RNG - 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	3.36%	2.42%	2.83%	4.79%	4.73%	3.17%	37.68	41.41	35.93	43.42
Weekly	7.35%	6.69%	7.33%	11.50%	10.41%	6.98%	35.59	43.83	32.07	48.65
Monthly	13.35%	12.57%	13.13%	24.32%	21.68%	14.53%	31.80	49.06	25.60	60.94
Quarterly	23.51%	22.82%	27.45%	41.46%	37.54%	25.17%	27.14	57.50	18.64	83.70
Annual	—	—	—	—	75.09%	50.34%	18.64	83.70	8.80	177.34
Option Time Horizon	—	—	—	—	28.88%	19.36%	29.59	52.73	22.17	70.38

Calculation Notes: Computed at 2026-02-22T14:22 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $39.50 | Strike (K): $40.00 (first OTM call ≥ spot) | Observed Mid-Price: $2.8500 (Bid $2.5000 / Ask $3.2000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 76.00% | Binomial IV (Annual): 75.09% (CRR binomial tree, American, n=20 steps) | BS IV: 83.06% (European Black-Scholes) | Yahoo IV (Annual): 50.34% → 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 0.90. Further, an Excess Kurtosis of 11.52 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.01%	12	0.38%	0.38%
-10.01% to -7.48%	29	0.93%	1.32%
-7.48% to -4.96%	125	4.01%	5.33%
-4.96% to -2.44%	348	11.16%	16.49%
-2.44% to 0.08%	1074	34.46%	50.95%
0.08% to 2.60%	1011	32.44%	83.38%
2.60% to 5.12%	371	11.90%	95.28%
5.12% to 7.65%	96	3.08%	98.36%
7.65% to 10.17%	29	0.93%	99.29%
Greater than 10.17%	22	0.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.28%	3.44%	77.70%	68.27%
±2σ	-6.64%	6.81%	95.32%	95.45%
±3σ	-10.01%	10.17%	98.91%	99.73%

Descriptive Statistics

Mean	0.081%
Standard Error	0.060%
Median	0.046%
Mode	0.000%
Standard Deviation	3.362%
Sample Variance	11.306%
Kurtosis	11.52
Skewness	0.90
Range	57.824%
Minimum	-23.425%
Maximum	34.399%
Count	3117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.369%	1568	50.30%	1.192%
Negative	-2.291%	1512	48.51%	-1.111%
Zero	0.000%	37	1.19%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	11.22%
20	1 Month	7.20%
60	1 Quarter	4.59%
250	1 Year	4.03%
750	3 Years	4.15%
1,250	5 Years	4.79%
2,500	10 Years	4.38%
5,000	20 Years	4.37%
12,500	50 Years	4.37%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.24. Further, an Excess Kurtosis of 8.69 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.02%	0	0.00%	0.00%
-3.02% to -1.20%	0	0.00%	0.00%
-1.20% to 0.61%	0	0.00%	0.00%
0.61% to 2.43%	620	19.88%	19.88%
2.43% to 4.25%	1289	41.34%	61.23%
4.25% to 6.06%	703	22.55%	83.77%
6.06% to 7.88%	278	8.92%	92.69%
7.88% to 9.69%	128	4.11%	96.79%
9.69% to 11.51%	47	1.51%	98.30%
Greater than 11.51%	53	1.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.83%	6.67%	80.92%	68.27%
±2σ	-0.59%	9.09%	95.70%	95.45%
±3σ	-3.02%	11.51%	98.30%	99.73%

Descriptive Statistics

Mean	4.247%
Standard Error	0.043%
Median	3.658%
Mode	2.558%
Standard Deviation	2.421%
Sample Variance	5.863%
Kurtosis	8.69
Skewness	2.24
Range	22.177%
Minimum	0.648%
Maximum	22.825%
Count	3118

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.247%	3118	100.00%	4.247%
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 0.28. Further, an Excess Kurtosis of 2.68 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.41%	16	0.51%	0.51%
-8.41% to -6.29%	34	1.09%	1.60%
-6.29% to -4.17%	111	3.56%	5.16%
-4.17% to -2.04%	408	13.09%	18.25%
-2.04% to 0.08%	1069	34.28%	52.53%
0.08% to 2.20%	902	28.93%	81.46%
2.20% to 4.33%	385	12.35%	93.81%
4.33% to 6.45%	124	3.98%	97.79%
6.45% to 8.57%	45	1.44%	99.23%
Greater than 8.57%	24	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.75%	2.91%	74.79%	68.27%
±2σ	-5.58%	5.74%	94.39%	95.45%
±3σ	-8.41%	8.57%	98.72%	99.73%

Descriptive Statistics

Mean	0.081%
Standard Error	0.051%
Median	-0.043%
Mode	0.000%
Standard Deviation	2.831%
Sample Variance	8.014%
Kurtosis	2.68
Skewness	0.28
Range	31.443%
Minimum	-13.629%
Maximum	17.814%
Count	3118

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.214%	1510	48.43%	1.072%
Negative	-1.961%	1576	50.55%	-0.991%
Zero	0.000%	32	1.03%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.31. Further, an Excess Kurtosis of 3.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.65%	2	0.31%	0.31%
-21.65% to -16.14%	10	1.55%	1.85%
-16.14% to -10.63%	25	3.86%	5.72%
-10.63% to -5.12%	84	12.98%	18.70%
-5.12% to 0.39%	205	31.68%	50.39%
0.39% to 5.90%	207	31.99%	82.38%
5.90% to 11.41%	81	12.52%	94.90%
11.41% to 16.92%	21	3.25%	98.15%
16.92% to 22.43%	8	1.24%	99.38%
Greater than 22.43%	4	0.62%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.96%	7.74%	76.04%	68.27%
±2σ	-14.31%	15.09%	95.05%	95.45%
±3σ	-21.65%	22.43%	99.07%	99.73%

Descriptive Statistics

Mean	0.390%
Standard Error	0.289%
Median	0.329%
Mode	-2.863%
Standard Deviation	7.348%
Sample Variance	53.997%
Kurtosis	3.38
Skewness	0.31
Range	71.538%
Minimum	-33.664%
Maximum	37.874%
Count	647

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.414%	343	53.01%	2.870%
Negative	-5.279%	304	46.99%	-2.480%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	19.34%
12	1 Quarter	11.47%
52	1 Year	9.83%
156	3 Years	10.05%
260	5 Years	11.50%
520	10 Years	10.46%
1,040	20 Years	10.33%
2,600	50 Years	10.33%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.40. Further, an Excess Kurtosis of 9.11 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.13%	0	0.00%	0.00%
-9.13% to -4.11%	0	0.00%	0.00%
-4.11% to 0.91%	0	0.00%	0.00%
0.91% to 5.93%	125	19.29%	19.29%
5.93% to 10.95%	277	42.75%	62.04%
10.95% to 15.97%	150	23.15%	85.19%
15.97% to 20.99%	50	7.72%	92.90%
20.99% to 26.01%	23	3.55%	96.45%
26.01% to 31.03%	13	2.01%	98.46%
Greater than 31.03%	10	1.54%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.26%	17.64%	84.26%	68.27%
±2σ	-2.44%	24.33%	95.37%	95.45%
±3σ	-9.13%	31.03%	98.46%	99.73%

Descriptive Statistics

Mean	10.948%
Standard Error	0.263%
Median	9.172%
Mode	3.111%
Standard Deviation	6.693%
Sample Variance	44.792%
Kurtosis	9.11
Skewness	2.40
Range	54.595%
Minimum	2.573%
Maximum	57.168%
Count	648

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.948%	648	100.00%	10.948%
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 0.87. 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 -21.75%	4	0.62%	0.62%
-21.75% to -16.25%	2	0.31%	0.93%
-16.25% to -10.75%	23	3.55%	4.48%
-10.75% to -5.25%	93	14.35%	18.83%
-5.25% to 0.25%	226	34.88%	53.70%
0.25% to 5.75%	188	29.01%	82.72%
5.75% to 11.25%	61	9.41%	92.13%
11.25% to 16.75%	36	5.56%	97.69%
16.75% to 22.25%	9	1.39%	99.07%
Greater than 22.25%	6	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.08%	7.58%	76.85%	68.27%
±2σ	-14.41%	14.91%	95.52%	95.45%
±3σ	-21.75%	22.24%	98.46%	99.73%

Descriptive Statistics

Mean	0.249%
Standard Error	0.288%
Median	-0.258%
Mode	-8.512%
Standard Deviation	7.332%
Sample Variance	53.756%
Kurtosis	4.98
Skewness	0.87
Range	79.662%
Minimum	-27.345%
Maximum	52.317%
Count	648

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.792%	308	47.53%	2.753%
Negative	-4.801%	338	52.16%	-2.504%
Zero	0.000%	2	0.31%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.41. Further, an Excess Kurtosis of 0.52 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -38.67%	0	0.00%	0.00%
-38.67% to -28.66%	0	0.00%	0.00%
-28.66% to -18.64%	6	4.03%	4.03%
-18.64% to -8.62%	29	19.46%	23.49%
-8.62% to 1.39%	41	27.52%	51.01%
1.39% to 11.41%	37	24.83%	75.84%
11.41% to 21.42%	24	16.11%	91.95%
21.42% to 31.44%	11	7.38%	99.33%
31.44% to 41.45%	0	0.00%	99.33%
Greater than 41.45%	1	0.67%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.96%	14.74%	67.79%	68.27%
±2σ	-25.32%	28.10%	96.64%	95.45%
±3σ	-38.67%	41.45%	99.33%	99.73%

Descriptive Statistics

Mean	1.391%
Standard Error	1.094%
Median	0.958%
Mode	7.073%
Standard Deviation	13.354%
Sample Variance	178.328%
Kurtosis	0.52
Skewness	0.41
Range	80.237%
Minimum	-27.609%
Maximum	52.628%
Count	149

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.769%	76	51.01%	6.003%
Negative	-9.414%	73	48.99%	-4.612%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	31.37%
12	1 Year	21.33%
36	3 Years	22.08%
60	5 Years	24.32%
120	10 Years	22.33%
240	20 Years	22.06%
600	50 Years	22.06%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.69. Further, an Excess Kurtosis of 4.65 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.42%	0	0.00%	0.00%
-12.42% to -2.99%	0	0.00%	0.00%
-2.99% to 6.44%	0	0.00%	0.00%
6.44% to 15.87%	34	22.67%	22.67%
15.87% to 25.29%	56	37.33%	60.00%
25.29% to 34.72%	31	20.67%	80.67%
34.72% to 44.15%	17	11.33%	92.00%
44.15% to 53.58%	9	6.00%	98.00%
53.58% to 63.01%	1	0.67%	98.67%
Greater than 63.01%	2	1.33%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.72%	37.87%	75.33%	68.27%
±2σ	0.15%	50.44%	94.67%	95.45%
±3σ	-12.42%	63.01%	98.67%	99.73%

Descriptive Statistics

Mean	25.294%
Standard Error	1.026%
Median	21.906%
Mode	6.999%
Standard Deviation	12.572%
Sample Variance	158.045%
Kurtosis	4.65
Skewness	1.69
Range	83.286%
Minimum	6.999%
Maximum	90.285%
Count	150

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	25.294%	150	100.00%	25.294%
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 0.17 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -38.98%	0	0.00%	0.00%
-38.98% to -29.13%	1	0.67%	0.67%
-29.13% to -19.28%	6	4.00%	4.67%
-19.28% to -9.42%	30	20.00%	24.67%
-9.42% to 0.43%	42	28.00%	52.67%
0.43% to 10.28%	37	24.67%	77.33%
10.28% to 20.13%	23	15.33%	92.67%
20.13% to 29.98%	7	4.67%	97.33%
29.98% to 39.83%	4	2.67%	100.00%
Greater than 39.83%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.71%	13.56%	68.00%	68.27%
±2σ	-25.84%	26.70%	96.00%	95.45%
±3σ	-38.98%	39.83%	100.00%	99.73%

Descriptive Statistics

Mean	0.427%
Standard Error	1.072%
Median	-0.635%
Mode	-34.481%
Standard Deviation	13.135%
Sample Variance	172.530%
Kurtosis	0.17
Skewness	0.39
Range	73.856%
Minimum	-34.481%
Maximum	39.375%
Count	150

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.958%	74	49.33%	5.406%
Negative	-9.828%	76	50.67%	-4.980%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.18. Further, an Excess Kurtosis of -0.31 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -66.25%	0	0.00%	0.00%
-66.25% to -48.61%	1	2.00%	2.00%
-48.61% to -30.98%	2	4.00%	6.00%
-30.98% to -13.35%	8	16.00%	22.00%
-13.35% to 4.28%	13	26.00%	48.00%
4.28% to 21.92%	14	28.00%	76.00%
21.92% to 39.55%	9	18.00%	94.00%
39.55% to 57.18%	3	6.00%	100.00%
57.18% to 74.81%	0	0.00%	100.00%
Greater than 74.81%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.23%	27.79%	76.00%	68.27%
±2σ	-42.74%	51.30%	96.00%	95.45%
±3σ	-66.25%	74.81%	100.00%	99.73%

Descriptive Statistics

Mean	4.284%
Standard Error	3.325%
Median	7.158%
Mode	-51.924%
Standard Deviation	23.510%
Sample Variance	552.718%
Kurtosis	-0.31
Skewness	-0.18
Range	104.552%
Minimum	-51.924%
Maximum	52.628%
Count	50

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.444%	28	56.00%	12.009%
Negative	-17.557%	22	44.00%	-7.725%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	35.15%
12	3 Years	37.96%
20	5 Years	41.46%
40	10 Years	38.56%
80	20 Years	37.64%
200	50 Years	37.64%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.16. Further, an Excess Kurtosis of 0.75 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.16%	0	0.00%	0.00%
-21.16% to -4.05%	0	0.00%	0.00%
-4.05% to 13.07%	0	0.00%	0.00%
13.07% to 30.18%	13	25.49%	25.49%
30.18% to 47.30%	20	39.22%	64.71%
47.30% to 64.41%	7	13.73%	78.43%
64.41% to 81.53%	5	9.80%	88.24%
81.53% to 98.64%	4	7.84%	96.08%
98.64% to 115.76%	2	3.92%	100.00%
Greater than 115.76%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	24.48%	70.12%	76.47%	68.27%
±2σ	1.66%	92.94%	96.08%	95.45%
±3σ	-21.17%	115.76%	100.00%	99.73%

Descriptive Statistics

Mean	47.298%
Standard Error	3.196%
Median	40.492%
Mode	19.818%
Standard Deviation	22.821%
Sample Variance	520.796%
Kurtosis	0.75
Skewness	1.16
Range	95.868%
Minimum	19.818%
Maximum	115.686%
Count	51

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	47.298%	51	100.00%	47.298%
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 1.67. Further, an Excess Kurtosis of 3.86 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -80.91%	0	0.00%	0.00%
-80.91% to -60.32%	0	0.00%	0.00%
-60.32% to -39.73%	0	0.00%	0.00%
-39.73% to -19.14%	11	21.57%	21.57%
-19.14% to 1.45%	20	39.22%	60.78%
1.45% to 22.04%	13	25.49%	86.27%
22.04% to 42.63%	4	7.84%	94.12%
42.63% to 63.22%	1	1.96%	96.08%
63.22% to 83.81%	1	1.96%	98.04%
Greater than 83.81%	1	1.96%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-26.01%	28.90%	78.43%	68.27%
±2σ	-53.46%	56.36%	96.08%	95.45%
±3σ	-80.91%	83.81%	98.04%	99.73%

Descriptive Statistics

Mean	1.448%
Standard Error	3.844%
Median	-5.867%
Mode	-34.481%
Standard Deviation	27.454%
Sample Variance	753.704%
Kurtosis	3.86
Skewness	1.67
Range	142.672%
Minimum	-34.481%
Maximum	108.191%
Count	51

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	25.074%	22	43.14%	10.816%
Negative	-16.476%	29	56.86%	-9.369%
Zero	0.000%	0	0.00%	0.000%

RNG Deep Analytical Report