deviation	4.008%
Sample Variance	16.065%
Kurtosis	28.85
Skewness	1.29
Range	105.308%
Minimum	-36.771%
Maximum	68.537%
Count	5041

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.577%	2536	50.31%	1.296%
Negative	-2.426%	2477	49.14%	-1.192%
Zero	0.000%	28	0.56%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.79%
20	1 Month	4.85%
60	1 Quarter	3.77%
250	1 Year	4.63%
750	3 Years	3.83%
1,250	5 Years	4.01%
2,500	10 Years	3.86%
5,000	20 Years	4.67%
12,500	50 Years	4.67%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 29.09. Further, an Excess Kurtosis of 1428.94 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.43%	0	0.00%	0.00%
-13.43% to -8.92%	0	0.00%	0.00%
-8.92% to -4.42%	0	0.00%	0.00%
-4.42% to 0.08%	0	0.00%	0.00%
0.08% to 4.58%	3527	69.95%	69.95%
4.58% to 9.08%	1122	22.25%	92.21%
9.08% to 13.58%	228	4.52%	96.73%
13.58% to 18.08%	83	1.65%	98.37%
18.08% to 22.59%	39	0.77%	99.15%
Greater than 22.59%	43	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.42%	10.58%	94.31%	68.27%
±2σ	-7.42%	16.58%	97.98%	95.45%
±3σ	-13.43%	22.59%	99.15%	99.73%

Descriptive Statistics

Mean	4.580%
Standard Error	0.085%
Median	3.432%
Mode	2.434%
Standard Deviation	6.002%
Sample Variance	36.022%
Kurtosis	1428.94
Skewness	29.09
Range	314.449%
Minimum	0.551%
Maximum	315.000%
Count	5042

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.580%	5042	100.00%	4.580%
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 27.32 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.43%	33	0.65%	0.65%
-10.43% to -7.79%	27	0.54%	1.19%
-7.79% to -5.15%	125	2.48%	3.67%
-5.15% to -2.52%	515	10.21%	13.88%
-2.52% to 0.12%	1918	38.04%	51.92%
0.12% to 2.76%	1776	35.22%	87.15%
2.76% to 5.39%	443	8.79%	95.93%
5.39% to 8.03%	110	2.18%	98.12%
8.03% to 10.67%	47	0.93%	99.05%
Greater than 10.67%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.40%	3.64%	82.61%	68.27%
±2σ	-6.91%	7.15%	95.76%	95.45%
±3σ	-10.43%	10.67%	98.39%	99.73%

Descriptive Statistics

Mean	0.120%
Standard Error	0.050%
Median	0.024%
Mode	0.000%
Standard Deviation	3.516%
Sample Variance	12.360%
Kurtosis	27.32
Skewness	1.61
Range	89.682%
Minimum	-36.029%
Maximum	53.653%
Count	5042

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.298%	2534	50.26%	1.155%
Negative	-2.100%	2484	49.27%	-1.035%
Zero	0.000%	24	0.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.54. Further, an Excess Kurtosis of 13.52 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.73%	6	0.57%	0.57%
-25.73% to -19.18%	14	1.34%	1.91%
-19.18% to -12.62%	29	2.77%	4.68%
-12.62% to -6.07%	92	8.80%	13.48%
-6.07% to 0.48%	398	38.05%	51.53%
0.48% to 7.04%	364	34.80%	86.33%
7.04% to 13.59%	99	9.46%	95.79%
13.59% to 20.14%	25	2.39%	98.18%
20.14% to 26.70%	6	0.57%	98.76%
Greater than 26.70%	13	1.24%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.25%	9.22%	82.12%	68.27%
±2σ	-16.99%	17.96%	95.22%	95.45%
±3σ	-25.73%	26.70%	98.18%	99.73%

Descriptive Statistics

Mean	0.484%
Standard Error	0.270%
Median	0.230%
Mode	0.000%
Standard Deviation	8.738%
Sample Variance	76.352%
Kurtosis	13.52
Skewness	1.54
Range	120.452%
Minimum	-39.452%
Maximum	81.000%
Count	1046

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.955%	535	51.15%	3.046%
Negative	-5.274%	508	48.57%	-2.561%
Zero	0.000%	3	0.29%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.57%
12	1 Quarter	7.44%
52	1 Year	10.50%
156	3 Years	9.15%
260	5 Years	9.67%
520	10 Years	9.20%
1,040	20 Years	11.02%
2,600	50 Years	11.01%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 12.19. Further, an Excess Kurtosis of 243.74 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -35.04%	0	0.00%	0.00%
-35.04% to -23.22%	0	0.00%	0.00%
-23.22% to -11.40%	0	0.00%	0.00%
-11.40% to 0.41%	0	0.00%	0.00%
0.41% to 12.23%	742	70.87%	70.87%
12.23% to 24.05%	227	21.68%	92.55%
24.05% to 35.86%	37	3.53%	96.08%
35.86% to 47.68%	17	1.62%	97.71%
47.68% to 59.50%	10	0.96%	98.66%
Greater than 59.50%	14	1.34%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.53%	27.98%	93.89%	68.27%
±2σ	-19.28%	43.74%	97.42%	95.45%
±3σ	-35.04%	59.50%	98.66%	99.73%

Descriptive Statistics

Mean	12.229%
Standard Error	0.487%
Median	8.760%
Mode	3.732%
Standard Deviation	15.756%
Sample Variance	248.259%
Kurtosis	243.74
Skewness	12.19
Range	362.352%
Minimum	1.981%
Maximum	364.333%
Count	1047

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.229%	1047	100.00%	12.229%
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 1.30. Further, an Excess Kurtosis of 9.09 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.85%	6	0.57%	0.57%
-24.85% to -18.52%	7	0.67%	1.24%
-18.52% to -12.19%	22	2.10%	3.34%
-12.19% to -5.85%	116	11.08%	14.42%
-5.85% to 0.48%	417	39.83%	54.25%
0.48% to 6.81%	342	32.66%	86.91%
6.81% to 13.14%	79	7.55%	94.46%
13.14% to 19.47%	28	2.67%	97.13%
19.47% to 25.80%	15	1.43%	98.57%
Greater than 25.80%	15	1.43%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.96%	8.92%	82.81%	68.27%
±2σ	-16.41%	17.36%	94.84%	95.45%
±3σ	-24.85%	25.80%	97.99%	99.73%

Descriptive Statistics

Mean	0.478%
Standard Error	0.261%
Median	-0.042%
Mode	3.311%
Standard Deviation	8.442%
Sample Variance	71.265%
Kurtosis	9.09
Skewness	1.30
Range	104.112%
Minimum	-40.016%
Maximum	64.096%
Count	1047

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.967%	519	49.57%	2.958%
Negative	-4.927%	527	50.33%	-2.480%
Zero	0.000%	1	0.10%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.60. Further, an Excess Kurtosis of 3.00 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -46.75%	2	0.83%	0.83%
-46.75% to -34.62%	2	0.83%	1.67%
-34.62% to -22.48%	8	3.33%	5.00%
-22.48% to -10.34%	27	11.25%	16.25%
-10.34% to 1.79%	83	34.58%	50.83%
1.79% to 13.93%	81	33.75%	84.58%
13.93% to 26.07%	22	9.17%	93.75%
26.07% to 38.20%	9	3.75%	97.50%
38.20% to 50.34%	2	0.83%	98.33%
Greater than 50.34%	4	1.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.39%	17.98%	78.33%	68.27%
±2σ	-30.57%	34.16%	94.17%	95.45%
±3σ	-46.75%	50.34%	97.50%	99.73%

Descriptive Statistics

Mean	1.794%
Standard Error	1.045%
Median	0.764%
Mode	2.749%
Standard Deviation	16.182%
Sample Variance	261.850%
Kurtosis	3.00
Skewness	0.60
Range	115.140%
Minimum	-48.774%
Maximum	66.366%
Count	240

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.504%	127	52.92%	6.617%
Negative	-10.243%	113	47.08%	-4.823%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	16.23%
12	1 Year	20.83%
36	3 Years	20.36%
60	5 Years	21.30%
120	10 Years	19.98%
240	20 Years	24.06%
600	50 Years	24.06%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.53. Further, an Excess Kurtosis of 37.19 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -87.51%	0	0.00%	0.00%
-87.51% to -57.93%	0	0.00%	0.00%
-57.93% to -28.35%	0	0.00%	0.00%
-28.35% to 1.23%	0	0.00%	0.00%
1.23% to 30.81%	178	73.86%	73.86%
30.81% to 60.39%	45	18.67%	92.53%
60.39% to 89.96%	8	3.32%	95.85%
89.96% to 119.54%	4	1.66%	97.51%
119.54% to 149.12%	2	0.83%	98.34%
Greater than 149.12%	4	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.63%	70.25%	93.78%	68.27%
±2σ	-48.07%	109.68%	97.51%	95.45%
±3σ	-87.51%	149.12%	98.34%	99.73%

Descriptive Statistics

Mean	30.807%
Standard Error	2.540%
Median	20.721%
Mode	4.836%
Standard Deviation	39.438%
Sample Variance	1555.362%
Kurtosis	37.19
Skewness	5.53
Range	371.497%
Minimum	4.836%
Maximum	376.333%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.807%	241	100.00%	30.807%
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 1.83. Further, an Excess Kurtosis of 7.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -51.79%	0	0.00%	0.00%
-51.79% to -38.58%	3	1.24%	1.24%
-38.58% to -25.36%	4	1.66%	2.90%
-25.36% to -12.14%	30	12.45%	15.35%
-12.14% to 1.07%	102	42.32%	57.68%
1.07% to 14.29%	69	28.63%	86.31%
14.29% to 27.50%	20	8.30%	94.61%
27.50% to 40.72%	6	2.49%	97.10%
40.72% to 53.93%	3	1.24%	98.34%
Greater than 53.93%	4	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.55%	18.69%	81.74%	68.27%
±2σ	-34.17%	36.31%	95.02%	95.45%
±3σ	-51.79%	53.93%	98.34%	99.73%

Descriptive Statistics

Mean	1.070%
Standard Error	1.135%
Median	-1.239%
Mode	7.527%
Standard Deviation	17.620%
Sample Variance	310.472%
Kurtosis	7.39
Skewness	1.83
Range	133.467%
Minimum	-39.711%
Maximum	93.756%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.932%	109	45.23%	6.301%
Negative	-9.552%	132	54.77%	-5.232%
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 1.15 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -78.88%	0	0.00%	0.00%
-78.88% to -57.80%	2	2.50%	2.50%
-57.80% to -36.72%	2	2.50%	5.00%
-36.72% to -15.63%	10	12.50%	17.50%
-15.63% to 5.45%	25	31.25%	48.75%
5.45% to 26.53%	28	35.00%	83.75%
26.53% to 47.62%	8	10.00%	93.75%
47.62% to 68.70%	1	1.25%	95.00%
68.70% to 89.78%	4	5.00%	100.00%
Greater than 89.78%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-22.66%	33.56%	75.00%	68.27%
±2σ	-50.77%	61.67%	92.50%	95.45%
±3σ	-78.88%	89.78%	100.00%	99.73%

Descriptive Statistics

Mean	5.449%
Standard Error	3.143%
Median	6.220%
Mode	-60.738%
Standard Deviation	28.111%
Sample Variance	790.247%
Kurtosis	1.15
Skewness	0.39
Range	148.603%
Minimum	-60.738%
Maximum	87.865%
Count	80

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.162%	46	57.50%	13.318%
Negative	-18.516%	34	42.50%	-7.869%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	29.62%
12	3 Years	40.43%
20	5 Years	38.16%
40	10 Years	36.23%
80	20 Years	42.62%
200	50 Years	42.62%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.64. Further, an Excess Kurtosis of 14.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -181.59%	0	0.00%	0.00%
-181.59% to -119.68%	0	0.00%	0.00%
-119.68% to -57.78%	0	0.00%	0.00%
-57.78% to 4.12%	0	0.00%	0.00%
4.12% to 66.02%	65	80.25%	80.25%
66.02% to 127.92%	8	9.88%	90.12%
127.92% to 189.82%	4	4.94%	95.06%
189.82% to 251.72%	1	1.23%	96.30%
251.72% to 313.62%	0	0.00%	96.30%
Greater than 313.62%	3	3.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.52%	148.56%	92.59%	68.27%
±2σ	-99.05%	231.09%	96.30%	95.45%
±3σ	-181.58%	313.62%	96.30%	99.73%

Descriptive Statistics

Mean	66.020%
Standard Error	9.171%
Median	39.568%
Mode	14.376%
Standard Deviation	82.535%
Sample Variance	6812.035%
Kurtosis	14.38
Skewness	3.64
Range	473.481%
Minimum	14.376%
Maximum	487.857%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	66.020%	81	100.00%	66.020%
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 2.36. Further, an Excess Kurtosis of 7.66 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -102.34%	0	0.00%	0.00%
-102.34% to -75.93%	0	0.00%	0.00%
-75.93% to -49.52%	0	0.00%	0.00%
-49.52% to -23.11%	12	14.81%	14.81%
-23.11% to 3.30%	40	49.38%	64.20%
3.30% to 29.71%	19	23.46%	87.65%
29.71% to 56.12%	5	6.17%	93.83%
56.12% to 82.53%	2	2.47%	96.30%
82.53% to 108.94%	1	1.23%	97.53%
Greater than 108.94%	2	2.47%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.91%	38.51%	85.19%	68.27%
±2σ	-67.12%	73.72%	95.06%	95.45%
±3σ	-102.34%	108.94%	97.53%	99.73%

Descriptive Statistics

Mean	3.299%
Standard Error	3.912%
Median	-4.089%
Mode	-46.361%
Standard Deviation	35.212%
Sample Variance	1239.874%
Kurtosis	7.66
Skewness	2.36
Range	202.267%
Minimum	-46.361%
Maximum	155.906%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	31.508%	33	40.74%	12.837%
Negative	-16.095%	48	59.26%	-9.538%
Zero	0.000%	0	0.00%	0.000%

UAL Deep Analytical Report