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	4.15%	3.35%	3.68%	5.29%	3.54%	3.55%	24.50	26.29	23.65	27.24
Weekly	9.29%	8.36%	9.14%	12.42%	7.78%	7.81%	23.48	27.43	21.72	29.66
Monthly	19.86%	25.83%	16.11%	24.48%	16.21%	16.27%	21.58	29.84	18.35	35.10
Quarterly	45.57%	50.76%	26.63%	38.27%	28.07%	28.17%	19.17	33.60	14.48	44.49
Annual	—	—	—	—	56.14%	56.35%	14.48	44.49	8.26	78.00
Option Time Horizon	—	—	—	—	21.19%	21.27%	20.53	31.37	16.61	38.77

Calculation Notes: Computed at 2026-03-23T16:43 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $25.38 | Strike (K): $27.50 (first OTM call ≥ spot) | Observed Mid-Price: $1.3750 (Bid $1.3000 / Ask $1.4500) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 56.14% (CRR binomial tree, American, n=20 steps) | BS IV: 56.15% (European Black-Scholes) | Yahoo IV (Annual): 56.35% → 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 8.97 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.34%	9	0.41%	0.41%
-12.34% to -9.23%	24	1.08%	1.49%
-9.23% to -6.11%	87	3.92%	5.41%
-6.11% to -3.00%	270	12.18%	17.59%
-3.00% to 0.11%	718	32.39%	49.98%
0.11% to 3.22%	744	33.56%	83.54%
3.22% to 6.33%	261	11.77%	95.31%
6.33% to 9.44%	69	3.11%	98.42%
9.44% to 12.56%	19	0.86%	99.28%
Greater than 12.56%	16	0.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.04%	4.26%	76.77%	68.27%
±2σ	-8.19%	8.41%	95.49%	95.45%
±3σ	-12.34%	12.56%	98.87%	99.73%

Descriptive Statistics

Mean	0.109%
Standard Error	0.088%
Median	0.111%
Mode	0.000%
Standard Deviation	4.149%
Sample Variance	17.218%
Kurtosis	8.97
Skewness	0.61
Range	66.374%
Minimum	-27.787%
Maximum	38.587%
Count	2217

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.934%	1139	51.38%	1.507%
Negative	-2.897%	1070	48.26%	-1.398%
Zero	0.000%	8	0.36%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	4.29%
20	1 Month	4.52%
60	1 Quarter	5.01%
250	1 Year	4.32%
750	3 Years	4.38%
1,250	5 Years	5.29%
2,500	10 Years	5.61%
5,000	20 Years	5.61%
12,500	50 Years	5.61%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.90. Further, an Excess Kurtosis of 15.48 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -4.55%	0	0.00%	0.00%
-4.55% to -2.04%	0	0.00%	0.00%
-2.04% to 0.48%	0	0.00%	0.00%
0.48% to 2.99%	396	17.85%	17.85%
2.99% to 5.51%	975	43.96%	61.81%
5.51% to 8.03%	521	23.49%	85.30%
8.03% to 10.54%	192	8.66%	93.96%
10.54% to 13.06%	69	3.11%	97.07%
13.06% to 15.57%	27	1.22%	98.29%
Greater than 15.57%	38	1.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.15%	8.87%	84.67%	68.27%
±2σ	-1.20%	12.22%	96.08%	95.45%
±3σ	-4.55%	15.57%	98.29%	99.73%

Descriptive Statistics

Mean	5.510%
Standard Error	0.071%
Median	4.705%
Mode	3.296%
Standard Deviation	3.355%
Sample Variance	11.254%
Kurtosis	15.48
Skewness	2.90
Range	38.380%
Minimum	1.075%
Maximum	39.455%
Count	2218

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.510%	2218	100.00%	5.510%
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 7.66 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.95%	13	0.59%	0.59%
-10.95% to -8.19%	20	0.90%	1.49%
-8.19% to -5.44%	67	3.02%	4.51%
-5.44% to -2.68%	325	14.65%	19.16%
-2.68% to 0.08%	712	32.10%	51.26%
0.08% to 2.84%	693	31.24%	82.51%
2.84% to 5.59%	253	11.41%	93.91%
5.59% to 8.35%	100	4.51%	98.42%
8.35% to 11.11%	24	1.08%	99.50%
Greater than 11.11%	11	0.50%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.60%	3.76%	75.20%	68.27%
±2σ	-7.28%	7.43%	95.40%	95.45%
±3σ	-10.95%	11.11%	98.92%	99.73%

Descriptive Statistics

Mean	0.079%
Standard Error	0.078%
Median	-0.036%
Mode	0.000%
Standard Deviation	3.677%
Sample Variance	13.520%
Kurtosis	7.66
Skewness	0.39
Range	60.531%
Minimum	-24.052%
Maximum	36.479%
Count	2218

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.759%	1091	49.19%	1.357%
Negative	-2.543%	1115	50.27%	-1.278%
Zero	0.000%	12	0.54%	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 3.03 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -27.37%	0	0.00%	0.00%
-27.37% to -20.40%	8	1.74%	1.74%
-20.40% to -13.43%	16	3.48%	5.22%
-13.43% to -6.46%	63	13.70%	18.91%
-6.46% to 0.51%	152	33.04%	51.96%
0.51% to 7.47%	133	28.91%	80.87%
7.47% to 14.44%	64	13.91%	94.78%
14.44% to 21.41%	10	2.17%	96.96%
21.41% to 28.38%	9	1.96%	98.91%
Greater than 28.38%	5	1.09%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.79%	9.80%	77.61%	68.27%
±2σ	-18.08%	19.09%	94.78%	95.45%
±3σ	-27.37%	28.38%	98.91%	99.73%

Descriptive Statistics

Mean	0.505%
Standard Error	0.433%
Median	-0.061%
Mode	0.000%
Standard Deviation	9.292%
Sample Variance	86.333%
Kurtosis	3.03
Skewness	0.71
Range	74.144%
Minimum	-26.655%
Maximum	47.489%
Count	460

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.435%	226	49.13%	3.653%
Negative	-6.269%	231	50.22%	-3.148%
Zero	0.000%	3	0.65%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	8.86%
12	1 Quarter	11.61%
52	1 Year	10.91%
156	3 Years	10.59%
260	5 Years	12.42%
520	10 Years	13.29%
1,040	20 Years	13.29%
2,600	50 Years	13.29%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.23. Further, an Excess Kurtosis of 8.06 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 -4.52%	0	0.00%	0.00%
-4.52% to 1.75%	0	0.00%	0.00%
1.75% to 8.02%	88	19.09%	19.09%
8.02% to 14.29%	196	42.52%	61.61%
14.29% to 20.56%	107	23.21%	84.82%
20.56% to 26.83%	37	8.03%	92.84%
26.83% to 33.09%	15	3.25%	96.10%
33.09% to 39.36%	9	1.95%	98.05%
Greater than 39.36%	9	1.95%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.93%	22.65%	81.56%	68.27%
±2σ	-2.43%	31.01%	95.01%	95.45%
±3σ	-10.79%	39.36%	98.05%	99.73%

Descriptive Statistics

Mean	14.289%
Standard Error	0.389%
Median	12.135%
Mode	7.418%
Standard Deviation	8.358%
Sample Variance	69.851%
Kurtosis	8.06
Skewness	2.23
Range	69.428%
Minimum	3.010%
Maximum	72.438%
Count	461

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.289%	461	100.00%	14.289%
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 2.50 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.98%	3	0.65%	0.65%
-26.98% to -20.12%	6	1.30%	1.95%
-20.12% to -13.27%	12	2.60%	4.56%
-13.27% to -6.41%	69	14.97%	19.52%
-6.41% to 0.45%	148	32.10%	51.63%
0.45% to 7.31%	142	30.80%	82.43%
7.31% to 14.17%	47	10.20%	92.62%
14.17% to 21.02%	24	5.21%	97.83%
21.02% to 27.88%	5	1.08%	98.92%
Greater than 27.88%	5	1.08%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.69%	9.60%	75.70%	68.27%
±2σ	-17.84%	18.74%	94.36%	95.45%
±3σ	-26.98%	27.88%	98.26%	99.73%

Descriptive Statistics

Mean	0.451%
Standard Error	0.426%
Median	0.065%
Mode	-9.086%
Standard Deviation	9.144%
Sample Variance	83.611%
Kurtosis	2.50
Skewness	0.52
Range	71.817%
Minimum	-31.588%
Maximum	40.229%
Count	461

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.080%	234	50.76%	3.594%
Negative	-6.382%	227	49.24%	-3.143%
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 2.44. Further, an Excess Kurtosis of 11.65 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -57.65%	0	0.00%	0.00%
-57.65% to -42.75%	0	0.00%	0.00%
-42.75% to -27.86%	0	0.00%	0.00%
-27.86% to -12.96%	20	18.87%	18.87%
-12.96% to 1.93%	41	38.68%	57.55%
1.93% to 16.83%	28	26.42%	83.96%
16.83% to 31.73%	10	9.43%	93.40%
31.73% to 46.62%	4	3.77%	97.17%
46.62% to 61.52%	2	1.89%	99.06%
Greater than 61.52%	1	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.93%	21.80%	80.19%	68.27%
±2σ	-37.79%	41.66%	96.23%	95.45%
±3σ	-57.65%	61.52%	99.06%	99.73%

Descriptive Statistics

Mean	1.934%
Standard Error	1.929%
Median	-0.799%
Mode	-25.334%
Standard Deviation	19.861%
Sample Variance	394.470%
Kurtosis	11.65
Skewness	2.44
Range	146.503%
Minimum	-25.334%
Maximum	121.169%
Count	106

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.104%	48	45.28%	7.745%
Negative	-10.620%	58	54.72%	-5.811%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	19.06%
12	1 Year	21.63%
36	3 Years	21.40%
60	5 Years	24.48%
120	10 Years	28.69%
240	20 Years	28.69%
600	50 Years	28.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.47. Further, an Excess Kurtosis of 41.53 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.37%	0	0.00%	0.00%
-44.37% to -25.00%	0	0.00%	0.00%
-25.00% to -5.63%	0	0.00%	0.00%
-5.63% to 13.74%	4	3.74%	3.74%
13.74% to 33.12%	72	67.29%	71.03%
33.12% to 52.49%	19	17.76%	88.79%
52.49% to 71.86%	7	6.54%	95.33%
71.86% to 91.23%	3	2.80%	98.13%
91.23% to 110.60%	1	0.93%	99.07%
Greater than 110.60%	1	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	7.29%	58.94%	91.59%	68.27%
±2σ	-18.54%	84.77%	98.13%	95.45%
±3σ	-44.37%	110.60%	99.07%	99.73%

Descriptive Statistics

Mean	33.115%
Standard Error	2.497%
Median	27.143%
Mode	11.327%
Standard Deviation	25.829%
Sample Variance	667.145%
Kurtosis	41.53
Skewness	5.47
Range	232.126%
Minimum	11.327%
Maximum	243.453%
Count	107

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	33.115%	107	100.00%	33.115%
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 0.50 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -47.37%	1	0.93%	0.93%
-47.37% to -35.28%	0	0.00%	0.93%
-35.28% to -23.20%	6	5.61%	6.54%
-23.20% to -11.11%	16	14.95%	21.50%
-11.11% to 0.97%	32	29.91%	51.40%
0.97% to 13.06%	26	24.30%	75.70%
13.06% to 25.14%	17	15.89%	91.59%
25.14% to 37.23%	9	8.41%	100.00%
37.23% to 49.31%	0	0.00%	100.00%
Greater than 49.31%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.14%	17.08%	73.83%	68.27%
±2σ	-31.25%	33.20%	95.33%	95.45%
±3σ	-47.37%	49.31%	99.07%	99.73%

Descriptive Statistics

Mean	0.971%
Standard Error	1.558%
Median	0.280%
Mode	-0.098%
Standard Deviation	16.113%
Sample Variance	259.620%
Kurtosis	0.50
Skewness	-0.24
Range	88.591%
Minimum	-53.766%
Maximum	34.825%
Count	107

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.313%	55	51.40%	6.843%
Negative	-12.082%	52	48.60%	-5.872%
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 23.43 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -130.63%	0	0.00%	0.00%
-130.63% to -96.45%	0	0.00%	0.00%
-96.45% to -62.28%	0	0.00%	0.00%
-62.28% to -28.10%	2	5.71%	5.71%
-28.10% to 6.08%	19	54.29%	60.00%
6.08% to 40.25%	13	37.14%	97.14%
40.25% to 74.43%	0	0.00%	97.14%
74.43% to 108.61%	0	0.00%	97.14%
108.61% to 142.78%	0	0.00%	97.14%
Greater than 142.78%	1	2.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-39.49%	51.65%	91.43%	68.27%
±2σ	-85.06%	97.21%	97.14%	95.45%
±3σ	-130.63%	142.78%	97.14%	99.73%

Descriptive Statistics

Mean	6.076%
Standard Error	7.703%
Median	-1.846%
Mode	-44.530%
Standard Deviation	45.569%
Sample Variance	2076.538%
Kurtosis	23.43
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	-14.307%	19	54.29%	-7.767%
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 3.43 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 -57.83%	1	2.78%	2.78%
-57.83% to -37.86%	0	0.00%	2.78%
-37.86% to -17.89%	5	13.89%	16.67%
-17.89% to 2.08%	13	36.11%	52.78%
2.08% to 22.05%	14	38.89%	91.67%
22.05% to 42.02%	1	2.78%	94.44%
42.02% to 61.99%	0	0.00%	94.44%
61.99% to 81.96%	2	5.56%	100.00%
Greater than 81.96%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.55%	28.70%	83.33%	68.27%
±2σ	-51.18%	55.33%	91.67%	95.45%
±3σ	-77.80%	81.96%	100.00%	99.73%

Descriptive Statistics

Mean	2.077%
Standard Error	4.438%
Median	0.501%
Mode	-70.371%
Standard Deviation	26.627%
Sample Variance	708.998%
Kurtosis	3.43
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	20.468%	18	50.00%	10.234%
Negative	-17.273%	17	47.22%	-8.157%
Zero	0.000%	1	2.78%	0.000%

APPN Deep Analytical Report