PENN - 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	3.38%	3.49%	3.11%	4.74%	3.74%	3.82%	12.54	13.52	12.08	14.03
Weekly	7.22%	9.87%	7.88%	10.94%	8.24%	8.41%	11.99	14.14	11.04	15.35
Monthly	14.90%	38.18%	15.55%	21.78%	17.15%	17.51%	10.97	15.46	9.24	18.35
Quarterly	26.11%	85.21%	22.97%	34.84%	29.70%	30.32%	9.67	17.52	7.19	23.58
Annual	—	—	—	—	59.40%	60.64%	7.19	23.58	3.97	42.71
Option Time Horizon	—	—	—	—	22.85%	23.33%	10.36	16.36	8.24	20.56

Calculation Notes: Computed at 2026-02-22T14:20 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $13.02 | Strike (K): $14.00 (first OTM call ≥ spot) | Observed Mid-Price: $0.8350 (Bid $0.7700 / Ask $0.9000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 59.40% (CRR binomial tree, American, n=20 steps) | BS IV: 59.87% (European Black-Scholes) | Yahoo IV (Annual): 60.64% → 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 Fairly Symmetrical with a Skewness factor of 0.46. Further, an Excess Kurtosis of 15.64 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.03%	43	0.54%	0.54%
-10.03% to -7.50%	72	0.91%	1.45%
-7.50% to -4.96%	199	2.50%	3.95%
-4.96% to -2.42%	886	11.15%	15.10%
-2.42% to 0.11%	3104	39.05%	54.15%
0.11% to 2.65%	2497	31.42%	85.57%
2.65% to 5.18%	724	9.11%	94.68%
5.18% to 7.71%	255	3.21%	97.89%
7.71% to 10.25%	87	1.09%	98.98%
Greater than 10.25%	81	1.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.27%	3.49%	80.37%	68.27%
±2σ	-6.65%	6.87%	95.19%	95.45%
±3σ	-10.03%	10.25%	98.44%	99.73%

Descriptive Statistics

Mean	0.110%
Standard Error	0.038%
Median	0.000%
Mode	0.000%
Standard Deviation	3.380%
Sample Variance	11.423%
Kurtosis	15.64
Skewness	0.46
Range	79.592%
Minimum	-44.796%
Maximum	34.796%
Count	7948

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.447%	3754	47.23%	1.156%
Negative	-2.138%	3885	48.88%	-1.045%
Zero	0.000%	309	3.89%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	6.05%
20	1 Month	5.51%
60	1 Quarter	4.44%
250	1 Year	4.43%
750	3 Years	4.38%
1,250	5 Years	4.74%
2,500	10 Years	4.70%
5,000	20 Years	4.05%
12,500	50 Years	4.36%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 6.18. Further, an Excess Kurtosis of 99.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.16%	0	0.00%	0.00%
-6.16% to -3.54%	0	0.00%	0.00%
-3.54% to -0.93%	0	0.00%	0.00%
-0.93% to 1.69%	892	11.22%	11.22%
1.69% to 4.31%	4221	53.10%	64.32%
4.31% to 6.92%	1762	22.17%	86.49%
6.92% to 9.54%	585	7.36%	93.85%
9.54% to 12.15%	258	3.25%	97.09%
12.15% to 14.77%	120	1.51%	98.60%
Greater than 14.77%	111	1.40%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.82%	7.79%	88.85%	68.27%
±2σ	-2.67%	11.28%	96.41%	95.45%
±3σ	-6.16%	14.77%	98.60%	99.73%

Descriptive Statistics

Mean	4.307%
Standard Error	0.039%
Median	3.416%
Mode	5.000%
Standard Deviation	3.488%
Sample Variance	12.166%
Kurtosis	99.46
Skewness	6.18
Range	87.507%
Minimum	0.243%
Maximum	87.750%
Count	7949

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.307%	7949	100.00%	4.307%
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 1.13. Further, an Excess Kurtosis of 20.71 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.25%	53	0.67%	0.67%
-9.25% to -6.92%	101	1.27%	1.94%
-6.92% to -4.59%	251	3.16%	5.09%
-4.59% to -2.25%	811	10.20%	15.30%
-2.25% to 0.08%	2938	36.96%	52.26%
0.08% to 2.41%	2575	32.39%	84.65%
2.41% to 4.74%	795	10.00%	94.65%
4.74% to 7.07%	282	3.55%	98.20%
7.07% to 9.40%	82	1.03%	99.23%
Greater than 9.40%	61	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.03%	3.18%	78.85%	68.27%
±2σ	-6.14%	6.29%	94.73%	95.45%
±3σ	-9.25%	9.40%	98.57%	99.73%

Descriptive Statistics

Mean	0.076%
Standard Error	0.035%
Median	0.000%
Mode	0.000%
Standard Deviation	3.108%
Sample Variance	9.657%
Kurtosis	20.71
Skewness	1.13
Range	74.936%
Minimum	-21.839%
Maximum	53.097%
Count	7949

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.197%	3880	48.81%	1.072%
Negative	-2.111%	3751	47.19%	-0.996%
Zero	0.000%	318	4.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.22. Further, an Excess Kurtosis of 6.15 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.14%	10	0.60%	0.60%
-21.14% to -15.73%	19	1.15%	1.75%
-15.73% to -10.31%	49	2.96%	4.71%
-10.31% to -4.90%	195	11.78%	16.49%
-4.90% to 0.52%	587	35.45%	51.93%
0.52% to 5.94%	538	32.49%	84.42%
5.94% to 11.35%	162	9.78%	94.20%
11.35% to 16.77%	48	2.90%	97.10%
16.77% to 22.18%	31	1.87%	98.97%
Greater than 22.18%	17	1.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.70%	7.74%	78.38%	68.27%
±2σ	-13.92%	14.96%	94.14%	95.45%
±3σ	-21.14%	22.18%	98.37%	99.73%

Descriptive Statistics

Mean	0.520%
Standard Error	0.177%
Median	0.120%
Mode	0.000%
Standard Deviation	7.221%
Sample Variance	52.144%
Kurtosis	6.15
Skewness	0.22
Range	108.536%
Minimum	-58.199%
Maximum	50.337%
Count	1656

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.376%	847	51.15%	2.750%
Negative	-4.680%	789	47.65%	-2.230%
Zero	0.000%	20	1.21%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	13.78%
12	1 Quarter	10.47%
52	1 Year	10.45%
156	3 Years	10.26%
260	5 Years	10.94%
520	10 Years	10.82%
1,040	20 Years	9.33%
2,600	50 Years	9.99%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 9.19. Further, an Excess Kurtosis of 159.79 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -18.84%	0	0.00%	0.00%
-18.84% to -11.44%	0	0.00%	0.00%
-11.44% to -4.04%	0	0.00%	0.00%
-4.04% to 3.36%	99	5.97%	5.97%
3.36% to 10.77%	966	58.30%	64.27%
10.77% to 18.17%	384	23.17%	87.45%
18.17% to 25.57%	128	7.72%	95.17%
25.57% to 32.97%	48	2.90%	98.07%
32.97% to 40.37%	18	1.09%	99.16%
Greater than 40.37%	14	0.84%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.90%	20.63%	90.77%	68.27%
±2σ	-8.97%	30.50%	97.59%	95.45%
±3σ	-18.84%	40.37%	99.16%	99.73%

Descriptive Statistics

Mean	10.765%
Standard Error	0.242%
Median	8.473%
Mode	12.500%
Standard Deviation	9.868%
Sample Variance	97.375%
Kurtosis	159.79
Skewness	9.19
Range	214.983%
Minimum	1.017%
Maximum	216.000%
Count	1657

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.765%	1657	100.00%	10.765%
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 4.74. Further, an Excess Kurtosis of 85.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.50%	2	0.12%	0.12%
-23.50% to -17.59%	17	1.03%	1.15%
-17.59% to -11.69%	48	2.90%	4.04%
-11.69% to -5.78%	176	10.62%	14.67%
-5.78% to 0.13%	647	39.05%	53.71%
0.13% to 6.04%	537	32.41%	86.12%
6.04% to 11.94%	154	9.29%	95.41%
11.94% to 17.85%	46	2.78%	98.19%
17.85% to 23.76%	18	1.09%	99.28%
Greater than 23.76%	12	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.75%	8.01%	81.35%	68.27%
±2σ	-15.62%	15.88%	95.41%	95.45%
±3σ	-23.50%	23.76%	99.16%	99.73%

Descriptive Statistics

Mean	0.129%
Standard Error	0.193%
Median	-0.219%
Mode	0.000%
Standard Deviation	7.877%
Sample Variance	62.039%
Kurtosis	85.46
Skewness	4.74
Range	182.718%
Minimum	-30.493%
Maximum	152.225%
Count	1657

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.427%	777	46.89%	2.545%
Negative	-4.693%	853	51.48%	-2.416%
Zero	0.000%	27	1.63%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.75. Further, an Excess Kurtosis of 3.51 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.53%	1	0.26%	0.26%
-42.53% to -31.35%	1	0.26%	0.52%
-31.35% to -20.18%	13	3.41%	3.94%
-20.18% to -9.00%	67	17.59%	21.52%
-9.00% to 2.17%	114	29.92%	51.44%
2.17% to 13.35%	124	32.55%	83.99%
13.35% to 24.52%	35	9.19%	93.18%
24.52% to 35.70%	16	4.20%	97.38%
35.70% to 46.87%	5	1.31%	98.69%
Greater than 46.87%	5	1.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.73%	17.07%	74.28%	68.27%
±2σ	-27.63%	31.97%	95.28%	95.45%
±3σ	-42.53%	46.87%	98.43%	99.73%

Descriptive Statistics

Mean	2.174%
Standard Error	0.763%
Median	1.402%
Mode	0.000%
Standard Deviation	14.900%
Sample Variance	222.011%
Kurtosis	3.51
Skewness	0.75
Range	141.339%
Minimum	-57.220%
Maximum	84.119%
Count	381

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.905%	210	55.12%	6.562%
Negative	-9.894%	169	44.36%	-4.389%
Zero	0.000%	2	0.53%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	16.63%
12	1 Year	19.81%
36	3 Years	20.19%
60	5 Years	21.78%
120	10 Years	23.35%
240	20 Years	19.93%
600	50 Years	21.58%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 14.38. Further, an Excess Kurtosis of 248.02 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -88.27%	0	0.00%	0.00%
-88.27% to -59.63%	0	0.00%	0.00%
-59.63% to -31.00%	0	0.00%	0.00%
-31.00% to -2.37%	0	0.00%	0.00%
-2.37% to 26.27%	256	67.02%	67.02%
26.27% to 54.90%	110	28.80%	95.81%
54.90% to 83.54%	10	2.62%	98.43%
83.54% to 112.17%	1	0.26%	98.69%
112.17% to 140.81%	3	0.79%	99.48%
Greater than 140.81%	2	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.91%	64.45%	97.12%	68.27%
±2σ	-50.09%	102.63%	98.69%	95.45%
±3σ	-88.27%	140.81%	99.48%	99.73%

Descriptive Statistics

Mean	26.268%
Standard Error	1.953%
Median	20.096%
Mode	13.274%
Standard Deviation	38.179%
Sample Variance	1457.637%
Kurtosis	248.02
Skewness	14.38
Range	690.917%
Minimum	3.750%
Maximum	694.667%
Count	382

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	26.268%	382	100.00%	26.268%
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 1.99. Further, an Excess Kurtosis of 14.82 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -46.56%	1	0.26%	0.26%
-46.56% to -34.90%	1	0.26%	0.52%
-34.90% to -23.24%	11	2.88%	3.40%
-23.24% to -11.58%	51	13.35%	16.75%
-11.58% to 0.09%	148	38.74%	55.50%
0.09% to 11.75%	103	26.96%	82.46%
11.75% to 23.41%	50	13.09%	95.55%
23.41% to 35.08%	7	1.83%	97.38%
35.08% to 46.74%	8	2.09%	99.48%
Greater than 46.74%	2	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.46%	15.64%	76.44%	68.27%
±2σ	-31.01%	31.19%	94.76%	95.45%
±3σ	-46.56%	46.74%	99.21%	99.73%

Descriptive Statistics

Mean	0.087%
Standard Error	0.796%
Median	-1.450%
Mode	-20.833%
Standard Deviation	15.550%
Sample Variance	241.812%
Kurtosis	14.82
Skewness	1.99
Range	184.036%
Minimum	-49.253%
Maximum	134.783%
Count	382

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.208%	171	44.76%	5.465%
Negative	-9.829%	209	54.71%	-5.378%
Zero	0.000%	2	0.52%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.10. Further, an Excess Kurtosis of 1.85 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -71.82%	0	0.00%	0.00%
-71.82% to -52.23%	0	0.00%	0.00%
-52.23% to -32.65%	5	3.94%	3.94%
-32.65% to -13.07%	25	19.69%	23.62%
-13.07% to 6.51%	36	28.35%	51.97%
6.51% to 26.10%	44	34.65%	86.61%
26.10% to 45.68%	7	5.51%	92.13%
45.68% to 65.26%	5	3.94%	96.06%
65.26% to 84.84%	2	1.57%	97.64%
Greater than 84.84%	3	2.36%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.60%	32.62%	74.80%	68.27%
±2σ	-45.71%	58.73%	94.49%	95.45%
±3σ	-71.82%	84.84%	97.64%	99.73%

Descriptive Statistics

Mean	6.513%
Standard Error	2.317%
Median	4.799%
Mode	19.877%
Standard Deviation	26.110%
Sample Variance	681.740%
Kurtosis	1.85
Skewness	1.10
Range	132.406%
Minimum	-40.261%
Maximum	92.145%
Count	127

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.695%	73	57.48%	13.045%
Negative	-15.363%	54	42.52%	-6.532%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	27.51%
12	3 Years	33.62%
20	5 Years	34.84%
40	10 Years	42.39%
80	20 Years	36.17%
200	50 Years	39.09%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 9.20. Further, an Excess Kurtosis of 95.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -201.14%	0	0.00%	0.00%
-201.14% to -137.24%	0	0.00%	0.00%
-137.24% to -73.33%	0	0.00%	0.00%
-73.33% to -9.42%	0	0.00%	0.00%
-9.42% to 54.48%	92	71.88%	71.88%
54.48% to 118.39%	30	23.44%	95.31%
118.39% to 182.29%	4	3.12%	98.44%
182.29% to 246.20%	1	0.78%	99.22%
246.20% to 310.11%	0	0.00%	99.22%
Greater than 310.11%	1	0.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-30.73%	139.69%	96.88%	68.27%
±2σ	-115.93%	224.90%	99.22%	95.45%
±3σ	-201.14%	310.11%	99.22%	99.73%

Descriptive Statistics

Mean	54.483%
Standard Error	7.531%
Median	39.190%
Mode	14.312%
Standard Deviation	85.208%
Sample Variance	7260.433%
Kurtosis	95.46
Skewness	9.20
Range	930.488%
Minimum	14.312%
Maximum	944.800%
Count	128

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	54.483%	128	100.00%	54.483%
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 0.46. Further, an Excess Kurtosis of 0.38 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -69.53%	0	0.00%	0.00%
-69.53% to -52.31%	0	0.00%	0.00%
-52.31% to -35.08%	9	7.03%	7.03%
-35.08% to -17.86%	10	7.81%	14.84%
-17.86% to -0.63%	55	42.97%	57.81%
-0.63% to 16.60%	25	19.53%	77.34%
16.60% to 33.82%	21	16.41%	93.75%
33.82% to 51.05%	5	3.91%	97.66%
51.05% to 68.28%	2	1.56%	99.22%
Greater than 68.28%	1	0.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-23.60%	22.34%	72.66%	68.27%
±2σ	-46.56%	45.31%	92.97%	95.45%
±3σ	-69.53%	68.27%	99.22%	99.73%

Descriptive Statistics

Mean	-0.629%
Standard Error	2.030%
Median	-3.932%
Mode	-50.812%
Standard Deviation	22.968%
Sample Variance	527.527%
Kurtosis	0.38
Skewness	0.46
Range	120.004%
Minimum	-50.812%
Maximum	69.192%
Count	128

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.035%	53	41.41%	8.710%
Negative	-16.153%	74	57.81%	-9.338%
Zero	0.000%	1	0.78%	0.000%

PENN Deep Analytical Report