ZS - 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	3.58%	2.53%	2.96%	4.31%	4.38%	4.44%	152.91	166.89	146.37	174.36
Weekly	8.17%	7.38%	8.02%	10.26%	9.63%	9.77%	145.08	175.90	131.76	193.69
Monthly	15.32%	13.15%	14.52%	22.27%	20.05%	20.35%	130.73	195.21	106.98	238.55
Quarterly	32.02%	24.00%	31.03%	39.57%	34.73%	35.24%	112.88	226.08	79.77	319.94
Annual	—	—	—	—	69.45%	70.48%	79.77	319.94	39.83	640.75
Option Time Horizon	—	—	—	—	22.70%	23.04%	127.31	200.46	101.45	251.55

Calculation Notes: Computed at 2026-02-22T14:22 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $159.75 | Strike (K): $170.00 (first OTM call ≥ spot) | Observed Mid-Price: $10.7250 (Bid $9.8500 / Ask $11.6000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 69.45% (CRR binomial tree, American, n=20 steps) | BS IV: 70.22% (European Black-Scholes) | Yahoo IV (Annual): 70.48% → 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 0.54. Further, an Excess Kurtosis of 8.36 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.59%	15	0.75%	0.75%
-10.59% to -7.91%	25	1.25%	2.01%
-7.91% to -5.23%	69	3.46%	5.47%
-5.23% to -2.54%	214	10.74%	16.21%
-2.54% to 0.14%	664	33.32%	49.52%
0.14% to 2.83%	689	34.57%	84.09%
2.83% to 5.51%	228	11.44%	95.53%
5.51% to 8.20%	62	3.11%	98.65%
8.20% to 10.88%	12	0.60%	99.25%
Greater than 10.88%	15	0.75%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.44%	3.72%	79.18%	68.27%
±2σ	-7.02%	7.30%	95.08%	95.45%
±3σ	-10.59%	10.88%	98.49%	99.73%

Descriptive Statistics

Mean	0.143%
Standard Error	0.080%
Median	0.181%
Mode	1.961%
Standard Deviation	3.579%
Sample Variance	12.807%
Kurtosis	8.36
Skewness	0.54
Range	48.773%
Minimum	-19.367%
Maximum	29.406%
Count	1993

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.470%	1051	52.73%	1.303%
Negative	-2.462%	939	47.12%	-1.160%
Zero	0.000%	3	0.15%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	5.18%
20	1 Month	5.28%
60	1 Quarter	3.80%
250	1 Year	3.54%
750	3 Years	3.69%
1,250	5 Years	4.31%
2,500	10 Years	4.68%
5,000	20 Years	4.68%
12,500	50 Years	4.68%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.30. Further, an Excess Kurtosis of 9.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.12%	0	0.00%	0.00%
-3.12% to -1.23%	0	0.00%	0.00%
-1.23% to 0.67%	0	0.00%	0.00%
0.67% to 2.56%	379	19.01%	19.01%
2.56% to 4.46%	863	43.28%	62.29%
4.46% to 6.35%	426	21.36%	83.65%
6.35% to 8.25%	178	8.93%	92.58%
8.25% to 10.14%	84	4.21%	96.79%
10.14% to 12.04%	28	1.40%	98.19%
Greater than 12.04%	36	1.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.93%	6.98%	81.49%	68.27%
±2σ	-0.59%	9.51%	95.69%	95.45%
±3σ	-3.12%	12.04%	98.19%	99.73%

Descriptive Statistics

Mean	4.457%
Standard Error	0.057%
Median	3.809%
Mode	2.235%
Standard Deviation	2.526%
Sample Variance	6.380%
Kurtosis	9.63
Skewness	2.30
Range	27.441%
Minimum	0.993%
Maximum	28.434%
Count	1994

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.457%	1994	100.00%	4.457%
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 0.17. Further, an Excess Kurtosis of 3.12 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.84%	11	0.55%	0.55%
-8.84% to -6.62%	22	1.10%	1.65%
-6.62% to -4.40%	69	3.46%	5.12%
-4.40% to -2.18%	257	12.89%	18.00%
-2.18% to 0.04%	675	33.85%	51.86%
0.04% to 2.26%	614	30.79%	82.65%
2.26% to 4.48%	220	11.03%	93.68%
4.48% to 6.70%	77	3.86%	97.54%
6.70% to 8.92%	32	1.60%	99.15%
Greater than 8.92%	17	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.92%	3.00%	75.93%	68.27%
±2σ	-5.88%	5.96%	94.58%	95.45%
±3σ	-8.84%	8.92%	98.60%	99.73%

Descriptive Statistics

Mean	0.040%
Standard Error	0.066%
Median	-0.094%
Mode	-0.990%
Standard Deviation	2.959%
Sample Variance	8.755%
Kurtosis	3.12
Skewness	0.17
Range	33.276%
Minimum	-16.667%
Maximum	16.609%
Count	1994

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.226%	973	48.80%	1.086%
Negative	-2.047%	1019	51.10%	-1.046%
Zero	0.000%	2	0.10%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.77. Further, an Excess Kurtosis of 4.50 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.77%	2	0.48%	0.48%
-23.77% to -17.64%	4	0.97%	1.45%
-17.64% to -11.51%	11	2.66%	4.11%
-11.51% to -5.38%	66	15.94%	20.05%
-5.38% to 0.75%	128	30.92%	50.97%
0.75% to 6.88%	134	32.37%	83.33%
6.88% to 13.01%	45	10.87%	94.20%
13.01% to 19.14%	18	4.35%	98.55%
19.14% to 25.27%	2	0.48%	99.03%
Greater than 25.27%	4	0.97%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.43%	8.92%	74.64%	68.27%
±2σ	-15.60%	17.09%	96.14%	95.45%
±3σ	-23.77%	25.26%	98.55%	99.73%

Descriptive Statistics

Mean	0.746%
Standard Error	0.402%
Median	0.613%
Mode	-1.801%
Standard Deviation	8.173%
Sample Variance	66.797%
Kurtosis	4.50
Skewness	0.77
Range	69.423%
Minimum	-26.480%
Maximum	42.943%
Count	414

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.181%	220	53.14%	3.285%
Negative	-5.446%	193	46.62%	-2.539%
Zero	0.000%	1	0.24%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	13.31%
12	1 Quarter	7.93%
52	1 Year	8.19%
156	3 Years	8.80%
260	5 Years	10.26%
520	10 Years	11.10%
1,040	20 Years	11.10%
2,600	50 Years	11.10%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.39. Further, an Excess Kurtosis of 8.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.29%	0	0.00%	0.00%
-10.29% to -4.76%	0	0.00%	0.00%
-4.76% to 0.78%	0	0.00%	0.00%
0.78% to 6.31%	76	18.31%	18.31%
6.31% to 11.84%	177	42.65%	60.96%
11.84% to 17.37%	103	24.82%	85.78%
17.37% to 22.90%	31	7.47%	93.25%
22.90% to 28.43%	10	2.41%	95.66%
28.43% to 33.96%	9	2.17%	97.83%
Greater than 33.96%	9	2.17%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.46%	19.21%	83.86%	68.27%
±2σ	-2.91%	26.59%	94.70%	95.45%
±3σ	-10.29%	33.96%	97.83%	99.73%

Descriptive Statistics

Mean	11.838%
Standard Error	0.362%
Median	9.837%
Mode	5.444%
Standard Deviation	7.375%
Sample Variance	54.393%
Kurtosis	8.85
Skewness	2.39
Range	59.270%
Minimum	2.165%
Maximum	61.435%
Count	415

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.838%	415	100.00%	11.838%
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 0.52. Further, an Excess Kurtosis of 3.21 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.07%	4	0.96%	0.96%
-24.07% to -18.05%	1	0.24%	1.20%
-18.05% to -12.04%	13	3.13%	4.34%
-12.04% to -6.03%	58	13.98%	18.31%
-6.03% to -0.01%	153	36.87%	55.18%
-0.01% to 6.00%	111	26.75%	81.93%
6.00% to 12.01%	53	12.77%	94.70%
12.01% to 18.02%	12	2.89%	97.59%
18.02% to 24.04%	4	0.96%	98.55%
Greater than 24.04%	6	1.45%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.03%	8.00%	74.46%	68.27%
±2σ	-16.05%	16.02%	95.90%	95.45%
±3σ	-24.06%	24.04%	97.59%	99.73%

Descriptive Statistics

Mean	-0.014%
Standard Error	0.394%
Median	-0.671%
Mode	-2.435%
Standard Deviation	8.017%
Sample Variance	64.272%
Kurtosis	3.21
Skewness	0.52
Range	64.880%
Minimum	-29.399%
Maximum	35.481%
Count	415

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.475%	184	44.34%	2.871%
Negative	-5.229%	229	55.18%	-2.885%
Zero	0.000%	2	0.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.50. Further, an Excess Kurtosis of 0.78 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -43.00%	0	0.00%	0.00%
-43.00% to -31.51%	0	0.00%	0.00%
-31.51% to -20.02%	6	6.32%	6.32%
-20.02% to -8.54%	16	16.84%	23.16%
-8.54% to 2.95%	28	29.47%	52.63%
2.95% to 14.44%	29	30.53%	83.16%
14.44% to 25.93%	11	11.58%	94.74%
25.93% to 37.42%	2	2.11%	96.84%
37.42% to 48.91%	2	2.11%	98.95%
Greater than 48.91%	1	1.05%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.37%	18.27%	71.58%	68.27%
±2σ	-27.68%	33.59%	94.74%	95.45%
±3σ	-43.00%	48.91%	98.95%	99.73%

Descriptive Statistics

Mean	2.954%
Standard Error	1.572%
Median	2.673%
Mode	-31.248%
Standard Deviation	15.319%
Sample Variance	234.667%
Kurtosis	0.78
Skewness	0.50
Range	81.614%
Minimum	-31.248%
Maximum	50.366%
Count	95

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.447%	53	55.79%	7.502%
Negative	-10.286%	42	44.21%	-4.548%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	15.86%
12	1 Year	17.43%
36	3 Years	20.43%
60	5 Years	22.27%
120	10 Years	24.22%
240	20 Years	24.22%
600	50 Years	24.22%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.93. Further, an Excess Kurtosis of 5.65 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.77%	0	0.00%	0.00%
-11.77% to -1.91%	0	0.00%	0.00%
-1.91% to 7.96%	0	0.00%	0.00%
7.96% to 17.82%	20	20.83%	20.83%
17.82% to 27.68%	38	39.58%	60.42%
27.68% to 37.54%	23	23.96%	84.38%
37.54% to 47.40%	8	8.33%	92.71%
47.40% to 57.27%	3	3.12%	95.83%
57.27% to 67.13%	2	2.08%	97.92%
Greater than 67.13%	2	2.08%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	14.53%	40.83%	82.29%	68.27%
±2σ	1.38%	53.98%	95.83%	95.45%
±3σ	-11.77%	67.13%	97.92%	99.73%

Descriptive Statistics

Mean	27.680%
Standard Error	1.342%
Median	25.731%
Mode	9.482%
Standard Deviation	13.149%
Sample Variance	172.889%
Kurtosis	5.65
Skewness	1.93
Range	80.518%
Minimum	9.482%
Maximum	90.000%
Count	96

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	27.680%	96	100.00%	27.680%
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 0.44. Further, an Excess Kurtosis of 0.25 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -43.92%	0	0.00%	0.00%
-43.92% to -33.04%	1	1.04%	1.04%
-33.04% to -22.15%	2	2.08%	3.12%
-22.15% to -11.26%	19	19.79%	22.92%
-11.26% to -0.37%	28	29.17%	52.08%
-0.37% to 10.52%	25	26.04%	78.12%
10.52% to 21.41%	13	13.54%	91.67%
21.41% to 32.30%	7	7.29%	98.96%
32.30% to 43.19%	0	0.00%	98.96%
Greater than 43.19%	1	1.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.89%	14.15%	69.79%	68.27%
±2σ	-29.41%	28.67%	94.79%	95.45%
±3σ	-43.93%	43.19%	98.96%	99.73%

Descriptive Statistics

Mean	-0.368%
Standard Error	1.482%
Median	-2.123%
Mode	-6.658%
Standard Deviation	14.519%
Sample Variance	210.803%
Kurtosis	0.25
Skewness	0.44
Range	78.028%
Minimum	-33.953%
Maximum	44.075%
Count	96

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.975%	45	46.88%	5.613%
Negative	-11.260%	51	53.12%	-5.982%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.55. Further, an Excess Kurtosis of 0.49 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -86.19%	0	0.00%	0.00%
-86.19% to -62.17%	0	0.00%	0.00%
-62.17% to -38.15%	2	6.25%	6.25%
-38.15% to -14.14%	7	21.88%	28.12%
-14.14% to 9.88%	7	21.88%	50.00%
9.88% to 33.90%	10	31.25%	81.25%
33.90% to 57.92%	4	12.50%	93.75%
57.92% to 81.93%	1	3.12%	96.88%
81.93% to 105.95%	1	3.12%	100.00%
Greater than 105.95%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-22.14%	41.90%	71.88%	68.27%
±2σ	-54.17%	73.93%	93.75%	95.45%
±3σ	-86.19%	105.95%	100.00%	99.73%

Descriptive Statistics

Mean	9.881%
Standard Error	5.661%
Median	8.092%
Mode	-47.811%
Standard Deviation	32.023%
Sample Variance	1025.479%
Kurtosis	0.49
Skewness	0.55
Range	141.386%
Minimum	-47.811%
Maximum	93.575%
Count	32

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	28.448%	20	62.50%	17.780%
Negative	-21.063%	12	37.50%	-7.899%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	32.53%
12	3 Years	37.94%
20	5 Years	39.57%
40	10 Years	45.51%
80	20 Years	45.51%
200	50 Years	45.51%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.94. Further, an Excess Kurtosis of 0.35 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.88%	0	0.00%	0.00%
-15.88% to 2.12%	0	0.00%	0.00%
2.12% to 20.11%	0	0.00%	0.00%
20.11% to 38.11%	8	24.24%	24.24%
38.11% to 56.11%	10	30.30%	54.55%
56.11% to 74.11%	9	27.27%	81.82%
74.11% to 92.11%	3	9.09%	90.91%
92.11% to 110.10%	1	3.03%	93.94%
110.10% to 128.10%	2	6.06%	100.00%
Greater than 128.10%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	32.11%	80.11%	75.76%	68.27%
±2σ	8.12%	104.10%	93.94%	95.45%
±3σ	-15.88%	128.10%	100.00%	99.73%

Descriptive Statistics

Mean	56.110%
Standard Error	4.177%
Median	53.022%
Mode	21.624%
Standard Deviation	23.997%
Sample Variance	575.860%
Kurtosis	0.35
Skewness	0.94
Range	94.131%
Minimum	21.624%
Maximum	115.755%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	56.110%	33	100.00%	56.110%
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 1.12. Further, an Excess Kurtosis of 1.48 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -93.94%	0	0.00%	0.00%
-93.94% to -70.67%	0	0.00%	0.00%
-70.67% to -47.40%	1	3.03%	3.03%
-47.40% to -24.13%	6	18.18%	21.21%
-24.13% to -0.86%	15	45.45%	66.67%
-0.86% to 22.41%	2	6.06%	72.73%
22.41% to 45.68%	7	21.21%	93.94%
45.68% to 68.95%	1	3.03%	96.97%
68.95% to 92.22%	0	0.00%	96.97%
Greater than 92.22%	1	3.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.89%	30.17%	78.79%	68.27%
±2σ	-62.91%	61.19%	93.94%	95.45%
±3σ	-93.94%	92.22%	96.97%	99.73%

Descriptive Statistics

Mean	-0.860%
Standard Error	5.401%
Median	-7.996%
Mode	-48.995%
Standard Deviation	31.026%
Sample Variance	962.603%
Kurtosis	1.48
Skewness	1.12
Range	141.878%
Minimum	-48.995%
Maximum	92.883%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	34.354%	11	33.33%	11.451%
Negative	-18.468%	22	66.67%	-12.312%
Zero	0.000%	0	0.00%	0.000%

ZS Deep Analytical Report