APPN - 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	4.16%	3.37%	3.69%	5.38%	3.60%	3.63%	24.25	26.06	23.39	27.02
Weekly	9.33%	8.37%	9.18%	12.65%	7.93%	8.00%	23.22	27.22	21.45	29.46
Monthly	19.97%	25.86%	16.21%	24.98%	16.52%	16.65%	21.31	29.65	18.07	34.98
Quarterly	45.58%	50.76%	26.64%	38.27%	28.61%	28.83%	18.89	33.47	14.19	44.55
Annual	—	—	—	—	57.21%	57.67%	14.19	44.55	8.01	78.94
Option Time Horizon	—	—	—	—	22.01%	22.18%	20.17	31.33	16.19	39.04

Calculation Notes: Computed at 2026-02-22T14:25 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $25.14 | Strike (K): $27.50 (first OTM call ≥ spot) | Observed Mid-Price: $1.3500 (Bid $1.2000 / Ask $1.5000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 57.21% (CRR binomial tree, American, n=20 steps) | BS IV: 56.93% (European Black-Scholes) | Yahoo IV (Annual): 57.67% → 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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.61. Further, an Excess Kurtosis of 8.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.38%	9	0.41%	0.41%
-12.38% to -9.26%	24	1.09%	1.50%
-9.26% to -6.14%	85	3.87%	5.37%
-6.14% to -3.01%	270	12.30%	17.67%
-3.01% to 0.11%	708	32.24%	49.91%
0.11% to 3.23%	737	33.56%	83.47%
3.23% to 6.35%	260	11.84%	95.31%
6.35% to 9.48%	69	3.14%	98.45%
9.48% to 12.60%	18	0.82%	99.27%
Greater than 12.60%	16	0.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.05%	4.27%	76.68%	68.27%
±2σ	-8.22%	8.44%	95.45%	95.45%
±3σ	-12.38%	12.60%	98.86%	99.73%

Descriptive Statistics

Mean	0.109%
Standard Error	0.089%
Median	0.119%
Mode	0.000%
Standard Deviation	4.163%
Sample Variance	17.334%
Kurtosis	8.93
Skewness	0.61
Range	66.374%
Minimum	-27.787%
Maximum	38.587%
Count	2196

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.940%	1130	51.46%	1.513%
Negative	-2.913%	1058	48.18%	-1.403%
Zero	0.000%	8	0.36%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	7.45%
20	1 Month	6.14%
60	1 Quarter	4.76%
250	1 Year	4.31%
750	3 Years	4.38%
1,250	5 Years	5.38%
2,500	10 Years	5.62%
5,000	20 Years	5.62%
12,500	50 Years	5.62%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.89. Further, an Excess Kurtosis of 15.37 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -4.58%	0	0.00%	0.00%
-4.58% to -2.05%	0	0.00%	0.00%
-2.05% to 0.47%	0	0.00%	0.00%
0.47% to 2.99%	392	17.84%	17.84%
2.99% to 5.52%	966	43.97%	61.81%
5.52% to 8.04%	513	23.35%	85.16%
8.04% to 10.57%	193	8.78%	93.95%
10.57% to 13.09%	68	3.10%	97.04%
13.09% to 15.62%	27	1.23%	98.27%
Greater than 15.62%	38	1.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.15%	8.88%	84.84%	68.27%
±2σ	-1.21%	12.25%	96.04%	95.45%
±3σ	-4.58%	15.62%	98.27%	99.73%

Descriptive Statistics

Mean	5.519%
Standard Error	0.072%
Median	4.717%
Mode	3.296%
Standard Deviation	3.366%
Sample Variance	11.331%
Kurtosis	15.37
Skewness	2.89
Range	38.380%
Minimum	1.075%
Maximum	39.455%
Count	2197

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.519%	2197	100.00%	5.519%
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 0.39. Further, an Excess Kurtosis of 7.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.98%	13	0.59%	0.59%
-10.98% to -8.21%	19	0.86%	1.46%
-8.21% to -5.45%	68	3.10%	4.55%
-5.45% to -2.68%	320	14.57%	19.12%
-2.68% to 0.09%	706	32.13%	51.25%
0.09% to 2.85%	685	31.18%	82.43%
2.85% to 5.62%	251	11.42%	93.86%
5.62% to 8.38%	100	4.55%	98.41%
8.38% to 11.15%	24	1.09%	99.50%
Greater than 11.15%	11	0.50%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.60%	3.78%	75.24%	68.27%
±2σ	-7.29%	7.46%	95.45%	95.45%
±3σ	-10.98%	11.15%	98.91%	99.73%

Descriptive Statistics

Mean	0.087%
Standard Error	0.079%
Median	-0.033%
Mode	0.000%
Standard Deviation	3.688%
Sample Variance	13.599%
Kurtosis	7.63
Skewness	0.39
Range	60.531%
Minimum	-24.052%
Maximum	36.479%
Count	2197

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.770%	1083	49.29%	1.365%
Negative	-2.549%	1102	50.16%	-1.279%
Zero	0.000%	12	0.55%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.71. Further, an Excess Kurtosis of 2.99 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.48%	0	0.00%	0.00%
-27.48% to -20.48%	8	1.75%	1.75%
-20.48% to -13.49%	16	3.51%	5.26%
-13.49% to -6.49%	63	13.82%	19.08%
-6.49% to 0.51%	150	32.89%	51.97%
0.51% to 7.50%	133	29.17%	81.14%
7.50% to 14.50%	62	13.60%	94.74%
14.50% to 21.50%	10	2.19%	96.93%
21.50% to 28.49%	9	1.97%	98.90%
Greater than 28.49%	5	1.10%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.82%	9.84%	77.41%	68.27%
±2σ	-18.15%	19.16%	94.74%	95.45%
±3σ	-27.48%	28.49%	98.90%	99.73%

Descriptive Statistics

Mean	0.506%
Standard Error	0.437%
Median	-0.061%
Mode	0.000%
Standard Deviation	9.329%
Sample Variance	87.027%
Kurtosis	2.99
Skewness	0.71
Range	74.144%
Minimum	-26.655%
Maximum	47.489%
Count	456

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.476%	224	49.12%	3.672%
Negative	-6.304%	229	50.22%	-3.166%
Zero	0.000%	3	0.66%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	15.81%
12	1 Quarter	12.04%
52	1 Year	10.92%
156	3 Years	10.65%
260	5 Years	12.65%
520	10 Years	13.32%
1,040	20 Years	13.32%
2,600	50 Years	13.32%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.22. Further, an Excess Kurtosis of 8.02 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.79%	0	0.00%	0.00%
-10.79% to -4.51%	0	0.00%	0.00%
-4.51% to 1.77%	0	0.00%	0.00%
1.77% to 8.05%	87	19.04%	19.04%
8.05% to 14.33%	195	42.67%	61.71%
14.33% to 20.61%	105	22.98%	84.68%
20.61% to 26.89%	37	8.10%	92.78%
26.89% to 33.18%	15	3.28%	96.06%
33.18% to 39.46%	9	1.97%	98.03%
Greater than 39.46%	9	1.97%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.96%	22.71%	81.40%	68.27%
±2σ	-2.41%	31.08%	94.97%	95.45%
±3σ	-10.79%	39.46%	98.03%	99.73%

Descriptive Statistics

Mean	14.334%
Standard Error	0.392%
Median	12.154%
Mode	7.418%
Standard Deviation	8.374%
Sample Variance	70.116%
Kurtosis	8.02
Skewness	2.22
Range	69.428%
Minimum	3.010%
Maximum	72.438%
Count	457

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.334%	457	100.00%	14.334%
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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.52. Further, an Excess Kurtosis of 2.46 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.08%	3	0.66%	0.66%
-27.08% to -20.19%	6	1.31%	1.97%
-20.19% to -13.31%	12	2.63%	4.60%
-13.31% to -6.42%	68	14.88%	19.47%
-6.42% to 0.46%	147	32.17%	51.64%
0.46% to 7.34%	140	30.63%	82.28%
7.34% to 14.23%	48	10.50%	92.78%
14.23% to 21.11%	23	5.03%	97.81%
21.11% to 28.00%	5	1.09%	98.91%
Greater than 28.00%	5	1.09%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.72%	9.64%	75.49%	68.27%
±2σ	-17.90%	18.82%	94.31%	95.45%
±3σ	-27.08%	28.00%	98.25%	99.73%

Descriptive Statistics

Mean	0.461%
Standard Error	0.429%
Median	0.065%
Mode	-9.086%
Standard Deviation	9.179%
Sample Variance	84.249%
Kurtosis	2.46
Skewness	0.52
Range	71.817%
Minimum	-31.588%
Maximum	40.229%
Count	457

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.121%	232	50.77%	3.615%
Negative	-6.406%	225	49.23%	-3.154%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.42. Further, an Excess Kurtosis of 11.47 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -57.96%	0	0.00%	0.00%
-57.96% to -42.99%	0	0.00%	0.00%
-42.99% to -28.01%	0	0.00%	0.00%
-28.01% to -13.03%	20	19.05%	19.05%
-13.03% to 1.95%	40	38.10%	57.14%
1.95% to 16.92%	29	27.62%	84.76%
16.92% to 31.90%	9	8.57%	93.33%
31.90% to 46.88%	4	3.81%	97.14%
46.88% to 61.86%	2	1.90%	99.05%
Greater than 61.86%	1	0.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-18.02%	21.92%	80.00%	68.27%
±2σ	-37.99%	41.89%	96.19%	95.45%
±3σ	-57.96%	61.86%	99.05%	99.73%

Descriptive Statistics

Mean	1.946%
Standard Error	1.949%
Median	-0.741%
Mode	-25.334%
Standard Deviation	19.970%
Sample Variance	398.792%
Kurtosis	11.47
Skewness	2.42
Range	146.503%
Minimum	-25.334%
Maximum	121.169%
Count	105

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.104%	48	45.71%	7.819%
Negative	-10.819%	57	54.29%	-5.873%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	22.58%
12	1 Year	21.93%
36	3 Years	21.61%
60	5 Years	24.98%
120	10 Years	28.86%
240	20 Years	28.86%
600	50 Years	28.86%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.48. Further, an Excess Kurtosis of 41.55 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.27%	0	0.00%	0.00%
-44.27% to -24.87%	0	0.00%	0.00%
-24.87% to -5.48%	0	0.00%	0.00%
-5.48% to 13.92%	3	2.83%	2.83%
13.92% to 33.32%	73	68.87%	71.70%
33.32% to 52.72%	18	16.98%	88.68%
52.72% to 72.12%	7	6.60%	95.28%
72.12% to 91.51%	3	2.83%	98.11%
91.51% to 110.91%	1	0.94%	99.06%
Greater than 110.91%	1	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	7.46%	59.18%	91.51%	68.27%
±2σ	-18.41%	85.05%	98.11%	95.45%
±3σ	-44.27%	110.91%	99.06%	99.73%

Descriptive Statistics

Mean	33.320%
Standard Error	2.512%
Median	27.272%
Mode	12.112%
Standard Deviation	25.864%
Sample Variance	668.935%
Kurtosis	41.55
Skewness	5.48
Range	231.341%
Minimum	12.112%
Maximum	243.453%
Count	106

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	33.320%	106	100.00%	33.320%
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 -0.24. Further, an Excess Kurtosis of 0.45 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -47.62%	1	0.94%	0.94%
-47.62% to -35.46%	0	0.00%	0.94%
-35.46% to -23.30%	6	5.66%	6.60%
-23.30% to -11.14%	16	15.09%	21.70%
-11.14% to 1.02%	32	30.19%	51.89%
1.02% to 13.18%	25	23.58%	75.47%
13.18% to 25.34%	17	16.04%	91.51%
25.34% to 37.50%	9	8.49%	100.00%
37.50% to 49.66%	0	0.00%	100.00%
Greater than 49.66%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.19%	17.24%	73.58%	68.27%
±2σ	-31.41%	33.45%	95.28%	95.45%
±3σ	-47.62%	49.66%	99.06%	99.73%

Descriptive Statistics

Mean	1.022%
Standard Error	1.575%
Median	0.235%
Mode	-0.098%
Standard Deviation	16.214%
Sample Variance	262.884%
Kurtosis	0.45
Skewness	-0.24
Range	88.591%
Minimum	-53.766%
Maximum	34.825%
Count	106

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.641%	54	50.94%	6.949%
Negative	-12.082%	52	49.06%	-5.927%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.41. Further, an Excess Kurtosis of 23.42 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -130.69%	0	0.00%	0.00%
-130.69% to -96.50%	0	0.00%	0.00%
-96.50% to -62.32%	0	0.00%	0.00%
-62.32% to -28.14%	2	5.71%	5.71%
-28.14% to 6.05%	19	54.29%	60.00%
6.05% to 40.23%	13	37.14%	97.14%
40.23% to 74.42%	0	0.00%	97.14%
74.42% to 108.60%	0	0.00%	97.14%
108.60% to 142.79%	0	0.00%	97.14%
Greater than 142.79%	1	2.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-39.53%	51.63%	91.43%	68.27%
±2σ	-85.11%	97.21%	97.14%	95.45%
±3σ	-130.69%	142.79%	97.14%	99.73%

Descriptive Statistics

Mean	6.049%
Standard Error	7.704%
Median	-1.846%
Mode	-44.530%
Standard Deviation	45.579%
Sample Variance	2077.402%
Kurtosis	23.42
Skewness	4.41
Range	289.648%
Minimum	-44.530%
Maximum	245.118%
Count	35

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.281%	16	45.71%	13.843%
Negative	-14.357%	19	54.29%	-7.794%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	32.94%
12	3 Years	35.33%
20	5 Years	38.27%
40	10 Years	51.72%
80	20 Years	51.72%
200	50 Years	51.72%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.59. Further, an Excess Kurtosis of 16.43 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -88.02%	0	0.00%	0.00%
-88.02% to -49.95%	0	0.00%	0.00%
-49.95% to -11.88%	0	0.00%	0.00%
-11.88% to 26.19%	1	2.78%	2.78%
26.19% to 64.26%	21	58.33%	61.11%
64.26% to 102.33%	9	25.00%	86.11%
102.33% to 140.40%	4	11.11%	97.22%
140.40% to 178.46%	0	0.00%	97.22%
178.46% to 216.53%	0	0.00%	97.22%
Greater than 216.53%	1	2.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	13.50%	115.02%	88.89%	68.27%
±2σ	-37.26%	165.78%	97.22%	95.45%
±3σ	-88.02%	216.53%	97.22%	99.73%

Descriptive Statistics

Mean	64.259%
Standard Error	8.460%
Median	53.580%
Mode	22.852%
Standard Deviation	50.758%
Sample Variance	2576.368%
Kurtosis	16.43
Skewness	3.59
Range	289.781%
Minimum	22.852%
Maximum	312.633%
Count	36

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	64.259%	36	100.00%	64.259%
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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.73. Further, an Excess Kurtosis of 3.41 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -77.80%	0	0.00%	0.00%
-77.80% to -57.82%	1	2.78%	2.78%
-57.82% to -37.85%	0	0.00%	2.78%
-37.85% to -17.87%	5	13.89%	16.67%
-17.87% to 2.11%	14	38.89%	55.56%
2.11% to 22.09%	13	36.11%	91.67%
22.09% to 42.06%	1	2.78%	94.44%
42.06% to 62.04%	0	0.00%	94.44%
62.04% to 82.02%	2	5.56%	100.00%
Greater than 82.02%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.53%	28.75%	83.33%	68.27%
±2σ	-51.16%	55.38%	91.67%	95.45%
±3σ	-77.80%	82.02%	100.00%	99.73%

Descriptive Statistics

Mean	2.109%
Standard Error	4.440%
Median	0.501%
Mode	-70.371%
Standard Deviation	26.637%
Sample Variance	709.534%
Kurtosis	3.41
Skewness	0.73
Range	150.772%
Minimum	-70.371%
Maximum	80.401%
Count	36

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.532%	18	50.00%	10.266%
Negative	-17.273%	17	47.22%	-8.157%
Zero	0.000%	1	2.78%	0.000%

APPN Deep Analytical Report