PYPL - 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	2.42%	1.58%	1.92%	3.33%	3.07%	3.20%	43.90	46.68	42.57	48.13
Weekly	5.06%	4.46%	5.15%	8.03%	6.76%	7.05%	42.31	48.43	39.54	51.82
Monthly	9.68%	10.09%	10.41%	17.23%	14.07%	14.67%	39.32	52.10	34.16	59.98
Quarterly	18.50%	17.29%	21.86%	28.94%	24.37%	25.40%	35.47	57.76	27.80	73.70
Annual	—	—	—	—	48.74%	50.81%	27.80	73.70	17.08	119.99
Option Time Horizon	—	—	—	—	18.40%	19.18%	37.66	54.41	31.33	65.40

Calculation Notes: Computed at 2026-03-23T16:43 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $45.26 | Strike (K): $47.50 (first OTM call ≥ spot) | Observed Mid-Price: $2.5100 (Bid $2.4200 / Ask $2.6000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.64% | Binomial IV (Annual): 48.74% (CRR binomial tree, American, n=20 steps) | BS IV: 49.31% (European Black-Scholes) | Yahoo IV (Annual): 50.81% → 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
2026-03-04	$0.1400
2025-11-19	$0.1400

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Left Skewed (Negative Asymmetry) with a Skewness factor of -0.56. Further, an Excess Kurtosis of 9.78 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -7.23%	17	0.63%	0.63%
-7.23% to -5.41%	28	1.04%	1.67%
-5.41% to -3.60%	91	3.38%	5.05%
-3.60% to -1.78%	313	11.62%	16.67%
-1.78% to 0.04%	859	31.89%	48.55%
0.04% to 1.85%	948	35.19%	83.74%
1.85% to 3.67%	310	11.51%	95.25%
3.67% to 5.49%	83	3.08%	98.33%
5.49% to 7.30%	26	0.97%	99.29%
Greater than 7.30%	19	0.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.39%	2.46%	77.58%	68.27%
±2σ	-4.81%	4.88%	95.29%	95.45%
±3σ	-7.23%	7.30%	98.66%	99.73%

Descriptive Statistics

Mean	0.037%
Standard Error	0.047%
Median	0.096%
Mode	0.000%
Standard Deviation	2.422%
Sample Variance	5.864%
Kurtosis	9.78
Skewness	-0.56
Range	38.700%
Minimum	-24.590%
Maximum	14.110%
Count	2694

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.644%	1401	52.00%	0.855%
Negative	-1.714%	1285	47.70%	-0.818%
Zero	0.000%	8	0.30%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.29%
20	1 Month	3.88%
60	1 Quarter	3.58%
250	1 Year	3.18%
750	3 Years	2.98%
1,250	5 Years	3.33%
2,500	10 Years	3.02%
5,000	20 Years	3.05%
12,500	50 Years	3.05%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.10. Further, an Excess Kurtosis of 7.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -1.92%	0	0.00%	0.00%
-1.92% to -0.73%	0	0.00%	0.00%
-0.73% to 0.46%	0	0.00%	0.00%
0.46% to 1.65%	556	20.63%	20.63%
1.65% to 2.83%	1091	40.48%	61.11%
2.83% to 4.02%	587	21.78%	82.89%
4.02% to 5.21%	266	9.87%	92.76%
5.21% to 6.40%	107	3.97%	96.73%
6.40% to 7.58%	46	1.71%	98.44%
Greater than 7.58%	42	1.56%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.25%	4.42%	79.96%	68.27%
±2σ	-0.33%	6.00%	95.88%	95.45%
±3σ	-1.91%	7.58%	98.44%	99.73%

Descriptive Statistics

Mean	2.834%
Standard Error	0.030%
Median	2.450%
Mode	2.039%
Standard Deviation	1.583%
Sample Variance	2.505%
Kurtosis	7.75
Skewness	2.10
Range	15.512%
Minimum	0.655%
Maximum	16.167%
Count	2695

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.834%	2695	100.00%	2.834%
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 0.11. Further, an Excess Kurtosis of 2.38 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.70%	13	0.48%	0.48%
-5.70% to -4.26%	25	0.93%	1.41%
-4.26% to -2.82%	109	4.04%	5.45%
-2.82% to -1.39%	356	13.21%	18.66%
-1.39% to 0.05%	886	32.88%	51.54%
0.05% to 1.48%	808	29.98%	81.52%
1.48% to 2.92%	325	12.06%	93.58%
2.92% to 4.36%	106	3.93%	97.51%
4.36% to 5.79%	49	1.82%	99.33%
Greater than 5.79%	18	0.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.87%	1.96%	74.43%	68.27%
±2σ	-3.78%	3.88%	94.36%	95.45%
±3σ	-5.70%	5.79%	98.85%	99.73%

Descriptive Statistics

Mean	0.048%
Standard Error	0.037%
Median	-0.023%
Mode	-0.785%
Standard Deviation	1.915%
Sample Variance	3.666%
Kurtosis	2.38
Skewness	0.11
Range	21.721%
Minimum	-11.384%
Maximum	10.337%
Count	2695

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.483%	1329	49.31%	0.731%
Negative	-1.352%	1362	50.54%	-0.683%
Zero	0.000%	4	0.15%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Left Skewed (Negative Asymmetry) with a Skewness factor of -0.65. Further, an Excess Kurtosis of 3.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.02%	7	1.25%	1.25%
-15.02% to -11.22%	8	1.43%	2.68%
-11.22% to -7.43%	23	4.11%	6.80%
-7.43% to -3.63%	48	8.59%	15.38%
-3.63% to 0.17%	184	32.92%	48.30%
0.17% to 3.97%	185	33.09%	81.40%
3.97% to 7.76%	78	13.95%	95.35%
7.76% to 11.56%	20	3.58%	98.93%
11.56% to 15.36%	2	0.36%	99.28%
Greater than 15.36%	4	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.90%	5.23%	77.28%	68.27%
±2σ	-9.96%	10.30%	94.63%	95.45%
±3σ	-15.02%	15.36%	98.03%	99.73%

Descriptive Statistics

Mean	0.169%
Standard Error	0.214%
Median	0.321%
Mode	-1.619%
Standard Deviation	5.064%
Sample Variance	25.646%
Kurtosis	3.75
Skewness	-0.65
Range	47.742%
Minimum	-29.388%
Maximum	18.354%
Count	559

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.413%	307	54.92%	1.874%
Negative	-3.798%	251	44.90%	-1.705%
Zero	0.000%	1	0.18%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	8.04%
12	1 Quarter	9.31%
52	1 Year	7.68%
156	3 Years	7.16%
260	5 Years	8.03%
520	10 Years	7.11%
1,040	20 Years	7.20%
2,600	50 Years	7.20%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.53. Further, an Excess Kurtosis of 12.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.96%	0	0.00%	0.00%
-5.96% to -2.61%	0	0.00%	0.00%
-2.61% to 0.73%	0	0.00%	0.00%
0.73% to 4.08%	106	18.93%	18.93%
4.08% to 7.42%	250	44.64%	63.57%
7.42% to 10.77%	115	20.54%	84.11%
10.77% to 14.12%	51	9.11%	93.21%
14.12% to 17.46%	17	3.04%	96.25%
17.46% to 20.81%	10	1.79%	98.04%
Greater than 20.81%	11	1.96%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.96%	11.89%	82.86%	68.27%
±2σ	-1.50%	16.35%	95.36%	95.45%
±3σ	-5.96%	20.81%	98.04%	99.73%

Descriptive Statistics

Mean	7.425%
Standard Error	0.189%
Median	6.337%
Mode	3.361%
Standard Deviation	4.461%
Sample Variance	19.903%
Kurtosis	12.42
Skewness	2.53
Range	44.269%
Minimum	1.955%
Maximum	46.224%
Count	560

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.425%	560	100.00%	7.425%
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 1.62. Further, an Excess Kurtosis of 9.14 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.30%	0	0.00%	0.00%
-15.30% to -11.44%	5	0.89%	0.89%
-11.44% to -7.58%	16	2.86%	3.75%
-7.58% to -3.71%	79	14.11%	17.86%
-3.71% to 0.15%	221	39.46%	57.32%
0.15% to 4.01%	150	26.79%	84.11%
4.01% to 7.88%	51	9.11%	93.21%
7.88% to 11.74%	23	4.11%	97.32%
11.74% to 15.60%	11	1.96%	99.29%
Greater than 15.60%	4	0.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.00%	5.30%	77.32%	68.27%
±2σ	-10.15%	10.45%	94.46%	95.45%
±3σ	-15.30%	15.60%	99.29%	99.73%

Descriptive Statistics

Mean	0.150%
Standard Error	0.218%
Median	-0.410%
Mode	-2.724%
Standard Deviation	5.150%
Sample Variance	26.522%
Kurtosis	9.14
Skewness	1.62
Range	57.379%
Minimum	-15.076%
Maximum	42.303%
Count	560

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.138%	250	44.64%	1.847%
Negative	-3.066%	310	55.36%	-1.697%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.25. Further, an Excess Kurtosis of 1.35 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -28.44%	1	0.78%	0.78%
-28.44% to -21.18%	1	0.78%	1.56%
-21.18% to -13.92%	5	3.91%	5.47%
-13.92% to -6.66%	19	14.84%	20.31%
-6.66% to 0.60%	37	28.91%	49.22%
0.60% to 7.86%	36	28.12%	77.34%
7.86% to 15.12%	24	18.75%	96.09%
15.12% to 22.38%	2	1.56%	97.66%
22.38% to 29.64%	3	2.34%	100.00%
Greater than 29.64%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.08%	10.28%	70.31%	68.27%
±2σ	-18.76%	19.96%	94.53%	95.45%
±3σ	-28.44%	29.64%	99.22%	99.73%

Descriptive Statistics

Mean	0.601%
Standard Error	0.856%
Median	0.845%
Mode	-34.902%
Standard Deviation	9.680%
Sample Variance	93.704%
Kurtosis	1.35
Skewness	-0.25
Range	63.375%
Minimum	-34.902%
Maximum	28.473%
Count	128

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.291%	70	54.69%	3.987%
Negative	-7.474%	58	45.31%	-3.387%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	17.51%
12	1 Year	14.44%
36	3 Years	15.40%
60	5 Years	17.23%
120	10 Years	15.43%
240	20 Years	15.49%
600	50 Years	15.49%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.13. Further, an Excess Kurtosis of 16.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.88%	0	0.00%	0.00%
-12.88% to -5.32%	0	0.00%	0.00%
-5.32% to 2.24%	0	0.00%	0.00%
2.24% to 9.81%	18	13.95%	13.95%
9.81% to 17.37%	61	47.29%	61.24%
17.37% to 24.94%	30	23.26%	84.50%
24.94% to 32.50%	12	9.30%	93.80%
32.50% to 40.07%	5	3.88%	97.67%
40.07% to 47.63%	1	0.78%	98.45%
Greater than 47.63%	2	1.55%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	7.29%	27.46%	81.40%	68.27%
±2σ	-2.80%	37.55%	96.12%	95.45%
±3σ	-12.89%	47.63%	98.45%	99.73%

Descriptive Statistics

Mean	17.373%
Standard Error	0.888%
Median	14.890%
Mode	13.779%
Standard Deviation	10.086%
Sample Variance	101.733%
Kurtosis	16.88
Skewness	3.13
Range	81.520%
Minimum	4.935%
Maximum	86.455%
Count	129

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.373%	129	100.00%	17.373%
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 1.39. Further, an Excess Kurtosis of 5.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -30.78%	0	0.00%	0.00%
-30.78% to -22.97%	1	0.78%	0.78%
-22.97% to -15.16%	2	1.55%	2.33%
-15.16% to -7.36%	23	17.83%	20.16%
-7.36% to 0.45%	46	35.66%	55.81%
0.45% to 8.26%	34	26.36%	82.17%
8.26% to 16.07%	16	12.40%	94.57%
16.07% to 23.88%	4	3.10%	97.67%
23.88% to 31.68%	1	0.78%	98.45%
Greater than 31.68%	2	1.55%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.96%	10.86%	75.97%	68.27%
±2σ	-20.37%	21.27%	93.02%	95.45%
±3σ	-30.78%	31.68%	98.45%	99.73%

Descriptive Statistics

Mean	0.451%
Standard Error	0.917%
Median	-0.720%
Mode	-24.618%
Standard Deviation	10.411%
Sample Variance	108.385%
Kurtosis	5.42
Skewness	1.39
Range	78.973%
Minimum	-24.618%
Maximum	54.355%
Count	129

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.721%	58	44.96%	3.921%
Negative	-6.305%	71	55.04%	-3.470%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.05. Further, an Excess Kurtosis of 1.99 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -53.38%	0	0.00%	0.00%
-53.38% to -39.51%	1	2.33%	2.33%
-39.51% to -25.64%	3	6.98%	9.30%
-25.64% to -11.76%	4	9.30%	18.60%
-11.76% to 2.11%	14	32.56%	51.16%
2.11% to 15.98%	12	27.91%	79.07%
15.98% to 29.86%	8	18.60%	97.67%
29.86% to 43.73%	0	0.00%	97.67%
43.73% to 57.60%	0	0.00%	97.67%
Greater than 57.60%	1	2.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.39%	20.61%	76.74%	68.27%
±2σ	-34.88%	39.10%	95.35%	95.45%
±3σ	-53.38%	57.60%	97.67%	99.73%

Descriptive Statistics

Mean	2.110%
Standard Error	2.821%
Median	0.742%
Mode	-48.860%
Standard Deviation	18.497%
Sample Variance	342.152%
Kurtosis	1.99
Skewness	0.05
Range	108.267%
Minimum	-48.860%
Maximum	59.407%
Count	43

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.615%	23	53.49%	7.817%
Negative	-12.271%	20	46.51%	-5.707%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	24.43%
12	3 Years	27.37%
20	5 Years	28.94%
40	10 Years	27.64%
80	20 Years	27.48%
200	50 Years	27.48%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.77. Further, an Excess Kurtosis of 10.87 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -18.86%	0	0.00%	0.00%
-18.86% to -5.90%	0	0.00%	0.00%
-5.90% to 7.07%	0	0.00%	0.00%
7.07% to 20.04%	6	13.64%	13.64%
20.04% to 33.00%	23	52.27%	65.91%
33.00% to 45.97%	9	20.45%	86.36%
45.97% to 58.94%	3	6.82%	93.18%
58.94% to 71.90%	2	4.55%	97.73%
71.90% to 84.87%	0	0.00%	97.73%
Greater than 84.87%	1	2.27%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	15.71%	50.29%	84.09%	68.27%
±2σ	-1.57%	67.58%	97.73%	95.45%
±3σ	-18.86%	84.87%	97.73%	99.73%

Descriptive Statistics

Mean	33.004%
Standard Error	2.606%
Median	28.172%
Mode	11.666%
Standard Deviation	17.289%
Sample Variance	298.925%
Kurtosis	10.87
Skewness	2.77
Range	102.898%
Minimum	11.666%
Maximum	114.564%
Count	44

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	33.004%	44	100.00%	33.004%
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 2.16. Further, an Excess Kurtosis of 7.70 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -64.20%	0	0.00%	0.00%
-64.20% to -47.80%	0	0.00%	0.00%
-47.80% to -31.41%	1	2.27%	2.27%
-31.41% to -15.02%	7	15.91%	18.18%
-15.02% to 1.38%	18	40.91%	59.09%
1.38% to 17.77%	12	27.27%	86.36%
17.77% to 34.16%	3	6.82%	93.18%
34.16% to 50.56%	2	4.55%	97.73%
50.56% to 66.95%	0	0.00%	97.73%
Greater than 66.95%	1	2.27%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-20.48%	23.24%	84.09%	68.27%
±2σ	-42.34%	45.09%	95.45%	95.45%
±3σ	-64.20%	66.95%	97.73%	99.73%

Descriptive Statistics

Mean	1.378%
Standard Error	3.295%
Median	-1.821%
Mode	-38.874%
Standard Deviation	21.858%
Sample Variance	477.756%
Kurtosis	7.70
Skewness	2.16
Range	135.360%
Minimum	-38.874%
Maximum	96.486%
Count	44

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	16.869%	20	45.45%	7.668%
Negative	-11.530%	24	54.55%	-6.289%
Zero	0.000%	0	0.00%	0.000%

PYPL Deep Analytical Report