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%	4.71%	3.76%	3.81%	14.00	15.10	13.49	15.67
Weekly	7.23%	9.87%	7.88%	10.81%	8.27%	8.38%	13.39	15.79	12.32	17.16
Monthly	14.93%	38.13%	15.57%	21.59%	17.22%	17.45%	12.24	17.27	10.30	20.52
Quarterly	26.11%	85.17%	22.97%	35.55%	29.83%	30.22%	10.79	19.59	8.01	26.40
Annual	—	—	—	—	59.65%	60.45%	8.01	26.40	4.41	47.94
Option Time Horizon	—	—	—	—	22.52%	22.82%	11.61	18.21	9.27	22.81

Calculation Notes: Computed at 2026-03-23T16:36 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $14.54 | Strike (K): $15.00 (first OTM call ≥ spot) | Observed Mid-Price: $1.1500 (Bid $0.9600 / Ask $1.3400) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 59.65% (CRR binomial tree, American, n=20 steps) | BS IV: 60.17% (European Black-Scholes) | Yahoo IV (Annual): 60.45% → 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 0.47. Further, an Excess Kurtosis of 15.56 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.04%	43	0.54%	0.54%
-10.04% to -7.50%	71	0.89%	1.43%
-7.50% to -4.97%	203	2.55%	3.98%
-4.97% to -2.43%	886	11.12%	15.10%
-2.43% to 0.11%	3114	39.08%	54.18%
0.11% to 2.65%	2501	31.39%	85.57%
2.65% to 5.19%	728	9.14%	94.70%
5.19% to 7.73%	255	3.20%	97.90%
7.73% to 10.27%	85	1.07%	98.97%
Greater than 10.27%	82	1.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.27%	3.50%	80.40%	68.27%
±2σ	-6.66%	6.88%	95.19%	95.45%
±3σ	-10.04%	10.27%	98.43%	99.73%

Descriptive Statistics

Mean	0.111%
Standard Error	0.038%
Median	0.000%
Mode	0.000%
Standard Deviation	3.385%
Sample Variance	11.458%
Kurtosis	15.56
Skewness	0.47
Range	79.592%
Minimum	-44.796%
Maximum	34.796%
Count	7968

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.451%	3763	47.23%	1.158%
Negative	-2.140%	3896	48.90%	-1.046%
Zero	0.000%	309	3.88%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	4.47%
20	1 Month	6.18%
60	1 Quarter	5.36%
250	1 Year	4.46%
750	3 Years	4.44%
1,250	5 Years	4.71%
2,500	10 Years	4.72%
5,000	20 Years	4.06%
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 6.17. Further, an Excess Kurtosis of 99.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.15%	0	0.00%	0.00%
-6.15% to -3.53%	0	0.00%	0.00%
-3.53% to -0.92%	0	0.00%	0.00%
-0.92% to 1.70%	902	11.32%	11.32%
1.70% to 4.31%	4222	52.98%	64.30%
4.31% to 6.93%	1766	22.16%	86.46%
6.93% to 9.54%	589	7.39%	93.85%
9.54% to 12.15%	258	3.24%	97.09%
12.15% to 14.77%	121	1.52%	98.61%
Greater than 14.77%	111	1.39%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.82%	7.80%	88.79%	68.27%
±2σ	-2.66%	11.28%	96.42%	95.45%
±3σ	-6.15%	14.77%	98.61%	99.73%

Descriptive Statistics

Mean	4.311%
Standard Error	0.039%
Median	3.422%
Mode	5.000%
Standard Deviation	3.486%
Sample Variance	12.154%
Kurtosis	99.38
Skewness	6.17
Range	87.507%
Minimum	0.243%
Maximum	87.750%
Count	7969

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.311%	7969	100.00%	4.311%
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 1.12. Further, an Excess Kurtosis of 20.60 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%	102	1.28%	1.95%
-6.92% to -4.59%	252	3.16%	5.11%
-4.59% to -2.26%	814	10.21%	15.32%
-2.26% to 0.08%	2941	36.91%	52.23%
0.08% to 2.41%	2581	32.39%	84.62%
2.41% to 4.74%	800	10.04%	94.65%
4.74% to 7.07%	283	3.55%	98.21%
7.07% to 9.41%	82	1.03%	99.23%
Greater than 9.41%	61	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.03%	3.19%	78.78%	68.27%
±2σ	-6.14%	6.30%	94.73%	95.45%
±3σ	-9.25%	9.41%	98.57%	99.73%

Descriptive Statistics

Mean	0.076%
Standard Error	0.035%
Median	0.000%
Mode	0.000%
Standard Deviation	3.110%
Sample Variance	9.669%
Kurtosis	20.60
Skewness	1.12
Range	74.936%
Minimum	-21.839%
Maximum	53.097%
Count	7969

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.198%	3893	48.85%	1.074%
Negative	-2.115%	3758	47.16%	-0.997%
Zero	0.000%	318	3.99%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.23. Further, an Excess Kurtosis of 6.13 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.16%	10	0.60%	0.60%
-21.16% to -15.74%	19	1.14%	1.75%
-15.74% to -10.32%	49	2.95%	4.70%
-10.32% to -4.90%	195	11.75%	16.45%
-4.90% to 0.52%	592	35.66%	52.11%
0.52% to 5.95%	537	32.35%	84.46%
5.95% to 11.37%	161	9.70%	94.16%
11.37% to 16.79%	48	2.89%	97.05%
16.79% to 22.21%	32	1.93%	98.98%
Greater than 22.21%	17	1.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.71%	7.75%	78.49%	68.27%
±2σ	-13.94%	14.98%	94.10%	95.45%
±3σ	-21.16%	22.21%	98.37%	99.73%

Descriptive Statistics

Mean	0.523%
Standard Error	0.177%
Median	0.115%
Mode	0.000%
Standard Deviation	7.229%
Sample Variance	52.264%
Kurtosis	6.13
Skewness	0.23
Range	108.536%
Minimum	-58.199%
Maximum	50.337%
Count	1660

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.387%	848	51.08%	2.752%
Negative	-4.673%	792	47.71%	-2.230%
Zero	0.000%	20	1.21%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	14.59%
12	1 Quarter	12.66%
52	1 Year	10.56%
156	3 Years	10.39%
260	5 Years	10.81%
520	10 Years	10.87%
1,040	20 Years	9.37%
2,600	50 Years	10.00%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 9.17. Further, an Excess Kurtosis of 159.33 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -18.83%	0	0.00%	0.00%
-18.83% to -11.43%	0	0.00%	0.00%
-11.43% to -4.03%	0	0.00%	0.00%
-4.03% to 3.38%	101	6.08%	6.08%
3.38% to 10.78%	968	58.28%	64.36%
10.78% to 18.18%	383	23.06%	87.42%
18.18% to 25.58%	128	7.71%	95.12%
25.58% to 32.98%	49	2.95%	98.07%
32.98% to 40.38%	18	1.08%	99.16%
Greater than 40.38%	14	0.84%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.91%	20.64%	90.79%	68.27%
±2σ	-8.96%	30.51%	97.59%	95.45%
±3σ	-18.83%	40.38%	99.16%	99.73%

Descriptive Statistics

Mean	10.776%
Standard Error	0.242%
Median	8.473%
Mode	12.500%
Standard Deviation	9.868%
Sample Variance	97.384%
Kurtosis	159.33
Skewness	9.17
Range	214.983%
Minimum	1.017%
Maximum	216.000%
Count	1661

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.776%	1661	100.00%	10.776%
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 4.72. Further, an Excess Kurtosis of 85.18 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.51%	2	0.12%	0.12%
-23.51% to -17.60%	18	1.08%	1.20%
-17.60% to -11.69%	48	2.89%	4.09%
-11.69% to -5.78%	175	10.54%	14.63%
-5.78% to 0.13%	648	39.01%	53.64%
0.13% to 6.04%	540	32.51%	86.15%
6.04% to 11.94%	154	9.27%	95.42%
11.94% to 17.85%	46	2.77%	98.19%
17.85% to 23.76%	18	1.08%	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%	81.40%	68.27%
±2σ	-15.63%	15.88%	95.36%	95.45%
±3σ	-23.51%	23.76%	99.16%	99.73%

Descriptive Statistics

Mean	0.127%
Standard Error	0.193%
Median	-0.194%
Mode	0.000%
Standard Deviation	7.878%
Sample Variance	62.070%
Kurtosis	85.18
Skewness	4.72
Range	182.718%
Minimum	-30.493%
Maximum	152.225%
Count	1661

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.418%	780	46.96%	2.544%
Negative	-4.702%	854	51.41%	-2.418%
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 0.74. Further, an Excess Kurtosis of 3.44 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -42.61%	1	0.26%	0.26%
-42.61% to -31.41%	1	0.26%	0.52%
-31.41% to -20.21%	13	3.40%	3.93%
-20.21% to -9.01%	68	17.80%	21.73%
-9.01% to 2.19%	114	29.84%	51.57%
2.19% to 13.39%	123	32.20%	83.77%
13.39% to 24.59%	36	9.42%	93.19%
24.59% to 35.79%	16	4.19%	97.38%
35.79% to 46.99%	5	1.31%	98.69%
Greater than 46.99%	5	1.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.75%	17.12%	74.08%	68.27%
±2σ	-27.68%	32.06%	95.29%	95.45%
±3σ	-42.61%	46.99%	98.43%	99.73%

Descriptive Statistics

Mean	2.188%
Standard Error	0.764%
Median	1.421%
Mode	0.000%
Standard Deviation	14.934%
Sample Variance	223.011%
Kurtosis	3.44
Skewness	0.74
Range	141.339%
Minimum	-57.220%
Maximum	84.119%
Count	382

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.003%	210	54.97%	6.599%
Negative	-9.909%	170	44.50%	-4.410%
Zero	0.000%	2	0.52%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	21.68%
12	1 Year	19.82%
36	3 Years	20.43%
60	5 Years	21.59%
120	10 Years	23.40%
240	20 Years	19.93%
600	50 Years	21.60%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 14.39. Further, an Excess Kurtosis of 248.53 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -88.12%	0	0.00%	0.00%
-88.12% to -59.52%	0	0.00%	0.00%
-59.52% to -30.92%	0	0.00%	0.00%
-30.92% to -2.32%	0	0.00%	0.00%
-2.32% to 26.28%	256	66.84%	66.84%
26.28% to 54.88%	111	28.98%	95.82%
54.88% to 83.48%	10	2.61%	98.43%
83.48% to 112.08%	1	0.26%	98.69%
112.08% to 140.68%	3	0.78%	99.48%
Greater than 140.68%	2	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.86%	64.41%	97.13%	68.27%
±2σ	-49.99%	102.55%	98.69%	95.45%
±3σ	-88.12%	140.68%	99.48%	99.73%

Descriptive Statistics

Mean	26.279%
Standard Error	1.949%
Median	20.035%
Mode	13.274%
Standard Deviation	38.134%
Sample Variance	1454.171%
Kurtosis	248.53
Skewness	14.39
Range	690.917%
Minimum	3.750%
Maximum	694.667%
Count	383

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	26.279%	383	100.00%	26.279%
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 1.98. Further, an Excess Kurtosis of 14.69 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -46.64%	1	0.26%	0.26%
-46.64% to -34.96%	1	0.26%	0.52%
-34.96% to -23.28%	11	2.87%	3.39%
-23.28% to -11.60%	52	13.58%	16.97%
-11.60% to 0.08%	146	38.12%	55.09%
0.08% to 11.76%	104	27.15%	82.25%
11.76% to 23.43%	51	13.32%	95.56%
23.43% to 35.11%	7	1.83%	97.39%
35.11% to 46.79%	8	2.09%	99.48%
Greater than 46.79%	2	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.49%	15.65%	76.24%	68.27%
±2σ	-31.07%	31.22%	94.78%	95.45%
±3σ	-46.64%	46.79%	99.22%	99.73%

Descriptive Statistics

Mean	0.077%
Standard Error	0.796%
Median	-1.411%
Mode	-20.833%
Standard Deviation	15.572%
Sample Variance	242.488%
Kurtosis	14.69
Skewness	1.98
Range	184.036%
Minimum	-49.253%
Maximum	134.783%
Count	383

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.210%	172	44.91%	5.483%
Negative	-9.907%	209	54.57%	-5.406%
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 -71.77%	0	0.00%	0.00%
-71.77% to -52.19%	0	0.00%	0.00%
-52.19% to -32.61%	5	3.94%	3.94%
-32.61% to -13.03%	25	19.69%	23.62%
-13.03% to 6.55%	36	28.35%	51.97%
6.55% to 26.13%	44	34.65%	86.61%
26.13% to 45.71%	7	5.51%	92.13%
45.71% to 65.29%	5	3.94%	96.06%
65.29% to 84.87%	2	1.57%	97.64%
Greater than 84.87%	3	2.36%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.55%	32.66%	74.80%	68.27%
±2σ	-45.66%	58.77%	94.49%	95.45%
±3σ	-71.77%	84.87%	97.64%	99.73%

Descriptive Statistics

Mean	6.553%
Standard Error	2.317%
Median	4.811%
Mode	19.877%
Standard Deviation	26.106%
Sample Variance	681.531%
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	22.764%	73	57.48%	13.085%
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	31.04%
12	3 Years	34.80%
20	5 Years	35.55%
40	10 Years	42.75%
80	20 Years	36.35%
200	50 Years	39.21%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 9.21. Further, an Excess Kurtosis of 95.58 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -200.91%	0	0.00%	0.00%
-200.91% to -137.03%	0	0.00%	0.00%
-137.03% to -73.15%	0	0.00%	0.00%
-73.15% to -9.27%	0	0.00%	0.00%
-9.27% to 54.60%	92	71.88%	71.88%
54.60% to 118.48%	30	23.44%	95.31%
118.48% to 182.36%	4	3.12%	98.44%
182.36% to 246.24%	1	0.78%	99.22%
246.24% 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	-30.57%	139.77%	96.88%	68.27%
±2σ	-115.74%	224.94%	99.22%	95.45%
±3σ	-200.91%	310.11%	99.22%	99.73%

Descriptive Statistics

Mean	54.604%
Standard Error	7.528%
Median	39.191%
Mode	14.312%
Standard Deviation	85.170%
Sample Variance	7253.950%
Kurtosis	95.58
Skewness	9.21
Range	930.488%
Minimum	14.312%
Maximum	944.800%
Count	128

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	54.604%	128	100.00%	54.604%
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 0.47. 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 -69.59%	0	0.00%	0.00%
-69.59% to -52.36%	0	0.00%	0.00%
-52.36% to -35.13%	9	7.03%	7.03%
-35.13% to -17.90%	10	7.81%	14.84%
-17.90% to -0.67%	55	42.97%	57.81%
-0.67% to 16.57%	25	19.53%	77.34%
16.57% to 33.80%	21	16.41%	93.75%
33.80% to 51.03%	5	3.91%	97.66%
51.03% to 68.26%	2	1.56%	99.22%
Greater than 68.26%	1	0.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-23.64%	22.31%	72.66%	68.27%
±2σ	-46.61%	45.28%	92.97%	95.45%
±3σ	-69.59%	68.26%	99.22%	99.73%

Descriptive Statistics

Mean	-0.665%
Standard Error	2.031%
Median	-4.107%
Mode	-50.812%
Standard Deviation	22.974%
Sample Variance	527.813%
Kurtosis	0.38
Skewness	0.47
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	-16.216%	74	57.81%	-9.375%
Zero	0.000%	1	0.78%	0.000%

PENN Deep Analytical Report