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	3.59%	2.52%	2.96%	4.32%	3.39%	3.44%	146.08	156.33	141.21	161.73
Weekly	8.20%	7.38%	8.06%	10.29%	7.47%	7.56%	140.25	162.83	130.16	175.46
Monthly	15.35%	13.26%	14.69%	22.27%	15.54%	15.74%	129.37	176.53	110.75	206.21
Quarterly	32.15%	23.53%	31.22%	40.04%	26.92%	27.27%	115.45	197.80	88.21	258.91
Annual	—	—	—	—	53.84%	54.54%	88.21	258.91	51.48	443.57
Option Time Horizon	—	—	—	—	20.32%	20.59%	123.33	185.17	100.65	226.90

Calculation Notes: Computed at 2026-03-23T16:39 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $151.12 | Strike (K): $155.00 (first OTM call ≥ spot) | Observed Mid-Price: $10.9750 (Bid $10.6500 / Ask $11.3000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 53.84% (CRR binomial tree, American, n=20 steps) | BS IV: 54.24% (European Black-Scholes) | Yahoo IV (Annual): 54.54% → 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 Fairly Symmetrical with a Skewness factor of 0.50. Further, an Excess Kurtosis of 8.24 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.63%	16	0.79%	0.79%
-10.63% to -7.93%	26	1.29%	2.09%
-7.93% to -5.24%	69	3.43%	5.51%
-5.24% to -2.55%	211	10.48%	15.99%
-2.55% to 0.14%	674	33.47%	49.45%
0.14% to 2.83%	696	34.56%	84.01%
2.83% to 5.52%	232	11.52%	95.53%
5.52% to 8.21%	63	3.13%	98.66%
8.21% to 10.91%	12	0.60%	99.26%
Greater than 10.91%	15	0.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.45%	3.73%	79.10%	68.27%
±2σ	-7.04%	7.32%	95.03%	95.45%
±3σ	-10.63%	10.91%	98.46%	99.73%

Descriptive Statistics

Mean	0.140%
Standard Error	0.080%
Median	0.180%
Mode	1.961%
Standard Deviation	3.589%
Sample Variance	12.879%
Kurtosis	8.24
Skewness	0.50
Range	48.773%
Minimum	-19.367%
Maximum	29.406%
Count	2014

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.473%	1062	52.73%	1.304%
Negative	-2.471%	949	47.12%	-1.164%
Zero	0.000%	3	0.15%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	4.01%
20	1 Month	5.50%
60	1 Quarter	4.69%
250	1 Year	3.64%
750	3 Years	3.71%
1,250	5 Years	4.32%
2,500	10 Years	4.69%
5,000	20 Years	4.69%
12,500	50 Years	4.69%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.29. Further, an Excess Kurtosis of 9.57 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.10%	0	0.00%	0.00%
-3.10% to -1.21%	0	0.00%	0.00%
-1.21% to 0.68%	0	0.00%	0.00%
0.68% to 2.57%	383	19.01%	19.01%
2.57% to 4.46%	873	43.33%	62.33%
4.46% to 6.36%	429	21.29%	83.62%
6.36% to 8.25%	181	8.98%	92.61%
8.25% to 10.14%	83	4.12%	96.72%
10.14% to 12.03%	30	1.49%	98.21%
Greater than 12.03%	36	1.79%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.94%	6.99%	81.24%	68.27%
±2σ	-0.58%	9.51%	95.68%	95.45%
±3σ	-3.10%	12.03%	98.21%	99.73%

Descriptive Statistics

Mean	4.465%
Standard Error	0.056%
Median	3.833%
Mode	2.235%
Standard Deviation	2.522%
Sample Variance	6.361%
Kurtosis	9.57
Skewness	2.29
Range	27.441%
Minimum	0.993%
Maximum	28.434%
Count	2015

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.465%	2015	100.00%	4.465%
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 0.18. Further, an Excess Kurtosis of 3.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.85%	11	0.55%	0.55%
-8.85% to -6.63%	21	1.04%	1.59%
-6.63% to -4.41%	72	3.57%	5.16%
-4.41% to -2.19%	259	12.85%	18.01%
-2.19% to 0.04%	681	33.80%	51.81%
0.04% to 2.26%	622	30.87%	82.68%
2.26% to 4.48%	222	11.02%	93.70%
4.48% to 6.70%	77	3.82%	97.52%
6.70% to 8.92%	32	1.59%	99.11%
Greater than 8.92%	18	0.89%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.93%	3.00%	75.98%	68.27%
±2σ	-5.89%	5.96%	94.59%	95.45%
±3σ	-8.85%	8.92%	98.56%	99.73%

Descriptive Statistics

Mean	0.035%
Standard Error	0.066%
Median	-0.098%
Mode	-0.990%
Standard Deviation	2.963%
Sample Variance	8.780%
Kurtosis	3.10
Skewness	0.18
Range	33.276%
Minimum	-16.667%
Maximum	16.609%
Count	2015

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.226%	982	48.73%	1.085%
Negative	-2.053%	1031	51.17%	-1.050%
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 0.74. Further, an Excess Kurtosis of 4.44 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -23.85%	1	0.24%	0.24%
-23.85% to -17.71%	6	1.44%	1.67%
-17.71% to -11.56%	11	2.63%	4.31%
-11.56% to -5.41%	65	15.55%	19.86%
-5.41% to 0.73%	130	31.10%	50.96%
0.73% to 6.88%	135	32.30%	83.25%
6.88% to 13.02%	46	11.00%	94.26%
13.02% to 19.17%	18	4.31%	98.56%
19.17% to 25.32%	2	0.48%	99.04%
Greater than 25.32%	4	0.96%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.46%	8.93%	74.64%	68.27%
±2σ	-15.66%	17.12%	95.93%	95.45%
±3σ	-23.85%	25.32%	98.80%	99.73%

Descriptive Statistics

Mean	0.732%
Standard Error	0.401%
Median	0.613%
Mode	-1.801%
Standard Deviation	8.195%
Sample Variance	67.151%
Kurtosis	4.44
Skewness	0.74
Range	69.423%
Minimum	-26.480%
Maximum	42.943%
Count	418

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.185%	222	53.11%	3.285%
Negative	-5.472%	195	46.65%	-2.553%
Zero	0.000%	1	0.24%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	12.77%
12	1 Quarter	11.62%
52	1 Year	8.67%
156	3 Years	8.90%
260	5 Years	10.29%
520	10 Years	11.14%
1,040	20 Years	11.14%
2,600	50 Years	11.14%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.36. Further, an Excess Kurtosis of 8.67 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.26%	0	0.00%	0.00%
-10.26% to -4.72%	0	0.00%	0.00%
-4.72% to 0.81%	0	0.00%	0.00%
0.81% to 6.35%	77	18.38%	18.38%
6.35% to 11.88%	180	42.96%	61.34%
11.88% to 17.42%	102	24.34%	85.68%
17.42% to 22.95%	31	7.40%	93.08%
22.95% to 28.49%	13	3.10%	96.18%
28.49% to 34.02%	7	1.67%	97.85%
Greater than 34.02%	9	2.15%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	4.50%	19.27%	84.01%	68.27%
±2σ	-2.88%	26.64%	94.75%	95.45%
±3σ	-10.26%	34.02%	97.85%	99.73%

Descriptive Statistics

Mean	11.885%
Standard Error	0.361%
Median	9.837%
Mode	5.444%
Standard Deviation	7.380%
Sample Variance	54.468%
Kurtosis	8.67
Skewness	2.36
Range	59.270%
Minimum	2.165%
Maximum	61.435%
Count	419

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.885%	419	100.00%	11.885%
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 0.54. Further, an Excess Kurtosis of 3.16 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -24.19%	4	0.95%	0.95%
-24.19% to -18.14%	1	0.24%	1.19%
-18.14% to -12.09%	12	2.86%	4.06%
-12.09% to -6.04%	58	13.84%	17.90%
-6.04% to 0.00%	158	37.71%	55.61%
0.00% to 6.05%	110	26.25%	81.86%
6.05% to 12.10%	53	12.65%	94.51%
12.10% to 18.14%	12	2.86%	97.37%
18.14% to 24.19%	5	1.19%	98.57%
Greater than 24.19%	6	1.43%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.06%	8.07%	74.46%	68.27%
±2σ	-16.12%	16.13%	95.70%	95.45%
±3σ	-24.19%	24.19%	97.61%	99.73%

Descriptive Statistics

Mean	0.002%
Standard Error	0.394%
Median	-0.671%
Mode	-2.435%
Standard Deviation	8.063%
Sample Variance	65.019%
Kurtosis	3.16
Skewness	0.54
Range	64.880%
Minimum	-29.399%
Maximum	35.481%
Count	419

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	6.514%	186	44.39%	2.892%
Negative	-5.241%	231	55.13%	-2.889%
Zero	0.000%	2	0.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.47. Further, an Excess Kurtosis of 0.81 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -43.17%	0	0.00%	0.00%
-43.17% to -31.65%	0	0.00%	0.00%
-31.65% to -20.14%	6	6.25%	6.25%
-20.14% to -8.62%	16	16.67%	22.92%
-8.62% to 2.89%	28	29.17%	52.08%
2.89% to 14.40%	30	31.25%	83.33%
14.40% to 25.92%	11	11.46%	94.79%
25.92% to 37.43%	2	2.08%	96.88%
37.43% to 48.95%	2	2.08%	98.96%
Greater than 48.95%	1	1.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.46%	18.24%	71.88%	68.27%
±2σ	-27.82%	33.60%	94.79%	95.45%
±3σ	-43.17%	48.95%	98.96%	99.73%

Descriptive Statistics

Mean	2.890%
Standard Error	1.567%
Median	2.684%
Mode	-31.248%
Standard Deviation	15.353%
Sample Variance	235.722%
Kurtosis	0.81
Skewness	0.47
Range	81.614%
Minimum	-31.248%
Maximum	50.366%
Count	96

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	13.256%	54	56.25%	7.456%
Negative	-10.438%	42	43.75%	-4.567%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	20.45%
12	1 Year	18.11%
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 1.87. Further, an Excess Kurtosis of 5.29 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.04%	0	0.00%	0.00%
-12.04% to -2.10%	0	0.00%	0.00%
-2.10% to 7.85%	0	0.00%	0.00%
7.85% to 17.79%	21	21.65%	21.65%
17.79% to 27.73%	37	38.14%	59.79%
27.73% to 37.67%	23	23.71%	83.51%
37.67% to 47.61%	9	9.28%	92.78%
47.61% to 57.55%	3	3.09%	95.88%
57.55% to 67.49%	2	2.06%	97.94%
Greater than 67.49%	2	2.06%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	14.47%	40.98%	82.47%	68.27%
±2σ	1.22%	54.24%	95.88%	95.45%
±3σ	-12.04%	67.49%	97.94%	99.73%

Descriptive Statistics

Mean	27.729%
Standard Error	1.346%
Median	25.710%
Mode	9.482%
Standard Deviation	13.255%
Sample Variance	175.685%
Kurtosis	5.29
Skewness	1.87
Range	80.518%
Minimum	9.482%
Maximum	90.000%
Count	97

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	27.729%	97	100.00%	27.729%
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 0.51. Further, an Excess Kurtosis of 0.37 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.35%	0	0.00%	0.00%
-44.35% to -33.33%	1	1.03%	1.03%
-33.33% to -22.32%	2	2.06%	3.09%
-22.32% to -11.30%	19	19.59%	22.68%
-11.30% to -0.29%	29	29.90%	52.58%
-0.29% to 10.72%	25	25.77%	78.35%
10.72% to 21.74%	14	14.43%	92.78%
21.74% to 32.75%	5	5.15%	97.94%
32.75% to 43.77%	1	1.03%	98.97%
Greater than 43.77%	1	1.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.98%	14.40%	71.13%	68.27%
±2σ	-29.66%	29.08%	93.81%	95.45%
±3σ	-44.35%	43.77%	98.97%	99.73%

Descriptive Statistics

Mean	-0.290%
Standard Error	1.491%
Median	-2.215%
Mode	-6.658%
Standard Deviation	14.686%
Sample Variance	215.685%
Kurtosis	0.37
Skewness	0.51
Range	78.028%
Minimum	-33.953%
Maximum	44.075%
Count	97

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	12.216%	45	46.39%	5.667%
Negative	-11.113%	52	53.61%	-5.957%
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 0.54. Further, an Excess Kurtosis of 0.45 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -86.70%	0	0.00%	0.00%
-86.70% to -62.59%	0	0.00%	0.00%
-62.59% to -38.47%	2	6.25%	6.25%
-38.47% to -14.36%	7	21.88%	28.12%
-14.36% to 9.76%	7	21.88%	50.00%
9.76% to 33.87%	10	31.25%	81.25%
33.87% to 57.98%	4	12.50%	93.75%
57.98% to 82.10%	1	3.12%	96.88%
82.10% to 106.21%	1	3.12%	100.00%
Greater than 106.21%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-22.40%	41.91%	68.75%	68.27%
±2σ	-54.55%	74.06%	93.75%	95.45%
±3σ	-86.70%	106.21%	100.00%	99.73%

Descriptive Statistics

Mean	9.755%
Standard Error	5.684%
Median	8.091%
Mode	-47.811%
Standard Deviation	32.153%
Sample Variance	1033.822%
Kurtosis	0.45
Skewness	0.54
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	-21.401%	12	37.50%	-8.025%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	34.92%
12	3 Years	38.74%
20	5 Years	40.04%
40	10 Years	45.81%
80	20 Years	45.81%
200	50 Years	45.81%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.99. Further, an Excess Kurtosis of 0.46 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -13.96%	0	0.00%	0.00%
-13.96% to 3.69%	0	0.00%	0.00%
3.69% to 21.34%	0	0.00%	0.00%
21.34% to 38.98%	7	21.21%	21.21%
38.98% to 56.63%	12	36.36%	57.58%
56.63% to 74.28%	8	24.24%	81.82%
74.28% to 91.93%	2	6.06%	87.88%
91.93% to 109.58%	2	6.06%	93.94%
109.58% to 127.23%	2	6.06%	100.00%
Greater than 127.23%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	33.10%	80.17%	75.76%	68.27%
±2σ	9.57%	103.70%	93.94%	95.45%
±3σ	-13.96%	127.23%	100.00%	99.73%

Descriptive Statistics

Mean	56.634%
Standard Error	4.096%
Median	53.022%
Mode	21.624%
Standard Deviation	23.532%
Sample Variance	553.749%
Kurtosis	0.46
Skewness	0.99
Range	94.131%
Minimum	21.624%
Maximum	115.755%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	56.634%	33	100.00%	56.634%
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 1.09. Further, an Excess Kurtosis of 1.36 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -94.32%	0	0.00%	0.00%
-94.32% to -70.91%	0	0.00%	0.00%
-70.91% to -47.49%	1	3.03%	3.03%
-47.49% to -24.07%	6	18.18%	21.21%
-24.07% to -0.66%	15	45.45%	66.67%
-0.66% to 22.76%	2	6.06%	72.73%
22.76% to 46.17%	7	21.21%	93.94%
46.17% to 69.59%	1	3.03%	96.97%
69.59% to 93.00%	1	3.03%	100.00%
Greater than 93.00%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-31.88%	30.56%	75.76%	68.27%
±2σ	-63.10%	61.78%	93.94%	95.45%
±3σ	-94.32%	93.00%	100.00%	99.73%

Descriptive Statistics

Mean	-0.658%
Standard Error	5.435%
Median	-7.996%
Mode	-48.995%
Standard Deviation	31.221%
Sample Variance	974.752%
Kurtosis	1.36
Skewness	1.09
Range	141.878%
Minimum	-48.995%
Maximum	92.883%
Count	33

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	34.961%	11	33.33%	11.654%
Negative	-18.468%	22	66.67%	-12.312%
Zero	0.000%	0	0.00%	0.000%

ZS Deep Analytical Report