AMZN - 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	3.48%	3.29%	2.92%	2.87%	2.47%	2.55%	205.26	215.64	200.26	221.03
Weekly	8.09%	9.11%	7.84%	6.93%	5.43%	5.61%	199.27	222.13	188.74	234.52
Monthly	16.14%	21.39%	14.80%	14.65%	11.30%	11.67%	187.91	235.56	167.83	263.75
Quarterly	31.68%	53.22%	25.08%	24.50%	19.57%	20.22%	172.99	255.88	142.23	311.21
Annual	—	—	—	—	39.15%	40.44%	142.23	311.21	96.16	460.34
Option Time Horizon	—	—	—	—	14.78%	15.26%	181.49	243.89	156.56	282.73

Calculation Notes: Computed at 2026-03-23T16:30 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $210.39 | Strike (K): $215.00 (first OTM call ≥ spot) | Observed Mid-Price: $10.9500 (Bid $10.9000 / Ask $11.0000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 39.15% (CRR binomial tree, American, n=20 steps) | BS IV: 39.51% (European Black-Scholes) | Yahoo IV (Annual): 40.44% → 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 Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.06. Further, an Excess Kurtosis of 11.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.27%	58	0.80%	0.80%
-10.27% to -7.66%	70	0.96%	1.76%
-7.66% to -5.05%	189	2.60%	4.37%
-5.05% to -2.44%	659	9.08%	13.45%
-2.44% to 0.17%	2845	39.21%	52.66%
0.17% to 2.77%	2534	34.92%	87.58%
2.77% to 5.38%	568	7.83%	95.41%
5.38% to 7.99%	164	2.26%	97.67%
7.99% to 10.60%	74	1.02%	98.69%
Greater than 10.60%	95	1.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.31%	3.64%	82.65%	68.27%
±2σ	-6.79%	7.12%	94.74%	95.45%
±3σ	-10.27%	10.60%	97.89%	99.73%

Descriptive Statistics

Mean	0.165%
Standard Error	0.041%
Median	0.043%
Mode	0.000%
Standard Deviation	3.477%
Sample Variance	12.091%
Kurtosis	11.49
Skewness	1.06
Range	59.237%
Minimum	-24.766%
Maximum	34.471%
Count	7256

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.302%	3699	50.98%	1.174%
Negative	-2.080%	3519	48.50%	-1.009%
Zero	0.000%	38	0.52%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.27%
20	1 Month	2.61%
60	1 Quarter	2.74%
250	1 Year	2.77%
750	3 Years	2.60%
1,250	5 Years	2.87%
2,500	10 Years	2.57%
5,000	20 Years	2.92%
12,500	50 Years	4.00%

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 12.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.08%	0	0.00%	0.00%
-6.08% to -3.61%	0	0.00%	0.00%
-3.61% to -1.14%	0	0.00%	0.00%
-1.14% to 1.33%	746	10.28%	10.28%
1.33% to 3.80%	4207	57.97%	68.25%
3.80% to 6.27%	1272	17.53%	85.78%
6.27% to 8.74%	476	6.56%	92.34%
8.74% to 11.22%	271	3.73%	96.07%
11.22% to 13.69%	129	1.78%	97.85%
Greater than 13.69%	156	2.15%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.51%	7.10%	88.38%	68.27%
±2σ	-2.78%	10.39%	95.12%	95.45%
±3σ	-6.08%	13.69%	97.85%	99.73%

Descriptive Statistics

Mean	3.804%
Standard Error	0.039%
Median	2.749%
Mode	1.911%
Standard Deviation	3.294%
Sample Variance	10.850%
Kurtosis	12.49
Skewness	2.89
Range	32.894%
Minimum	0.375%
Maximum	33.269%
Count	7257

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.804%	7257	100.00%	3.804%
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 8.99 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.76%	71	0.98%	0.98%
-8.76% to -6.57%	81	1.12%	2.09%
-6.57% to -4.38%	182	2.51%	4.60%
-4.38% to -2.20%	659	9.08%	13.68%
-2.20% to -0.01%	2626	36.19%	49.87%
-0.01% to 2.18%	2644	36.43%	86.30%
2.18% to 4.37%	651	8.97%	95.27%
4.37% to 6.55%	197	2.71%	97.99%
6.55% to 8.74%	71	0.98%	98.97%
Greater than 8.74%	75	1.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.93%	2.91%	81.44%	68.27%
±2σ	-5.84%	5.82%	94.56%	95.45%
±3σ	-8.76%	8.74%	97.99%	99.73%

Descriptive Statistics

Mean	-0.009%
Standard Error	0.034%
Median	-0.002%
Mode	0.000%
Standard Deviation	2.917%
Sample Variance	8.512%
Kurtosis	8.99
Skewness	0.06
Range	48.315%
Minimum	-22.420%
Maximum	25.895%
Count	7257

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.892%	3591	49.48%	0.936%
Negative	-1.888%	3631	50.03%	-0.945%
Zero	0.000%	35	0.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.35. Further, an Excess Kurtosis of 10.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.43%	7	0.47%	0.47%
-23.43% to -17.36%	20	1.33%	1.79%
-17.36% to -11.30%	48	3.19%	4.98%
-11.30% to -5.24%	136	9.04%	14.02%
-5.24% to 0.83%	575	38.21%	52.23%
0.83% to 6.89%	528	35.08%	87.31%
6.89% to 12.96%	120	7.97%	95.28%
12.96% to 19.02%	37	2.46%	97.74%
19.02% to 25.09%	15	1.00%	98.74%
Greater than 25.09%	19	1.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.26%	8.92%	82.19%	68.27%
±2σ	-15.34%	17.00%	94.55%	95.45%
±3σ	-23.43%	25.09%	98.27%	99.73%

Descriptive Statistics

Mean	0.830%
Standard Error	0.208%
Median	0.558%
Mode	0.433%
Standard Deviation	8.086%
Sample Variance	65.384%
Kurtosis	10.38
Skewness	1.35
Range	110.521%
Minimum	-37.848%
Maximum	72.673%
Count	1505

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.512%	820	54.48%	3.003%
Negative	-4.789%	683	45.38%	-2.173%
Zero	0.000%	2	0.13%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	5.82%
12	1 Quarter	7.02%
52	1 Year	6.86%
156	3 Years	6.18%
260	5 Years	6.93%
520	10 Years	6.29%
1,040	20 Years	7.00%
2,600	50 Years	9.62%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.69. Further, an Excess Kurtosis of 10.28 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.03%	0	0.00%	0.00%
-17.03% to -10.20%	0	0.00%	0.00%
-10.20% to -3.37%	0	0.00%	0.00%
-3.37% to 3.46%	144	9.56%	9.56%
3.46% to 10.29%	888	58.96%	68.53%
10.29% to 17.12%	253	16.80%	85.33%
17.12% to 23.95%	108	7.17%	92.50%
23.95% to 30.78%	49	3.25%	95.75%
30.78% to 37.60%	30	1.99%	97.74%
Greater than 37.60%	34	2.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.18%	19.39%	87.72%	68.27%
±2σ	-7.92%	28.50%	95.09%	95.45%
±3σ	-17.03%	37.60%	97.74%	99.73%

Descriptive Statistics

Mean	10.287%
Standard Error	0.235%
Median	7.115%
Mode	2.483%
Standard Deviation	9.106%
Sample Variance	82.919%
Kurtosis	10.28
Skewness	2.69
Range	85.985%
Minimum	0.987%
Maximum	86.972%
Count	1506

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.287%	1506	100.00%	10.287%
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 0.90. Further, an Excess Kurtosis of 8.00 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.60%	11	0.73%	0.73%
-23.60% to -17.72%	19	1.26%	1.99%
-17.72% to -11.84%	33	2.19%	4.18%
-11.84% to -5.96%	134	8.90%	13.08%
-5.96% to -0.08%	615	40.84%	53.92%
-0.08% to 5.80%	494	32.80%	86.72%
5.80% to 11.68%	115	7.64%	94.36%
11.68% to 17.56%	42	2.79%	97.14%
17.56% to 23.44%	25	1.66%	98.80%
Greater than 23.44%	18	1.20%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.92%	7.76%	82.01%	68.27%
±2σ	-15.76%	15.60%	94.09%	95.45%
±3σ	-23.60%	23.44%	98.07%	99.73%

Descriptive Statistics

Mean	-0.079%
Standard Error	0.202%
Median	-0.544%
Mode	-2.437%
Standard Deviation	7.839%
Sample Variance	61.451%
Kurtosis	8.00
Skewness	0.90
Range	104.761%
Minimum	-40.657%
Maximum	64.104%
Count	1506

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.488%	681	45.22%	2.482%
Negative	-4.686%	823	54.65%	-2.561%
Zero	0.000%	2	0.13%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.10. Further, an Excess Kurtosis of 12.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.97%	0	0.00%	0.00%
-44.97% to -32.87%	3	0.87%	0.87%
-32.87% to -20.77%	10	2.89%	3.76%
-20.77% to -8.67%	39	11.27%	15.03%
-8.67% to 3.44%	135	39.02%	54.05%
3.44% to 15.54%	119	34.39%	88.44%
15.54% to 27.64%	25	7.23%	95.66%
27.64% to 39.74%	7	2.02%	97.69%
39.74% to 51.84%	0	0.00%	97.69%
Greater than 51.84%	8	2.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.70%	19.57%	82.37%	68.27%
±2σ	-28.84%	35.71%	95.38%	95.45%
±3σ	-44.97%	51.84%	97.69%	99.73%

Descriptive Statistics

Mean	3.435%
Standard Error	0.867%
Median	2.392%
Mode	-3.276%
Standard Deviation	16.136%
Sample Variance	260.365%
Kurtosis	12.63
Skewness	2.10
Range	167.538%
Minimum	-41.155%
Maximum	126.383%
Count	346

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.935%	204	58.96%	7.037%
Negative	-8.775%	142	41.04%	-3.601%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	13.47%
12	1 Year	13.20%
36	3 Years	12.89%
60	5 Years	14.65%
120	10 Years	13.60%
240	20 Years	15.17%
600	50 Years	21.34%

High-to-Low 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 7.25 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -39.53%	0	0.00%	0.00%
-39.53% to -23.48%	0	0.00%	0.00%
-23.48% to -7.44%	0	0.00%	0.00%
-7.44% to 8.61%	39	11.24%	11.24%
8.61% to 24.65%	198	57.06%	68.30%
24.65% to 40.70%	58	16.71%	85.01%
40.70% to 56.74%	21	6.05%	91.07%
56.74% to 72.79%	15	4.32%	95.39%
72.79% to 88.83%	8	2.31%	97.69%
Greater than 88.83%	8	2.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.26%	46.05%	87.03%	68.27%
±2σ	-18.13%	67.44%	94.52%	95.45%
±3σ	-39.53%	88.83%	97.69%	99.73%

Descriptive Statistics

Mean	24.652%
Standard Error	1.148%
Median	17.304%
Mode	6.227%
Standard Deviation	21.393%
Sample Variance	457.663%
Kurtosis	7.25
Skewness	2.42
Range	150.657%
Minimum	3.282%
Maximum	153.939%
Count	347

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.652%	347	100.00%	24.652%
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 0.89. Further, an Excess Kurtosis of 4.20 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -45.38%	2	0.58%	0.58%
-45.38% to -34.28%	5	1.44%	2.02%
-34.28% to -23.18%	5	1.44%	3.46%
-23.18% to -12.08%	42	12.10%	15.56%
-12.08% to -0.98%	138	39.77%	55.33%
-0.98% to 10.12%	104	29.97%	85.30%
10.12% to 21.22%	27	7.78%	93.08%
21.22% to 32.32%	14	4.03%	97.12%
32.32% to 43.42%	5	1.44%	98.56%
Greater than 43.42%	5	1.44%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.78%	13.82%	80.98%	68.27%
±2σ	-30.58%	28.62%	93.66%	95.45%
±3σ	-45.38%	43.42%	97.98%	99.73%

Descriptive Statistics

Mean	-0.976%
Standard Error	0.795%
Median	-2.320%
Mode	-7.626%
Standard Deviation	14.800%
Sample Variance	219.042%
Kurtosis	4.20
Skewness	0.89
Range	124.602%
Minimum	-55.890%
Maximum	68.712%
Count	347

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.473%	140	40.35%	4.629%
Negative	-9.395%	207	59.65%	-5.604%
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 2.28. Further, an Excess Kurtosis of 9.17 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -84.67%	0	0.00%	0.00%
-84.67% to -60.91%	0	0.00%	0.00%
-60.91% to -37.14%	4	3.48%	3.48%
-37.14% to -13.38%	14	12.17%	15.65%
-13.38% to 10.38%	47	40.87%	56.52%
10.38% to 34.14%	36	31.30%	87.83%
34.14% to 57.91%	9	7.83%	95.65%
57.91% to 81.67%	1	0.87%	96.52%
81.67% to 105.43%	1	0.87%	97.39%
Greater than 105.43%	3	2.61%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.30%	42.06%	84.35%	68.27%
±2σ	-52.98%	73.75%	96.52%	95.45%
±3σ	-84.67%	105.43%	97.39%	99.73%

Descriptive Statistics

Mean	10.381%
Standard Error	2.954%
Median	6.508%
Mode	-52.730%
Standard Deviation	31.683%
Sample Variance	1003.810%
Kurtosis	9.17
Skewness	2.28
Range	230.189%
Minimum	-52.730%
Maximum	177.459%
Count	115

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	25.208%	71	61.74%	15.563%
Negative	-13.545%	44	38.26%	-5.182%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	20.98%
12	3 Years	23.56%
20	5 Years	24.50%
40	10 Years	23.77%
80	20 Years	27.85%
200	50 Years	40.92%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.57. Further, an Excess Kurtosis of 16.92 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -108.36%	0	0.00%	0.00%
-108.36% to -68.45%	0	0.00%	0.00%
-68.45% to -28.54%	0	0.00%	0.00%
-28.54% to 11.38%	0	0.00%	0.00%
11.38% to 51.29%	88	75.86%	75.86%
51.29% to 91.20%	12	10.34%	86.21%
91.20% to 131.11%	8	6.90%	93.10%
131.11% to 171.02%	4	3.45%	96.55%
171.02% to 210.94%	2	1.72%	98.28%
Greater than 210.94%	2	1.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.93%	104.50%	87.93%	68.27%
±2σ	-55.15%	157.72%	95.69%	95.45%
±3σ	-108.36%	210.94%	98.28%	99.73%

Descriptive Statistics

Mean	51.287%
Standard Error	4.941%
Median	34.467%
Mode	23.021%
Standard Deviation	53.216%
Sample Variance	2831.985%
Kurtosis	16.92
Skewness	3.57
Range	383.117%
Minimum	11.603%
Maximum	394.720%
Count	116

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	51.287%	116	100.00%	51.287%
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 1.36. Further, an Excess Kurtosis of 4.81 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -78.50%	0	0.00%	0.00%
-78.50% to -59.69%	1	0.86%	0.86%
-59.69% to -40.88%	3	2.59%	3.45%
-40.88% to -22.07%	17	14.66%	18.10%
-22.07% to -3.26%	41	35.34%	53.45%
-3.26% to 15.55%	37	31.90%	85.34%
15.55% to 34.35%	13	11.21%	96.55%
34.35% to 53.16%	0	0.00%	96.55%
53.16% to 71.97%	1	0.86%	97.41%
Greater than 71.97%	3	2.59%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.34%	21.82%	80.17%	68.27%
±2σ	-53.42%	46.89%	93.97%	95.45%
±3σ	-78.50%	71.97%	97.41%	99.73%

Descriptive Statistics

Mean	-3.262%
Standard Error	2.328%
Median	-5.460%
Mode	-63.478%
Standard Deviation	25.078%
Sample Variance	628.907%
Kurtosis	4.81
Skewness	1.36
Range	170.337%
Minimum	-63.478%
Maximum	106.859%
Count	116

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.902%	47	40.52%	7.253%
Negative	-17.677%	69	59.48%	-10.515%
Zero	0.000%	0	0.00%	0.000%

AMZN Deep Analytical Report