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%	3.30%	2.92%	2.87%	2.03%	2.14%	205.90	214.41	201.76	218.80
Weekly	8.10%	9.12%	7.85%	6.94%	4.46%	4.71%	200.94	219.70	192.18	229.72
Monthly	16.16%	21.41%	14.82%	14.66%	9.29%	9.81%	191.48	230.56	174.50	252.99
Quarterly	31.67%	53.22%	25.06%	24.50%	16.08%	16.99%	178.89	246.78	152.31	289.84
Annual	—	—	—	—	32.17%	33.97%	152.31	289.84	110.42	399.82
Option Time Horizon	—	—	—	—	12.37%	13.07%	185.66	237.78	164.05	269.10

Calculation Notes: Computed at 2026-02-22T14:16 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $210.11 | Strike (K): $215.00 (first OTM call ≥ spot) | Observed Mid-Price: $8.8250 (Bid $8.7500 / Ask $8.9000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 32.17% (CRR binomial tree, American, n=20 steps) | BS IV: 32.54% (European Black-Scholes) | Yahoo IV (Annual): 33.97% → 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 11.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.28%	58	0.80%	0.80%
-10.28% to -7.67%	70	0.97%	1.77%
-7.67% to -5.05%	188	2.60%	4.37%
-5.05% to -2.44%	658	9.09%	13.46%
-2.44% to 0.17%	2837	39.21%	52.67%
0.17% to 2.78%	2528	34.94%	87.60%
2.78% to 5.39%	565	7.81%	95.41%
5.39% to 8.00%	164	2.27%	97.68%
8.00% to 10.61%	73	1.01%	98.69%
Greater than 10.61%	95	1.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.31%	3.65%	82.64%	68.27%
±2σ	-6.80%	7.13%	94.73%	95.45%
±3σ	-10.28%	10.61%	97.89%	99.73%

Descriptive Statistics

Mean	0.166%
Standard Error	0.041%
Median	0.043%
Mode	0.000%
Standard Deviation	3.481%
Sample Variance	12.116%
Kurtosis	11.46
Skewness	1.06
Range	59.237%
Minimum	-24.766%
Maximum	34.471%
Count	7236

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.305%	3689	50.98%	1.175%
Negative	-2.082%	3509	48.49%	-1.010%
Zero	0.000%	38	0.53%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.54%
20	1 Month	3.39%
60	1 Quarter	2.53%
250	1 Year	2.85%
750	3 Years	2.62%
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 12.46 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -6.09%	0	0.00%	0.00%
-6.09% to -3.61%	0	0.00%	0.00%
-3.61% to -1.14%	0	0.00%	0.00%
-1.14% to 1.33%	745	10.29%	10.29%
1.33% to 3.81%	4193	57.94%	68.23%
3.81% to 6.28%	1267	17.51%	85.74%
6.28% to 8.76%	477	6.59%	92.33%
8.76% to 11.23%	271	3.74%	96.08%
11.23% to 13.70%	128	1.77%	97.84%
Greater than 13.70%	156	2.16%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.51%	7.11%	88.38%	68.27%
±2σ	-2.79%	10.40%	95.11%	95.45%
±3σ	-6.09%	13.70%	97.84%	99.73%

Descriptive Statistics

Mean	3.808%
Standard Error	0.039%
Median	2.750%
Mode	1.911%
Standard Deviation	3.298%
Sample Variance	10.874%
Kurtosis	12.46
Skewness	2.89
Range	32.894%
Minimum	0.375%
Maximum	33.269%
Count	7237

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.808%	7237	100.00%	3.808%
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 8.97 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.77%	71	0.98%	0.98%
-8.77% to -6.58%	80	1.11%	2.09%
-6.58% to -4.39%	183	2.53%	4.62%
-4.39% to -2.20%	657	9.08%	13.69%
-2.20% to -0.01%	2618	36.18%	49.87%
-0.01% to 2.18%	2634	36.40%	86.27%
2.18% to 4.37%	652	9.01%	95.27%
4.37% to 6.56%	196	2.71%	97.98%
6.56% to 8.75%	71	0.98%	98.96%
Greater than 8.75%	75	1.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.93%	2.91%	81.44%	68.27%
±2σ	-5.85%	5.83%	94.54%	95.45%
±3σ	-8.77%	8.75%	97.98%	99.73%

Descriptive Statistics

Mean	-0.008%
Standard Error	0.034%
Median	-0.001%
Mode	0.000%
Standard Deviation	2.920%
Sample Variance	8.529%
Kurtosis	8.97
Skewness	0.06
Range	48.315%
Minimum	-22.420%
Maximum	25.895%
Count	7237

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.894%	3583	49.51%	0.938%
Negative	-1.890%	3619	50.01%	-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 1.34. Further, an Excess Kurtosis of 10.34 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.46%	7	0.47%	0.47%
-23.46% to -17.38%	20	1.33%	1.80%
-17.38% to -11.31%	48	3.20%	5.00%
-11.31% to -5.24%	135	8.99%	13.99%
-5.24% to 0.83%	574	38.24%	52.23%
0.83% to 6.91%	526	35.04%	87.28%
6.91% to 12.98%	120	7.99%	95.27%
12.98% to 19.05%	37	2.47%	97.73%
19.05% to 25.12%	15	1.00%	98.73%
Greater than 25.12%	19	1.27%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.26%	8.93%	82.15%	68.27%
±2σ	-15.36%	17.03%	94.60%	95.45%
±3σ	-23.46%	25.12%	98.27%	99.73%

Descriptive Statistics

Mean	0.834%
Standard Error	0.209%
Median	0.558%
Mode	0.433%
Standard Deviation	8.097%
Sample Variance	65.556%
Kurtosis	10.34
Skewness	1.34
Range	110.521%
Minimum	-37.848%
Maximum	72.673%
Count	1501

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.524%	818	54.50%	3.010%
Negative	-4.798%	681	45.37%	-2.177%
Zero	0.000%	2	0.13%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	9.60%
12	1 Quarter	6.42%
52	1 Year	6.99%
156	3 Years	6.23%
260	5 Years	6.94%
520	10 Years	6.29%
1,040	20 Years	7.00%
2,600	50 Years	9.63%

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

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -17.05%	0	0.00%	0.00%
-17.05% to -10.21%	0	0.00%	0.00%
-10.21% to -3.37%	0	0.00%	0.00%
-3.37% to 3.46%	144	9.59%	9.59%
3.46% to 10.30%	884	58.85%	68.44%
10.30% to 17.13%	254	16.91%	85.35%
17.13% to 23.97%	107	7.12%	92.48%
23.97% to 30.81%	49	3.26%	95.74%
30.81% to 37.64%	30	2.00%	97.74%
Greater than 37.64%	34	2.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.18%	19.41%	87.68%	68.27%
±2σ	-7.93%	28.53%	95.07%	95.45%
±3σ	-17.05%	37.64%	97.74%	99.73%

Descriptive Statistics

Mean	10.298%
Standard Error	0.235%
Median	7.123%
Mode	2.483%
Standard Deviation	9.115%
Sample Variance	83.081%
Kurtosis	10.25
Skewness	2.69
Range	85.985%
Minimum	0.987%
Maximum	86.972%
Count	1502

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.298%	1502	100.00%	10.298%
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 Right Skewed (Positive Asymmetry) with a Skewness factor of 0.90. Further, an Excess Kurtosis of 7.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.63%	11	0.73%	0.73%
-23.63% to -17.74%	19	1.26%	2.00%
-17.74% to -11.85%	33	2.20%	4.19%
-11.85% to -5.97%	134	8.92%	13.12%
-5.97% to -0.08%	613	40.81%	53.93%
-0.08% to 5.81%	493	32.82%	86.75%
5.81% to 11.69%	114	7.59%	94.34%
11.69% to 17.58%	42	2.80%	97.14%
17.58% to 23.47%	25	1.66%	98.80%
Greater than 23.47%	18	1.20%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.93%	7.77%	81.96%	68.27%
±2σ	-15.78%	15.62%	94.07%	95.45%
±3σ	-23.63%	23.47%	98.07%	99.73%

Descriptive Statistics

Mean	-0.080%
Standard Error	0.203%
Median	-0.544%
Mode	-2.437%
Standard Deviation	7.849%
Sample Variance	61.601%
Kurtosis	7.98
Skewness	0.90
Range	104.761%
Minimum	-40.657%
Maximum	64.104%
Count	1502

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.497%	679	45.21%	2.485%
Negative	-4.692%	821	54.66%	-2.565%
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 12.59 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -45.02%	0	0.00%	0.00%
-45.02% to -32.90%	3	0.87%	0.87%
-32.90% to -20.78%	10	2.90%	3.77%
-20.78% to -8.66%	39	11.30%	15.07%
-8.66% to 3.45%	134	38.84%	53.91%
3.45% to 15.57%	119	34.49%	88.41%
15.57% to 27.69%	25	7.25%	95.65%
27.69% to 39.80%	7	2.03%	97.68%
39.80% to 51.92%	1	0.29%	97.97%
Greater than 51.92%	7	2.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.70%	19.61%	82.32%	68.27%
±2σ	-28.86%	35.76%	95.36%	95.45%
±3σ	-45.02%	51.92%	97.97%	99.73%

Descriptive Statistics

Mean	3.452%
Standard Error	0.870%
Median	2.437%
Mode	-3.276%
Standard Deviation	16.156%
Sample Variance	261.025%
Kurtosis	12.59
Skewness	2.10
Range	167.538%
Minimum	-41.155%
Maximum	126.383%
Count	345

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.935%	204	59.13%	7.057%
Negative	-8.821%	141	40.87%	-3.605%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	13.11%
12	1 Year	13.63%
36	3 Years	13.10%
60	5 Years	14.66%
120	10 Years	13.62%
240	20 Years	15.16%
600	50 Years	21.38%

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

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -39.52%	0	0.00%	0.00%
-39.52% to -23.47%	0	0.00%	0.00%
-23.47% to -7.41%	0	0.00%	0.00%
-7.41% to 8.64%	40	11.56%	11.56%
8.64% to 24.70%	197	56.94%	68.50%
24.70% to 40.75%	57	16.47%	84.97%
40.75% to 56.81%	21	6.07%	91.04%
56.81% to 72.86%	15	4.34%	95.38%
72.86% to 88.92%	8	2.31%	97.69%
Greater than 88.92%	8	2.31%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.29%	46.10%	86.71%	68.27%
±2σ	-18.12%	67.51%	94.51%	95.45%
±3σ	-39.52%	88.92%	97.69%	99.73%

Descriptive Statistics

Mean	24.697%
Standard Error	1.151%
Median	17.373%
Mode	6.227%
Standard Deviation	21.407%
Sample Variance	458.267%
Kurtosis	7.23
Skewness	2.42
Range	150.657%
Minimum	3.282%
Maximum	153.939%
Count	346

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.697%	346	100.00%	24.697%
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 4.18 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -45.44%	2	0.58%	0.58%
-45.44% to -34.33%	5	1.45%	2.02%
-34.33% to -23.21%	5	1.45%	3.47%
-23.21% to -12.09%	42	12.14%	15.61%
-12.09% to -0.98%	138	39.88%	55.49%
-0.98% to 10.14%	103	29.77%	85.26%
10.14% to 21.25%	27	7.80%	93.06%
21.25% to 32.37%	14	4.05%	97.11%
32.37% to 43.48%	5	1.45%	98.55%
Greater than 43.48%	5	1.45%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-15.80%	13.84%	80.92%	68.27%
±2σ	-30.62%	28.66%	93.64%	95.45%
±3σ	-45.44%	43.48%	97.98%	99.73%

Descriptive Statistics

Mean	-0.978%
Standard Error	0.797%
Median	-2.335%
Mode	-7.626%
Standard Deviation	14.821%
Sample Variance	219.671%
Kurtosis	4.18
Skewness	0.89
Range	124.602%
Minimum	-55.890%
Maximum	68.712%
Count	346

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.472%	140	40.46%	4.642%
Negative	-9.439%	206	59.54%	-5.620%
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 9.19 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -84.61%	0	0.00%	0.00%
-84.61% to -60.86%	0	0.00%	0.00%
-60.86% to -37.11%	4	3.48%	3.48%
-37.11% to -13.36%	13	11.30%	14.78%
-13.36% to 10.40%	48	41.74%	56.52%
10.40% to 34.15%	36	31.30%	87.83%
34.15% to 57.90%	9	7.83%	95.65%
57.90% to 81.66%	1	0.87%	96.52%
81.66% to 105.41%	1	0.87%	97.39%
Greater than 105.41%	3	2.61%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.27%	42.07%	84.35%	68.27%
±2σ	-52.94%	73.74%	96.52%	95.45%
±3σ	-84.61%	105.41%	97.39%	99.73%

Descriptive Statistics

Mean	10.398%
Standard Error	2.953%
Median	6.508%
Mode	-52.730%
Standard Deviation	31.670%
Sample Variance	1002.990%
Kurtosis	9.19
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	-13.500%	44	38.26%	-5.165%
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 4.84 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -78.47%	0	0.00%	0.00%
-78.47% to -59.67%	1	0.86%	0.86%
-59.67% to -40.88%	3	2.59%	3.45%
-40.88% to -22.08%	17	14.66%	18.10%
-22.08% to -3.28%	41	35.34%	53.45%
-3.28% to 15.51%	38	32.76%	86.21%
15.51% to 34.31%	12	10.34%	96.55%
34.31% to 53.11%	0	0.00%	96.55%
53.11% to 71.90%	1	0.86%	97.41%
Greater than 71.90%	3	2.59%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.35%	21.78%	80.17%	68.27%
±2σ	-53.41%	46.84%	93.97%	95.45%
±3σ	-78.47%	71.90%	97.41%	99.73%

Descriptive Statistics

Mean	-3.284%
Standard Error	2.327%
Median	-5.460%
Mode	-63.478%
Standard Deviation	25.062%
Sample Variance	628.087%
Kurtosis	4.84
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	17.846%	47	40.52%	7.231%
Negative	-17.677%	69	59.48%	-10.515%
Zero	0.000%	0	0.00%	0.000%

AMZN Deep Analytical Report