DLTR - 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	2.61%	2.25%	2.25%	3.00%	2.76%	3.10%	104.45	110.37	101.61	113.46
Weekly	5.64%	5.58%	5.67%	7.01%	6.07%	6.83%	101.05	114.09	95.10	121.23
Monthly	10.59%	13.00%	11.06%	15.86%	12.64%	14.21%	94.62	121.83	83.39	138.24
Quarterly	17.21%	21.02%	16.99%	29.70%	21.89%	24.62%	86.26	133.64	69.31	166.34
Annual	—	—	—	—	43.77%	49.23%	69.31	166.34	44.74	257.69
Option Time Horizon	—	—	—	—	16.52%	18.58%	91.02	126.66	77.16	149.41

Calculation Notes: Computed at 2026-03-23T16:38 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $107.37 | Strike (K): $110.00 (first OTM call ≥ spot) | Observed Mid-Price: $6.2250 (Bid $5.5000 / Ask $6.9500) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 43.77% (CRR binomial tree, American, n=20 steps) | BS IV: 44.18% (European Black-Scholes) | Yahoo IV (Annual): 49.23% → 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 Left Skewed (Negative Asymmetry) with a Skewness factor of -0.51. Further, an Excess Kurtosis of 17.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -7.73%	57	0.73%	0.73%
-7.73% to -5.78%	67	0.86%	1.59%
-5.78% to -3.82%	221	2.83%	4.42%
-3.82% to -1.86%	797	10.21%	14.62%
-1.86% to 0.09%	2908	37.24%	51.86%
0.09% to 2.05%	2612	33.45%	85.31%
2.05% to 4.01%	745	9.54%	94.85%
4.01% to 5.96%	232	2.97%	97.82%
5.96% to 7.92%	89	1.14%	98.96%
Greater than 7.92%	81	1.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.52%	2.70%	80.51%	68.27%
±2σ	-5.12%	5.31%	94.92%	95.45%
±3σ	-7.73%	7.92%	98.23%	99.73%

Descriptive Statistics

Mean	0.093%
Standard Error	0.030%
Median	0.000%
Mode	0.000%
Standard Deviation	2.609%
Sample Variance	6.809%
Kurtosis	17.75
Skewness	-0.51
Range	65.679%
Minimum	-43.814%
Maximum	21.865%
Count	7809

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.805%	3899	49.93%	0.901%
Negative	-1.672%	3778	48.38%	-0.809%
Zero	0.000%	132	1.69%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.87%
20	1 Month	3.91%
60	1 Quarter	3.54%
250	1 Year	3.59%
750	3 Years	3.13%
1,250	5 Years	3.00%
2,500	10 Years	2.77%
5,000	20 Years	2.73%
12,500	50 Years	3.30%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.81. Further, an Excess Kurtosis of 14.70 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.55%	0	0.00%	0.00%
-3.55% to -1.86%	0	0.00%	0.00%
-1.86% to -0.17%	0	0.00%	0.00%
-0.17% to 1.51%	1186	15.19%	15.19%
1.51% to 3.20%	3918	50.17%	65.35%
3.20% to 4.89%	1522	19.49%	84.84%
4.89% to 6.58%	615	7.87%	92.71%
6.58% to 8.27%	267	3.42%	96.13%
8.27% to 9.96%	153	1.96%	98.09%
Greater than 9.96%	149	1.91%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.95%	5.46%	86.33%	68.27%
±2σ	-1.30%	7.71%	95.02%	95.45%
±3σ	-3.55%	9.96%	98.09%	99.73%

Descriptive Statistics

Mean	3.203%
Standard Error	0.025%
Median	2.548%
Mode	4.167%
Standard Deviation	2.252%
Sample Variance	5.070%
Kurtosis	14.70
Skewness	2.81
Range	29.241%
Minimum	0.420%
Maximum	29.661%
Count	7810

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.203%	7810	100.00%	3.203%
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 -0.06. Further, an Excess Kurtosis of 4.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.75%	80	1.02%	1.02%
-6.75% to -5.07%	89	1.14%	2.16%
-5.07% to -3.38%	261	3.34%	5.51%
-3.38% to -1.69%	843	10.79%	16.30%
-1.69% to -0.01%	2572	32.93%	49.23%
-0.01% to 1.68%	2697	34.53%	83.76%
1.68% to 3.36%	838	10.73%	94.49%
3.36% to 5.05%	269	3.44%	97.94%
5.05% to 6.74%	94	1.20%	99.14%
Greater than 6.74%	67	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.26%	2.24%	78.03%	68.27%
±2σ	-4.50%	4.49%	94.39%	95.45%
±3σ	-6.75%	6.74%	98.12%	99.73%

Descriptive Statistics

Mean	-0.008%
Standard Error	0.025%
Median	0.000%
Mode	0.000%
Standard Deviation	2.248%
Sample Variance	5.055%
Kurtosis	4.10
Skewness	-0.06
Range	28.888%
Minimum	-14.366%
Maximum	14.522%
Count	7810

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.588%	3827	49.00%	0.778%
Negative	-1.598%	3845	49.23%	-0.787%
Zero	0.000%	138	1.77%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.11. Further, an Excess Kurtosis of 3.58 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.49%	13	0.80%	0.80%
-16.49% to -12.26%	20	1.24%	2.04%
-12.26% to -8.02%	64	3.95%	5.99%
-8.02% to -3.79%	162	10.01%	16.00%
-3.79% to 0.44%	575	35.52%	51.51%
0.44% to 4.67%	519	32.06%	83.57%
4.67% to 8.91%	165	10.19%	93.76%
8.91% to 13.14%	63	3.89%	97.65%
13.14% to 17.37%	24	1.48%	99.14%
Greater than 17.37%	14	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.20%	6.08%	77.89%	68.27%
±2σ	-10.84%	11.73%	93.33%	95.45%
±3σ	-16.49%	17.37%	98.33%	99.73%

Descriptive Statistics

Mean	0.441%
Standard Error	0.140%
Median	0.281%
Mode	0.000%
Standard Deviation	5.643%
Sample Variance	31.847%
Kurtosis	3.58
Skewness	-0.11
Range	62.730%
Minimum	-35.797%
Maximum	26.933%
Count	1619

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.118%	858	53.00%	2.182%
Negative	-3.733%	755	46.63%	-1.741%
Zero	0.000%	6	0.37%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	10.08%
12	1 Quarter	8.45%
52	1 Year	8.06%
156	3 Years	7.26%
260	5 Years	7.01%
520	10 Years	6.45%
1,040	20 Years	6.26%
2,600	50 Years	7.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.37. Further, an Excess Kurtosis of 9.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.72%	0	0.00%	0.00%
-8.72% to -4.54%	0	0.00%	0.00%
-4.54% to -0.35%	0	0.00%	0.00%
-0.35% to 3.83%	307	18.95%	18.95%
3.83% to 8.02%	726	44.81%	63.77%
8.02% to 12.20%	323	19.94%	83.70%
12.20% to 16.39%	131	8.09%	91.79%
16.39% to 20.57%	76	4.69%	96.48%
20.57% to 24.75%	29	1.79%	98.27%
Greater than 24.75%	28	1.73%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.44%	13.60%	84.63%	68.27%
±2σ	-3.14%	19.18%	95.37%	95.45%
±3σ	-8.73%	24.76%	98.27%	99.73%

Descriptive Statistics

Mean	8.015%
Standard Error	0.139%
Median	6.295%
Mode	3.231%
Standard Deviation	5.580%
Sample Variance	31.136%
Kurtosis	9.46
Skewness	2.37
Range	56.127%
Minimum	1.537%
Maximum	57.664%
Count	1620

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.015%	1620	100.00%	8.015%
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 1.11. Further, an Excess Kurtosis of 8.50 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.10%	6	0.37%	0.37%
-17.10% to -12.85%	19	1.17%	1.54%
-12.85% to -8.60%	63	3.89%	5.43%
-8.60% to -4.35%	176	10.86%	16.30%
-4.35% to -0.10%	590	36.42%	52.72%
-0.10% to 4.15%	526	32.47%	85.19%
4.15% to 8.40%	146	9.01%	94.20%
8.40% to 12.65%	53	3.27%	97.47%
12.65% to 16.90%	23	1.42%	98.89%
Greater than 16.90%	18	1.11%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.77%	5.57%	78.52%	68.27%
±2σ	-11.43%	11.23%	93.95%	95.45%
±3σ	-17.10%	16.90%	98.52%	99.73%

Descriptive Statistics

Mean	-0.099%
Standard Error	0.141%
Median	-0.326%
Mode	0.000%
Standard Deviation	5.666%
Sample Variance	32.106%
Kurtosis	8.50
Skewness	1.11
Range	77.699%
Minimum	-22.303%
Maximum	55.396%
Count	1620

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.090%	745	45.99%	1.881%
Negative	-3.699%	867	53.52%	-1.980%
Zero	0.000%	8	0.49%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.07. Further, an Excess Kurtosis of 1.23 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -30.01%	3	0.81%	0.81%
-30.01% to -22.07%	3	0.81%	1.61%
-22.07% to -14.13%	17	4.57%	6.18%
-14.13% to -6.19%	46	12.37%	18.55%
-6.19% to 1.75%	109	29.30%	47.85%
1.75% to 9.69%	122	32.80%	80.65%
9.69% to 17.63%	49	13.17%	93.82%
17.63% to 25.57%	18	4.84%	98.66%
25.57% to 33.51%	3	0.81%	99.46%
Greater than 33.51%	2	0.54%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.84%	12.33%	72.31%	68.27%
±2σ	-19.43%	22.92%	93.82%	95.45%
±3σ	-30.01%	33.51%	98.66%	99.73%

Descriptive Statistics

Mean	1.747%
Standard Error	0.549%
Median	2.087%
Mode	0.000%
Standard Deviation	10.587%
Sample Variance	112.077%
Kurtosis	1.23
Skewness	-0.07
Range	78.512%
Minimum	-36.088%
Maximum	42.424%
Count	372

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.509%	216	58.06%	4.941%
Negative	-7.765%	153	41.13%	-3.194%
Zero	0.000%	3	0.81%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	19.59%
12	1 Year	18.70%
36	3 Years	16.47%
60	5 Years	15.86%
120	10 Years	14.62%
240	20 Years	13.67%
600	50 Years	16.70%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.21. Further, an Excess Kurtosis of 19.70 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.12%	0	0.00%	0.00%
-20.12% to -10.37%	0	0.00%	0.00%
-10.37% to -0.62%	0	0.00%	0.00%
-0.62% to 9.13%	60	16.09%	16.09%
9.13% to 18.88%	176	47.18%	63.27%
18.88% to 28.63%	83	22.25%	85.52%
28.63% to 38.38%	29	7.77%	93.30%
38.38% to 48.13%	12	3.22%	96.51%
48.13% to 57.88%	9	2.41%	98.93%
Greater than 57.88%	4	1.07%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.88%	31.88%	86.86%	68.27%
±2σ	-7.12%	44.88%	95.98%	95.45%
±3σ	-20.12%	57.88%	98.93%	99.73%

Descriptive Statistics

Mean	18.881%
Standard Error	0.673%
Median	15.400%
Mode	14.925%
Standard Deviation	12.999%
Sample Variance	168.980%
Kurtosis	19.70
Skewness	3.21
Range	131.940%
Minimum	3.511%
Maximum	135.451%
Count	373

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.881%	373	100.00%	18.881%
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 1.18. Further, an Excess Kurtosis of 3.69 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -33.74%	0	0.00%	0.00%
-33.74% to -25.44%	1	0.27%	0.27%
-25.44% to -17.14%	15	4.02%	4.29%
-17.14% to -8.85%	56	15.01%	19.30%
-8.85% to -0.55%	136	36.46%	55.76%
-0.55% to 7.74%	102	27.35%	83.11%
7.74% to 16.04%	40	10.72%	93.83%
16.04% to 24.34%	11	2.95%	96.78%
24.34% to 32.63%	8	2.14%	98.93%
Greater than 32.63%	4	1.07%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.61%	10.51%	75.07%	68.27%
±2σ	-22.68%	21.57%	95.17%	95.45%
±3σ	-33.74%	32.63%	98.93%	99.73%

Descriptive Statistics

Mean	-0.552%
Standard Error	0.573%
Median	-1.849%
Mode	-12.500%
Standard Deviation	11.062%
Sample Variance	122.369%
Kurtosis	3.69
Skewness	1.18
Range	84.404%
Minimum	-28.369%
Maximum	56.035%
Count	373

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.174%	154	41.29%	3.788%
Negative	-7.459%	217	58.18%	-4.339%
Zero	0.000%	2	0.54%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.24. Further, an Excess Kurtosis of 0.19 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -46.72%	0	0.00%	0.00%
-46.72% to -33.81%	1	0.81%	0.81%
-33.81% to -20.90%	6	4.84%	5.65%
-20.90% to -7.98%	25	20.16%	25.81%
-7.98% to 4.93%	31	25.00%	50.81%
4.93% to 17.84%	36	29.03%	79.84%
17.84% to 30.75%	18	14.52%	94.35%
30.75% to 43.66%	4	3.23%	97.58%
43.66% to 56.57%	2	1.61%	99.19%
Greater than 56.57%	1	0.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.29%	22.14%	66.13%	68.27%
±2σ	-29.50%	39.36%	95.16%	95.45%
±3σ	-46.72%	56.57%	99.19%	99.73%

Descriptive Statistics

Mean	4.927%
Standard Error	1.546%
Median	4.428%
Mode	-38.049%
Standard Deviation	17.215%
Sample Variance	296.365%
Kurtosis	0.19
Skewness	0.24
Range	96.593%
Minimum	-38.049%
Maximum	58.544%
Count	124

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.405%	77	62.10%	9.566%
Negative	-12.238%	47	37.90%	-4.639%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	37.04%
12	3 Years	31.88%
20	5 Years	29.70%
40	10 Years	27.29%
80	20 Years	24.92%
200	50 Years	29.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.89. Further, an Excess Kurtosis of 4.78 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.92%	0	0.00%	0.00%
-26.92% to -11.16%	0	0.00%	0.00%
-11.16% to 4.61%	0	0.00%	0.00%
4.61% to 20.37%	26	20.80%	20.80%
20.37% to 36.13%	50	40.00%	60.80%
36.13% to 51.90%	28	22.40%	83.20%
51.90% to 67.66%	11	8.80%	92.00%
67.66% to 83.42%	4	3.20%	95.20%
83.42% to 99.19%	4	3.20%	98.40%
Greater than 99.19%	2	1.60%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	15.11%	57.15%	81.60%	68.27%
±2σ	-5.90%	78.17%	95.20%	95.45%
±3σ	-26.92%	99.19%	98.40%	99.73%

Descriptive Statistics

Mean	36.132%
Standard Error	1.880%
Median	31.359%
Mode	10.965%
Standard Deviation	21.018%
Sample Variance	441.737%
Kurtosis	4.78
Skewness	1.89
Range	124.486%
Minimum	10.965%
Maximum	135.451%
Count	125

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	36.132%	125	100.00%	36.132%
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 0.77. Further, an Excess Kurtosis of 1.03 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -53.05%	0	0.00%	0.00%
-53.05% to -40.31%	0	0.00%	0.00%
-40.31% to -27.57%	5	4.00%	4.00%
-27.57% to -14.82%	24	19.20%	23.20%
-14.82% to -2.08%	39	31.20%	54.40%
-2.08% to 10.66%	29	23.20%	77.60%
10.66% to 23.41%	20	16.00%	93.60%
23.41% to 36.15%	5	4.00%	97.60%
36.15% to 48.90%	2	1.60%	99.20%
Greater than 48.90%	1	0.80%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.07%	14.91%	67.20%	68.27%
±2σ	-36.06%	31.91%	96.00%	95.45%
±3σ	-53.05%	48.90%	99.20%	99.73%

Descriptive Statistics

Mean	-2.079%
Standard Error	1.520%
Median	-3.910%
Mode	-37.525%
Standard Deviation	16.992%
Sample Variance	288.715%
Kurtosis	1.03
Skewness	0.77
Range	98.787%
Minimum	-37.525%
Maximum	61.262%
Count	125

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.392%	47	37.60%	5.787%
Negative	-12.606%	78	62.40%	-7.866%
Zero	0.000%	0	0.00%	0.000%

DLTR Deep Analytical Report