FIVE - 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	2.79%	1.97%	2.25%	3.91%	2.87%	3.16%	223.52	236.71	217.21	243.60
Weekly	5.79%	5.94%	6.00%	9.07%	6.31%	6.95%	215.96	245.01	202.75	260.97
Monthly	11.83%	12.03%	12.04%	19.37%	13.14%	14.47%	201.71	262.32	176.88	299.15
Quarterly	20.35%	23.21%	21.65%	36.48%	22.75%	25.07%	183.21	288.80	145.93	362.59
Annual	—	—	—	—	45.51%	50.13%	145.93	362.59	92.58	571.55
Option Time Horizon	—	—	—	—	17.18%	18.92%	193.72	273.13	163.15	324.32

Calculation Notes: Computed at 2026-03-23T16:34 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $230.02 | Strike (K): $240.00 (first OTM call ≥ spot) | Observed Mid-Price: $12.2500 (Bid $11.0000 / Ask $13.5000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 45.51% (CRR binomial tree, American, n=20 steps) | BS IV: 46.06% (European Black-Scholes) | Yahoo IV (Annual): 50.13% → 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 -0.21. Further, an Excess Kurtosis of 14.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.26%	19	0.55%	0.55%
-8.26% to -6.17%	15	0.44%	0.99%
-6.17% to -4.08%	102	2.97%	3.96%
-4.08% to -1.99%	431	12.54%	16.50%
-1.99% to 0.10%	1158	33.69%	50.19%
0.10% to 2.19%	1166	33.92%	84.11%
2.19% to 4.28%	396	11.52%	95.64%
4.28% to 6.37%	94	2.73%	98.37%
6.37% to 8.46%	30	0.87%	99.24%
Greater than 8.46%	26	0.76%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.68%	2.89%	79.28%	68.27%
±2σ	-5.47%	5.67%	96.30%	95.45%
±3σ	-8.26%	8.46%	98.69%	99.73%

Descriptive Statistics

Mean	0.102%
Standard Error	0.048%
Median	0.095%
Mode	-0.275%
Standard Deviation	2.786%
Sample Variance	7.760%
Kurtosis	14.49
Skewness	-0.21
Range	50.269%
Minimum	-27.807%
Maximum	22.462%
Count	3437

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.928%	1776	51.67%	0.996%
Negative	-1.856%	1656	48.18%	-0.894%
Zero	0.000%	5	0.14%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.65%
20	1 Month	4.20%
60	1 Quarter	3.78%
250	1 Year	4.34%
750	3 Years	3.86%
1,250	5 Years	3.91%
2,500	10 Years	3.74%
5,000	20 Years	3.69%
12,500	50 Years	3.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.82. Further, an Excess Kurtosis of 29.30 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -2.37%	0	0.00%	0.00%
-2.37% to -0.90%	0	0.00%	0.00%
-0.90% to 0.58%	0	0.00%	0.00%
0.58% to 2.06%	537	15.62%	15.62%
2.06% to 3.53%	1586	46.13%	61.75%
3.53% to 5.01%	829	24.11%	85.86%
5.01% to 6.49%	297	8.64%	94.50%
6.49% to 7.96%	98	2.85%	97.35%
7.96% to 9.44%	34	0.99%	98.34%
Greater than 9.44%	57	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.56%	5.50%	85.72%	68.27%
±2σ	-0.40%	7.47%	96.92%	95.45%
±3σ	-2.37%	9.44%	98.34%	99.73%

Descriptive Statistics

Mean	3.534%
Standard Error	0.034%
Median	3.093%
Mode	3.130%
Standard Deviation	1.969%
Sample Variance	3.876%
Kurtosis	29.30
Skewness	3.82
Range	29.515%
Minimum	0.741%
Maximum	30.256%
Count	3438

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.534%	3438	100.00%	3.534%
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 0.43. Further, an Excess Kurtosis of 6.18 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.73%	20	0.58%	0.58%
-6.73% to -5.04%	28	0.81%	1.40%
-5.04% to -3.35%	139	4.04%	5.44%
-3.35% to -1.66%	463	13.47%	18.91%
-1.66% to 0.04%	1084	31.53%	50.44%
0.04% to 1.73%	1050	30.54%	80.98%
1.73% to 3.42%	454	13.21%	94.18%
3.42% to 5.11%	147	4.28%	98.46%
5.11% to 6.80%	33	0.96%	99.42%
Greater than 6.80%	20	0.58%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.22%	2.29%	75.39%	68.27%
±2σ	-4.47%	4.55%	95.46%	95.45%
±3σ	-6.73%	6.80%	98.84%	99.73%

Descriptive Statistics

Mean	0.036%
Standard Error	0.038%
Median	0.020%
Mode	0.000%
Standard Deviation	2.255%
Sample Variance	5.084%
Kurtosis	6.18
Skewness	0.43
Range	34.635%
Minimum	-12.193%
Maximum	22.442%
Count	3438

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.676%	1727	50.23%	0.842%
Negative	-1.631%	1699	49.42%	-0.806%
Zero	0.000%	12	0.35%	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 4.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.90%	5	0.70%	0.70%
-16.90% to -12.56%	11	1.54%	2.24%
-12.56% to -8.22%	21	2.95%	5.19%
-8.22% to -3.88%	84	11.78%	16.97%
-3.88% to 0.47%	241	33.80%	50.77%
0.47% to 4.81%	230	32.26%	83.03%
4.81% to 9.15%	86	12.06%	95.09%
9.15% to 13.50%	22	3.09%	98.18%
13.50% to 17.84%	8	1.12%	99.30%
Greater than 17.84%	5	0.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.32%	6.26%	77.84%	68.27%
±2σ	-11.11%	12.05%	94.81%	95.45%
±3σ	-16.90%	17.84%	98.60%	99.73%

Descriptive Statistics

Mean	0.468%
Standard Error	0.217%
Median	0.322%
Mode	1.332%
Standard Deviation	5.790%
Sample Variance	33.529%
Kurtosis	4.98
Skewness	-0.05
Range	68.137%
Minimum	-37.065%
Maximum	31.072%
Count	713

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.463%	371	52.03%	2.322%
Negative	-3.866%	342	47.97%	-1.854%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.29%
12	1 Quarter	7.99%
52	1 Year	9.76%
156	3 Years	9.10%
260	5 Years	9.07%
520	10 Years	8.62%
1,040	20 Years	8.47%
2,600	50 Years	8.47%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.56. Further, an Excess Kurtosis of 36.32 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.90%	0	0.00%	0.00%
-8.90% to -4.45%	0	0.00%	0.00%
-4.45% to 0.00%	0	0.00%	0.00%
0.00% to 4.46%	63	8.82%	8.82%
4.46% to 8.91%	393	55.04%	63.87%
8.91% to 13.36%	173	24.23%	88.10%
13.36% to 17.82%	47	6.58%	94.68%
17.82% to 22.27%	17	2.38%	97.06%
22.27% to 26.72%	8	1.12%	98.18%
Greater than 26.72%	13	1.82%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.97%	14.85%	89.92%	68.27%
±2σ	-2.97%	20.79%	96.92%	95.45%
±3σ	-8.90%	26.72%	98.18%	99.73%

Descriptive Statistics

Mean	8.910%
Standard Error	0.222%
Median	7.479%
Mode	7.914%
Standard Deviation	5.938%
Sample Variance	35.265%
Kurtosis	36.32
Skewness	4.56
Range	74.839%
Minimum	2.868%
Maximum	77.707%
Count	714

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.910%	714	100.00%	8.910%
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 2.05. Further, an Excess Kurtosis of 19.58 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -18.01%	3	0.42%	0.42%
-18.01% to -13.51%	4	0.56%	0.98%
-13.51% to -9.01%	19	2.66%	3.64%
-9.01% to -4.51%	102	14.29%	17.93%
-4.51% to -0.01%	238	33.33%	51.26%
-0.01% to 4.49%	251	35.15%	86.41%
4.49% to 8.99%	59	8.26%	94.68%
8.99% to 13.49%	21	2.94%	97.62%
13.49% to 17.99%	10	1.40%	99.02%
Greater than 17.99%	7	0.98%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.01%	5.99%	80.95%	68.27%
±2σ	-12.01%	11.99%	95.52%	95.45%
±3σ	-18.01%	17.99%	98.60%	99.73%

Descriptive Statistics

Mean	-0.014%
Standard Error	0.225%
Median	-0.178%
Mode	-6.604%
Standard Deviation	6.000%
Sample Variance	35.999%
Kurtosis	19.58
Skewness	2.05
Range	88.058%
Minimum	-24.091%
Maximum	63.967%
Count	714

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.184%	348	48.74%	2.039%
Negative	-4.005%	366	51.26%	-2.053%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.27. Further, an Excess Kurtosis of 1.98 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -33.55%	0	0.00%	0.00%
-33.55% to -24.67%	2	1.22%	1.22%
-24.67% to -15.80%	10	6.10%	7.32%
-15.80% to -6.92%	21	12.80%	20.12%
-6.92% to 1.95%	48	29.27%	49.39%
1.95% to 10.82%	48	29.27%	78.66%
10.82% to 19.70%	29	17.68%	96.34%
19.70% to 28.57%	4	2.44%	98.78%
28.57% to 37.45%	0	0.00%	98.78%
Greater than 37.45%	2	1.22%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-9.88%	13.78%	71.34%	68.27%
±2σ	-21.71%	25.62%	96.95%	95.45%
±3σ	-33.55%	37.45%	98.78%	99.73%

Descriptive Statistics

Mean	1.951%
Standard Error	0.924%
Median	2.212%
Mode	-33.248%
Standard Deviation	11.832%
Sample Variance	139.998%
Kurtosis	1.98
Skewness	0.27
Range	86.852%
Minimum	-33.248%
Maximum	53.604%
Count	164

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.249%	99	60.37%	5.583%
Negative	-9.164%	65	39.63%	-3.632%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	15.31%
12	1 Year	22.33%
36	3 Years	20.59%
60	5 Years	19.37%
120	10 Years	18.70%
240	20 Years	18.44%
600	50 Years	18.44%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.76. Further, an Excess Kurtosis of 22.51 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.36%	0	0.00%	0.00%
-15.36% to -6.34%	0	0.00%	0.00%
-6.34% to 2.69%	0	0.00%	0.00%
2.69% to 11.71%	16	9.70%	9.70%
11.71% to 20.73%	90	54.55%	64.24%
20.73% to 29.76%	40	24.24%	88.48%
29.76% to 38.78%	10	6.06%	94.55%
38.78% to 47.80%	3	1.82%	96.36%
47.80% to 56.83%	2	1.21%	97.58%
Greater than 56.83%	4	2.42%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	8.70%	32.77%	89.09%	68.27%
±2σ	-3.33%	44.80%	95.76%	95.45%
±3σ	-15.36%	56.83%	97.58%	99.73%

Descriptive Statistics

Mean	20.734%
Standard Error	0.937%
Median	18.346%
Mode	14.873%
Standard Deviation	12.031%
Sample Variance	144.740%
Kurtosis	22.51
Skewness	3.76
Range	107.942%
Minimum	5.544%
Maximum	113.486%
Count	165

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.734%	165	100.00%	20.734%
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 0.83. Further, an Excess Kurtosis of 1.97 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.54%	0	0.00%	0.00%
-36.54% to -27.52%	2	1.21%	1.21%
-27.52% to -18.49%	2	1.21%	2.42%
-18.49% to -9.46%	34	20.61%	23.03%
-9.46% to -0.44%	57	34.55%	57.58%
-0.44% to 8.59%	39	23.64%	81.21%
8.59% to 17.61%	19	11.52%	92.73%
17.61% to 26.64%	8	4.85%	97.58%
26.64% to 35.67%	2	1.21%	98.79%
Greater than 35.67%	2	1.21%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.47%	11.60%	72.73%	68.27%
±2σ	-24.51%	23.63%	95.15%	95.45%
±3σ	-36.54%	35.67%	98.79%	99.73%

Descriptive Statistics

Mean	-0.439%
Standard Error	0.937%
Median	-2.098%
Mode	-34.786%
Standard Deviation	12.035%
Sample Variance	144.852%
Kurtosis	1.97
Skewness	0.83
Range	85.350%
Minimum	-34.786%
Maximum	50.564%
Count	165

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.051%	65	39.39%	4.353%
Negative	-7.908%	100	60.61%	-4.793%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.40. Further, an Excess Kurtosis of 2.62 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -55.24%	0	0.00%	0.00%
-55.24% to -39.98%	1	1.82%	1.82%
-39.98% to -24.72%	1	1.82%	3.64%
-24.72% to -9.46%	10	18.18%	21.82%
-9.46% to 5.80%	13	23.64%	45.45%
5.80% to 21.06%	20	36.36%	81.82%
21.06% to 36.33%	8	14.55%	96.36%
36.33% to 51.59%	1	1.82%	98.18%
51.59% to 66.85%	0	0.00%	98.18%
Greater than 66.85%	1	1.82%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.55%	26.15%	70.91%	68.27%
±2σ	-34.90%	46.50%	96.36%	95.45%
±3σ	-55.24%	66.85%	98.18%	99.73%

Descriptive Statistics

Mean	5.802%
Standard Error	2.744%
Median	7.974%
Mode	-50.294%
Standard Deviation	20.349%
Sample Variance	414.085%
Kurtosis	2.62
Skewness	0.40
Range	130.186%
Minimum	-50.294%
Maximum	79.892%
Count	55

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	16.804%	36	65.45%	10.999%
Negative	-15.044%	19	34.55%	-5.197%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	46.42%
12	3 Years	42.16%
20	5 Years	36.48%
40	10 Years	35.43%
80	20 Years	33.53%
200	50 Years	33.53%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.70. Further, an Excess Kurtosis of 9.26 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.03%	0	0.00%	0.00%
-29.03% to -11.62%	0	0.00%	0.00%
-11.62% to 5.79%	0	0.00%	0.00%
5.79% to 23.20%	8	14.29%	14.29%
23.20% to 40.60%	27	48.21%	62.50%
40.60% to 58.01%	17	30.36%	92.86%
58.01% to 75.42%	0	0.00%	92.86%
75.42% to 92.83%	1	1.79%	94.64%
92.83% to 110.24%	2	3.57%	98.21%
Greater than 110.24%	1	1.79%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	17.39%	63.81%	89.29%	68.27%
±2σ	-5.82%	87.03%	92.86%	95.45%
±3σ	-29.03%	110.24%	98.21%	99.73%

Descriptive Statistics

Mean	40.604%
Standard Error	3.102%
Median	34.997%
Mode	16.428%
Standard Deviation	23.211%
Sample Variance	538.762%
Kurtosis	9.26
Skewness	2.70
Range	132.762%
Minimum	16.428%
Maximum	149.190%
Count	56

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	40.604%	56	100.00%	40.604%
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 Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.07. Further, an Excess Kurtosis of 8.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -66.80%	0	0.00%	0.00%
-66.80% to -50.57%	0	0.00%	0.00%
-50.57% to -34.33%	1	1.79%	1.79%
-34.33% to -18.10%	9	16.07%	17.86%
-18.10% to -1.86%	24	42.86%	60.71%
-1.86% to 14.37%	11	19.64%	80.36%
14.37% to 30.60%	8	14.29%	94.64%
30.60% to 46.84%	2	3.57%	98.21%
46.84% to 63.07%	0	0.00%	98.21%
Greater than 63.07%	1	1.79%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-23.51%	19.78%	78.57%	68.27%
±2σ	-45.16%	41.43%	98.21%	95.45%
±3σ	-66.80%	63.07%	98.21%	99.73%

Descriptive Statistics

Mean	-1.865%
Standard Error	2.893%
Median	-7.491%
Mode	-44.316%
Standard Deviation	21.646%
Sample Variance	468.552%
Kurtosis	8.10
Skewness	2.07
Range	145.031%
Minimum	-44.316%
Maximum	100.715%
Count	56

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	20.410%	19	33.93%	6.925%
Negative	-13.304%	37	66.07%	-8.790%
Zero	0.000%	0	0.00%	0.000%

FIVE Deep Analytical Report