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%	5.99%	3.51%	4.02%	3.77%	3.80%	90.04	97.10	86.71	100.83
Weekly	8.73%	15.73%	8.43%	9.70%	8.30%	8.36%	86.05	101.60	79.20	110.40
Monthly	16.16%	39.36%	17.58%	21.15%	17.28%	17.41%	78.66	111.15	66.18	132.12
Quarterly	28.15%	82.38%	35.20%	39.00%	29.94%	30.15%	69.31	126.14	51.38	170.16
Annual	—	—	—	—	59.87%	60.30%	51.38	170.16	28.24	309.66
Option Time Horizon	—	—	—	—	22.60%	22.76%	74.59	117.21	59.50	146.93

Calculation Notes: Computed at 2026-03-23T16:42 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $93.50 | Strike (K): $95.00 (first OTM call ≥ spot) | Observed Mid-Price: $7.9750 (Bid $7.8500 / Ask $8.1000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 59.87% (CRR binomial tree, American, n=20 steps) | BS IV: 59.91% (European Black-Scholes) | Yahoo IV (Annual): 60.30% → 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 1.28. Further, an Excess Kurtosis of 28.76 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 -8.92%	47	0.93%	1.70%
-8.92% to -5.91%	108	2.13%	3.83%
-5.91% to -2.90%	487	9.62%	13.45%
-2.90% to 0.10%	1915	37.83%	51.28%
0.10% to 3.10%	1795	35.46%	86.74%
3.10% to 6.11%	463	9.15%	95.89%
6.11% to 9.12%	121	2.39%	98.28%
9.12% to 12.12%	32	0.63%	98.91%
Greater than 12.12%	55	1.09%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.91%	4.11%	82.93%	68.27%
±2σ	-7.91%	8.11%	95.24%	95.45%
±3σ	-11.92%	12.12%	98.14%	99.73%

Descriptive Statistics

Mean	0.100%
Standard Error	0.056%
Median	0.028%
Mode	0.000%
Standard Deviation	4.007%
Sample Variance	16.058%
Kurtosis	28.76
Skewness	1.28
Range	105.308%
Minimum	-36.771%
Maximum	68.537%
Count	5062

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.577%	2545	50.28%	1.296%
Negative	-2.431%	2489	49.17%	-1.195%
Zero	0.000%	28	0.55%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.50%
20	1 Month	6.06%
60	1 Quarter	4.73%
250	1 Year	4.62%
750	3 Years	3.88%
1,250	5 Years	4.02%
2,500	10 Years	3.88%
5,000	20 Years	4.68%
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 29.12. Further, an Excess Kurtosis of 1432.95 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.39%	0	0.00%	0.00%
-13.39% to -8.90%	0	0.00%	0.00%
-8.90% to -4.40%	0	0.00%	0.00%
-4.40% to 0.09%	0	0.00%	0.00%
0.09% to 4.58%	3542	69.96%	69.96%
4.58% to 9.08%	1127	22.26%	92.22%
9.08% to 13.57%	228	4.50%	96.72%
13.57% to 18.06%	82	1.62%	98.34%
18.06% to 22.56%	41	0.81%	99.15%
Greater than 22.56%	43	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.41%	10.57%	94.31%	68.27%
±2σ	-7.40%	16.56%	97.99%	95.45%
±3σ	-13.39%	22.56%	99.15%	99.73%

Descriptive Statistics

Mean	4.583%
Standard Error	0.084%
Median	3.437%
Mode	2.434%
Standard Deviation	5.991%
Sample Variance	35.896%
Kurtosis	1432.95
Skewness	29.12
Range	314.449%
Minimum	0.551%
Maximum	315.000%
Count	5063

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.583%	5063	100.00%	4.583%
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 1.60. Further, an Excess Kurtosis of 27.26 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.42%	33	0.65%	0.65%
-10.42% to -7.79%	27	0.53%	1.19%
-7.79% to -5.15%	127	2.51%	3.69%
-5.15% to -2.51%	519	10.25%	13.94%
-2.51% to 0.12%	1921	37.94%	51.89%
0.12% to 2.76%	1782	35.20%	87.08%
2.76% to 5.39%	448	8.85%	95.93%
5.39% to 8.03%	111	2.19%	98.12%
8.03% to 10.66%	47	0.93%	99.05%
Greater than 10.66%	48	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.39%	3.63%	82.54%	68.27%
±2σ	-6.91%	7.15%	95.75%	95.45%
±3σ	-10.42%	10.66%	98.40%	99.73%

Descriptive Statistics

Mean	0.121%
Standard Error	0.049%
Median	0.024%
Mode	0.000%
Standard Deviation	3.514%
Sample Variance	12.348%
Kurtosis	27.26
Skewness	1.60
Range	89.682%
Minimum	-36.029%
Maximum	53.653%
Count	5063

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.299%	2546	50.29%	1.156%
Negative	-2.102%	2493	49.24%	-1.035%
Zero	0.000%	24	0.47%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.55. Further, an Excess Kurtosis of 13.54 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.72%	6	0.57%	0.57%
-25.72% to -19.17%	14	1.33%	1.90%
-19.17% to -12.63%	29	2.76%	4.67%
-12.63% to -6.08%	93	8.86%	13.52%
-6.08% to 0.47%	401	38.19%	51.71%
0.47% to 7.01%	363	34.57%	86.29%
7.01% to 13.56%	100	9.52%	95.81%
13.56% to 20.10%	25	2.38%	98.19%
20.10% to 26.65%	5	0.48%	98.67%
Greater than 26.65%	14	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.26%	9.19%	82.00%	68.27%
±2σ	-16.99%	17.92%	95.24%	95.45%
±3σ	-25.72%	26.65%	98.10%	99.73%

Descriptive Statistics

Mean	0.465%
Standard Error	0.269%
Median	0.220%
Mode	0.000%
Standard Deviation	8.728%
Sample Variance	76.180%
Kurtosis	13.54
Skewness	1.55
Range	120.452%
Minimum	-39.452%
Maximum	81.000%
Count	1050

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.954%	535	50.95%	3.034%
Negative	-5.269%	512	48.76%	-2.569%
Zero	0.000%	3	0.29%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	14.83%
12	1 Quarter	10.26%
52	1 Year	10.56%
156	3 Years	9.25%
260	5 Years	9.70%
520	10 Years	9.26%
1,040	20 Years	11.05%
2,600	50 Years	11.02%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 12.20. Further, an Excess Kurtosis of 244.21 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -34.95%	0	0.00%	0.00%
-34.95% to -23.15%	0	0.00%	0.00%
-23.15% to -11.35%	0	0.00%	0.00%
-11.35% to 0.45%	0	0.00%	0.00%
0.45% to 12.25%	743	70.69%	70.69%
12.25% to 24.05%	229	21.79%	92.48%
24.05% to 35.85%	37	3.52%	96.00%
35.85% to 47.65%	18	1.71%	97.72%
47.65% to 59.45%	9	0.86%	98.57%
Greater than 59.45%	15	1.43%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.49%	27.98%	93.91%	68.27%
±2σ	-19.22%	43.71%	97.43%	95.45%
±3σ	-34.95%	59.45%	98.57%	99.73%

Descriptive Statistics

Mean	12.247%
Standard Error	0.485%
Median	8.764%
Mode	3.732%
Standard Deviation	15.733%
Sample Variance	247.519%
Kurtosis	244.21
Skewness	12.20
Range	362.352%
Minimum	1.981%
Maximum	364.333%
Count	1051

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.247%	1051	100.00%	12.247%
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 9.11 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.79%	6	0.57%	0.57%
-24.79% to -18.47%	7	0.67%	1.24%
-18.47% to -12.15%	22	2.09%	3.33%
-12.15% to -5.83%	118	11.23%	14.56%
-5.83% to 0.49%	415	39.49%	54.04%
0.49% to 6.82%	346	32.92%	86.96%
6.82% to 13.14%	79	7.52%	94.48%
13.14% to 19.46%	28	2.66%	97.15%
19.46% to 25.78%	15	1.43%	98.57%
Greater than 25.78%	15	1.43%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.93%	8.92%	82.68%	68.27%
±2σ	-16.36%	17.35%	94.86%	95.45%
±3σ	-24.79%	25.78%	98.00%	99.73%

Descriptive Statistics

Mean	0.494%
Standard Error	0.260%
Median	-0.035%
Mode	3.311%
Standard Deviation	8.428%
Sample Variance	71.025%
Kurtosis	9.11
Skewness	1.30
Range	104.112%
Minimum	-40.016%
Maximum	64.096%
Count	1051

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.940%	524	49.86%	2.962%
Negative	-4.931%	526	50.05%	-2.468%
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 0.61. Further, an Excess Kurtosis of 3.02 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -46.79%	2	0.83%	0.83%
-46.79% to -34.66%	2	0.83%	1.66%
-34.66% to -22.54%	8	3.32%	4.98%
-22.54% to -10.42%	27	11.20%	16.18%
-10.42% to 1.71%	84	34.85%	51.04%
1.71% to 13.83%	81	33.61%	84.65%
13.83% to 25.95%	22	9.13%	93.78%
25.95% to 38.08%	9	3.73%	97.51%
38.08% to 50.20%	2	0.83%	98.34%
Greater than 50.20%	4	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.46%	17.87%	78.42%	68.27%
±2σ	-30.62%	34.04%	94.19%	95.45%
±3σ	-46.79%	50.20%	97.51%	99.73%

Descriptive Statistics

Mean	1.707%
Standard Error	1.041%
Median	0.623%
Mode	2.749%
Standard Deviation	16.165%
Sample Variance	261.304%
Kurtosis	3.02
Skewness	0.61
Range	115.140%
Minimum	-48.774%
Maximum	66.366%
Count	241

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.452%	127	52.70%	6.562%
Negative	-10.263%	114	47.30%	-4.855%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	18.01%
12	1 Year	19.76%
36	3 Years	20.17%
60	5 Years	21.15%
120	10 Years	20.07%
240	20 Years	24.05%
600	50 Years	24.05%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.54. Further, an Excess Kurtosis of 37.36 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -87.29%	0	0.00%	0.00%
-87.29% to -57.77%	0	0.00%	0.00%
-57.77% to -28.25%	0	0.00%	0.00%
-28.25% to 1.26%	0	0.00%	0.00%
1.26% to 30.78%	179	73.97%	73.97%
30.78% to 60.30%	45	18.60%	92.56%
60.30% to 89.82%	8	3.31%	95.87%
89.82% to 119.34%	4	1.65%	97.52%
119.34% to 148.86%	2	0.83%	98.35%
Greater than 148.86%	4	1.65%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.58%	70.14%	93.80%	68.27%
±2σ	-47.93%	109.50%	97.52%	95.45%
±3σ	-87.29%	148.86%	98.35%	99.73%

Descriptive Statistics

Mean	30.782%
Standard Error	2.530%
Median	20.727%
Mode	4.836%
Standard Deviation	39.358%
Sample Variance	1549.056%
Kurtosis	37.36
Skewness	5.54
Range	371.497%
Minimum	4.836%
Maximum	376.333%
Count	242

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.782%	242	100.00%	30.782%
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 7.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -51.62%	0	0.00%	0.00%
-51.62% to -38.44%	3	1.24%	1.24%
-38.44% to -25.25%	4	1.65%	2.89%
-25.25% to -12.06%	31	12.81%	15.70%
-12.06% to 1.12%	101	41.74%	57.44%
1.12% to 14.30%	70	28.93%	86.36%
14.30% to 27.49%	20	8.26%	94.63%
27.49% to 40.67%	6	2.48%	97.11%
40.67% to 53.86%	3	1.24%	98.35%
Greater than 53.86%	4	1.65%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.46%	18.70%	81.82%	68.27%
±2σ	-34.04%	36.28%	95.04%	95.45%
±3σ	-51.62%	53.86%	98.35%	99.73%

Descriptive Statistics

Mean	1.120%
Standard Error	1.130%
Median	-1.059%
Mode	7.527%
Standard Deviation	17.580%
Sample Variance	309.063%
Kurtosis	7.42
Skewness	1.83
Range	133.467%
Minimum	-39.711%
Maximum	93.756%
Count	242

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.874%	110	45.45%	6.306%
Negative	-9.508%	132	54.55%	-5.186%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.42. 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 -79.25%	0	0.00%	0.00%
-79.25% to -58.14%	2	2.50%	2.50%
-58.14% to -37.02%	2	2.50%	5.00%
-37.02% to -15.91%	10	12.50%	17.50%
-15.91% to 5.21%	26	32.50%	50.00%
5.21% to 26.32%	27	33.75%	83.75%
26.32% to 47.43%	8	10.00%	93.75%
47.43% to 68.55%	1	1.25%	95.00%
68.55% to 89.66%	4	5.00%	100.00%
Greater than 89.66%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-22.95%	33.36%	76.25%	68.27%
±2σ	-51.10%	61.51%	92.50%	95.45%
±3σ	-79.25%	89.66%	100.00%	99.73%

Descriptive Statistics

Mean	5.205%
Standard Error	3.147%
Median	4.758%
Mode	-60.738%
Standard Deviation	28.152%
Sample Variance	792.533%
Kurtosis	1.15
Skewness	0.42
Range	148.603%
Minimum	-60.738%
Maximum	87.865%
Count	80

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.444%	45	56.25%	13.187%
Negative	-18.246%	35	43.75%	-7.983%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	33.85%
12	3 Years	41.84%
20	5 Years	39.00%
40	10 Years	36.66%
80	20 Years	42.83%
200	50 Years	42.83%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.65. Further, an Excess Kurtosis of 14.45 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -180.81%	0	0.00%	0.00%
-180.81% to -119.02%	0	0.00%	0.00%
-119.02% to -57.23%	0	0.00%	0.00%
-57.23% to 4.55%	0	0.00%	0.00%
4.55% to 66.34%	65	80.25%	80.25%
66.34% to 128.13%	8	9.88%	90.12%
128.13% to 189.91%	4	4.94%	95.06%
189.91% to 251.70%	1	1.23%	96.30%
251.70% to 313.49%	0	0.00%	96.30%
Greater than 313.49%	3	3.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.04%	148.72%	92.59%	68.27%
±2σ	-98.42%	231.11%	96.30%	95.45%
±3σ	-180.81%	313.49%	96.30%	99.73%

Descriptive Statistics

Mean	66.341%
Standard Error	9.154%
Median	39.798%
Mode	15.177%
Standard Deviation	82.382%
Sample Variance	6786.780%
Kurtosis	14.45
Skewness	3.65
Range	472.680%
Minimum	15.177%
Maximum	487.857%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	66.341%	81	100.00%	66.341%
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 2.34. Further, an Excess Kurtosis of 7.60 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -102.06%	0	0.00%	0.00%
-102.06% to -75.66%	0	0.00%	0.00%
-75.66% to -49.26%	0	0.00%	0.00%
-49.26% to -22.86%	12	14.81%	14.81%
-22.86% to 3.54%	39	48.15%	62.96%
3.54% to 29.94%	20	24.69%	87.65%
29.94% to 56.35%	5	6.17%	93.83%
56.35% to 82.75%	2	2.47%	96.30%
82.75% to 109.15%	1	1.23%	97.53%
Greater than 109.15%	2	2.47%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.66%	38.74%	85.19%	68.27%
±2σ	-66.86%	73.95%	95.06%	95.45%
±3σ	-102.06%	109.15%	97.53%	99.73%

Descriptive Statistics

Mean	3.543%
Standard Error	3.911%
Median	-3.305%
Mode	-46.361%
Standard Deviation	35.202%
Sample Variance	1239.213%
Kurtosis	7.60
Skewness	2.34
Range	202.267%
Minimum	-46.361%
Maximum	155.906%
Count	81

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.934%	34	41.98%	12.985%
Negative	-16.271%	47	58.02%	-9.441%
Zero	0.000%	0	0.00%	0.000%

UAL Deep Analytical Report