BSX - 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	2.38%	1.96%	2.00%	2.08%	2.71%	2.90%	67.69	71.46	65.88	73.43
Weekly	5.30%	5.16%	5.35%	4.79%	5.97%	6.38%	65.52	73.83	61.72	78.38
Monthly	9.97%	10.87%	10.02%	10.09%	12.44%	13.29%	61.42	78.76	54.23	89.19
Quarterly	18.42%	20.81%	19.38%	17.08%	21.54%	23.01%	56.07	86.27	45.21	107.00
Annual	—	—	—	—	43.08%	46.02%	45.21	107.00	29.38	164.63
Option Time Horizon	—	—	—	—	16.26%	17.37%	59.11	81.83	50.24	96.28

Calculation Notes: Computed at 2026-03-23T16:33 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $69.55 | Strike (K): $70.00 (first OTM call ≥ spot) | Observed Mid-Price: $4.4500 (Bid $4.2000 / Ask $4.7000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 43.08% (CRR binomial tree, American, n=20 steps) | BS IV: 42.89% (European Black-Scholes) | Yahoo IV (Annual): 46.02% → 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 -0.21. Further, an Excess Kurtosis of 11.78 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -7.07%	54	0.63%	0.63%
-7.07% to -5.29%	88	1.03%	1.67%
-5.29% to -3.50%	232	2.72%	4.39%
-3.50% to -1.72%	953	11.19%	15.58%
-1.72% to 0.06%	3097	36.36%	51.94%
0.06% to 1.84%	2745	32.23%	84.16%
1.84% to 3.62%	918	10.78%	94.94%
3.62% to 5.41%	266	3.12%	98.06%
5.41% to 7.19%	92	1.08%	99.14%
Greater than 7.19%	73	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.31%	2.44%	79.77%	68.27%
±2σ	-4.69%	4.81%	95.10%	95.45%
±3σ	-7.07%	7.19%	98.51%	99.73%

Descriptive Statistics

Mean	0.061%
Standard Error	0.026%
Median	0.000%
Mode	0.000%
Standard Deviation	2.376%
Sample Variance	5.646%
Kurtosis	11.78
Skewness	-0.21
Range	52.828%
Minimum	-26.699%
Maximum	26.129%
Count	8518

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.691%	4160	48.84%	0.826%
Negative	-1.596%	4081	47.91%	-0.765%
Zero	0.000%	277	3.25%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.65%
20	1 Month	2.77%
60	1 Quarter	2.82%
250	1 Year	2.37%
750	3 Years	1.98%
1,250	5 Years	2.08%
2,500	10 Years	2.19%
5,000	20 Years	2.65%
12,500	50 Years	3.00%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.09. Further, an Excess Kurtosis of 19.57 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.01%	0	0.00%	0.00%
-3.01% to -1.54%	0	0.00%	0.00%
-1.54% to -0.08%	0	0.00%	0.00%
-0.08% to 1.39%	1382	16.22%	16.22%
1.39% to 2.86%	4008	47.05%	63.27%
2.86% to 4.32%	1885	22.13%	85.40%
4.32% to 5.79%	698	8.19%	93.59%
5.79% to 7.26%	263	3.09%	96.68%
7.26% to 8.72%	119	1.40%	98.07%
Greater than 8.72%	164	1.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.90%	4.81%	85.57%	68.27%
±2σ	-1.05%	6.77%	95.86%	95.45%
±3σ	-3.01%	8.73%	98.07%	99.73%

Descriptive Statistics

Mean	2.857%
Standard Error	0.021%
Median	2.365%
Mode	1.754%
Standard Deviation	1.956%
Sample Variance	3.826%
Kurtosis	19.57
Skewness	3.09
Range	32.460%
Minimum	0.334%
Maximum	32.794%
Count	8519

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.857%	8519	100.00%	2.857%
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 0.13. Further, an Excess Kurtosis of 5.70 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.99%	64	0.75%	0.75%
-5.99% to -4.49%	102	1.20%	1.95%
-4.49% to -2.99%	280	3.29%	5.24%
-2.99% to -1.49%	1003	11.77%	17.01%
-1.49% to 0.02%	2937	34.48%	51.48%
0.02% to 1.52%	2708	31.79%	83.27%
1.52% to 3.02%	988	11.60%	94.87%
3.02% to 4.52%	267	3.13%	98.00%
4.52% to 6.02%	104	1.22%	99.23%
Greater than 6.02%	66	0.77%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-1.99%	2.02%	77.65%	68.27%
±2σ	-3.99%	4.02%	94.81%	95.45%
±3σ	-5.99%	6.02%	98.47%	99.73%

Descriptive Statistics

Mean	0.016%
Standard Error	0.022%
Median	0.000%
Mode	0.000%
Standard Deviation	2.002%
Sample Variance	4.009%
Kurtosis	5.70
Skewness	0.13
Range	36.061%
Minimum	-17.083%
Maximum	18.978%
Count	8519

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.461%	4133	48.52%	0.709%
Negative	-1.443%	4090	48.01%	-0.693%
Zero	0.000%	296	3.48%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of -0.18. Further, an Excess Kurtosis of 4.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.60%	17	0.96%	0.96%
-15.60% to -11.63%	17	0.96%	1.93%
-11.63% to -7.65%	50	2.83%	4.76%
-7.65% to -3.68%	212	12.01%	16.77%
-3.68% to 0.30%	589	33.37%	50.14%
0.30% to 4.27%	591	33.48%	83.63%
4.27% to 8.25%	194	10.99%	94.62%
8.25% to 12.22%	60	3.40%	98.02%
12.22% to 16.19%	20	1.13%	99.15%
Greater than 16.19%	15	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.00%	5.60%	77.05%	68.27%
±2σ	-10.30%	10.90%	94.73%	95.45%
±3σ	-15.60%	16.19%	98.19%	99.73%

Descriptive Statistics

Mean	0.297%
Standard Error	0.126%
Median	0.292%
Mode	0.000%
Standard Deviation	5.299%
Sample Variance	28.077%
Kurtosis	4.39
Skewness	-0.18
Range	63.176%
Minimum	-35.765%
Maximum	27.411%
Count	1765

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.822%	928	52.58%	2.010%
Negative	-3.704%	816	46.23%	-1.712%
Zero	0.000%	21	1.19%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	5.97%
12	1 Quarter	6.88%
52	1 Year	5.59%
156	3 Years	4.53%
260	5 Years	4.79%
520	10 Years	5.06%
1,040	20 Years	6.06%
2,600	50 Years	7.09%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.38. Further, an Excess Kurtosis of 21.75 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.11%	0	0.00%	0.00%
-8.11% to -4.24%	0	0.00%	0.00%
-4.24% to -0.37%	0	0.00%	0.00%
-0.37% to 3.50%	254	14.38%	14.38%
3.50% to 7.37%	872	49.38%	63.76%
7.37% to 11.24%	389	22.03%	85.79%
11.24% to 15.11%	142	8.04%	93.83%
15.11% to 18.98%	56	3.17%	97.00%
18.98% to 22.85%	21	1.19%	98.19%
Greater than 22.85%	32	1.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.21%	12.53%	87.37%	68.27%
±2σ	-2.95%	17.69%	95.87%	95.45%
±3σ	-8.11%	22.85%	98.19%	99.73%

Descriptive Statistics

Mean	7.369%
Standard Error	0.123%
Median	6.106%
Mode	9.091%
Standard Deviation	5.161%
Sample Variance	26.641%
Kurtosis	21.75
Skewness	3.38
Range	67.702%
Minimum	1.068%
Maximum	68.770%
Count	1766

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.369%	1766	100.00%	7.369%
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 1.45. Further, an Excess Kurtosis of 10.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.02%	7	0.40%	0.40%
-16.02% to -12.00%	15	0.85%	1.25%
-12.00% to -7.99%	50	2.83%	4.08%
-7.99% to -3.98%	220	12.46%	16.53%
-3.98% to 0.04%	666	37.71%	54.25%
0.04% to 4.05%	538	30.46%	84.71%
4.05% to 8.07%	170	9.63%	94.34%
8.07% to 12.08%	60	3.40%	97.73%
12.08% to 16.10%	22	1.25%	98.98%
Greater than 16.10%	18	1.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.31%	5.39%	79.39%	68.27%
±2σ	-10.67%	10.75%	94.85%	95.45%
±3σ	-16.02%	16.10%	98.58%	99.73%

Descriptive Statistics

Mean	0.040%
Standard Error	0.127%
Median	-0.289%
Mode	0.000%
Standard Deviation	5.353%
Sample Variance	28.650%
Kurtosis	10.39
Skewness	1.45
Range	74.898%
Minimum	-20.319%
Maximum	54.579%
Count	1766

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.985%	812	45.98%	1.832%
Negative	-3.382%	936	53.00%	-1.792%
Zero	0.000%	18	1.02%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.18. Further, an Excess Kurtosis of 1.98 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -28.75%	1	0.25%	0.25%
-28.75% to -21.27%	9	2.22%	2.46%
-21.27% to -13.79%	13	3.20%	5.67%
-13.79% to -6.31%	62	15.27%	20.94%
-6.31% to 1.17%	118	29.06%	50.00%
1.17% to 8.64%	124	30.54%	80.54%
8.64% to 16.12%	53	13.05%	93.60%
16.12% to 23.60%	20	4.93%	98.52%
23.60% to 31.08%	5	1.23%	99.75%
Greater than 31.08%	1	0.25%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-8.80%	11.14%	71.43%	68.27%
±2σ	-18.78%	21.11%	94.09%	95.45%
±3σ	-28.75%	31.08%	99.51%	99.73%

Descriptive Statistics

Mean	1.166%
Standard Error	0.495%
Median	1.127%
Mode	-1.210%
Standard Deviation	9.971%
Sample Variance	99.413%
Kurtosis	1.98
Skewness	0.18
Range	84.974%
Minimum	-31.672%
Maximum	53.302%
Count	406

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.977%	224	55.17%	4.401%
Negative	-7.255%	181	44.58%	-3.234%
Zero	0.000%	1	0.25%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	16.58%
12	1 Year	11.87%
36	3 Years	9.86%
60	5 Years	10.09%
120	10 Years	10.83%
240	20 Years	12.72%
600	50 Years	15.19%

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

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.71%	0	0.00%	0.00%
-15.71% to -7.55%	0	0.00%	0.00%
-7.55% to 0.60%	0	0.00%	0.00%
0.60% to 8.76%	71	17.44%	17.44%
8.76% to 16.91%	187	45.95%	63.39%
16.91% to 25.06%	88	21.62%	85.01%
25.06% to 33.22%	34	8.35%	93.37%
33.22% to 41.37%	11	2.70%	96.07%
41.37% to 49.52%	9	2.21%	98.28%
Greater than 49.52%	7	1.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	6.04%	27.78%	84.28%	68.27%
±2σ	-4.84%	38.65%	94.59%	95.45%
±3σ	-15.71%	49.52%	98.28%	99.73%

Descriptive Statistics

Mean	16.909%
Standard Error	0.539%
Median	13.878%
Mode	8.889%
Standard Deviation	10.872%
Sample Variance	118.207%
Kurtosis	8.30
Skewness	2.36
Range	83.488%
Minimum	4.209%
Maximum	87.697%
Count	407

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	16.909%	407	100.00%	16.909%
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 0.80. Further, an Excess Kurtosis of 2.03 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -30.21%	1	0.25%	0.25%
-30.21% to -22.69%	1	0.25%	0.49%
-22.69% to -15.18%	12	2.95%	3.44%
-15.18% to -7.66%	67	16.46%	19.90%
-7.66% to -0.15%	140	34.40%	54.30%
-0.15% to 7.37%	108	26.54%	80.84%
7.37% to 14.89%	52	12.78%	93.61%
14.89% to 22.40%	12	2.95%	96.56%
22.40% to 29.92%	8	1.97%	98.53%
Greater than 29.92%	6	1.47%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-10.17%	9.88%	74.94%	68.27%
±2σ	-20.19%	19.90%	94.84%	95.45%
±3σ	-30.21%	29.92%	98.28%	99.73%

Descriptive Statistics

Mean	-0.146%
Standard Error	0.497%
Median	-1.047%
Mode	0.000%
Standard Deviation	10.021%
Sample Variance	100.424%
Kurtosis	2.03
Skewness	0.80
Range	77.851%
Minimum	-32.308%
Maximum	45.543%
Count	407

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	8.266%	182	44.72%	3.696%
Negative	-7.044%	222	54.55%	-3.842%
Zero	0.000%	3	0.74%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.44. Further, an Excess Kurtosis of 1.86 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -51.65%	0	0.00%	0.00%
-51.65% to -37.83%	1	0.74%	0.74%
-37.83% to -24.02%	9	6.67%	7.41%
-24.02% to -10.20%	16	11.85%	19.26%
-10.20% to 3.61%	44	32.59%	51.85%
3.61% to 17.43%	42	31.11%	82.96%
17.43% to 31.24%	15	11.11%	94.07%
31.24% to 45.06%	2	1.48%	95.56%
45.06% to 58.87%	5	3.70%	99.26%
Greater than 58.87%	1	0.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.81%	22.03%	74.81%	68.27%
±2σ	-33.23%	40.45%	91.85%	95.45%
±3σ	-51.65%	58.87%	99.26%	99.73%

Descriptive Statistics

Mean	3.612%
Standard Error	1.585%
Median	2.797%
Mode	-50.231%
Standard Deviation	18.420%
Sample Variance	339.309%
Kurtosis	1.86
Skewness	0.44
Range	124.846%
Minimum	-50.231%
Maximum	74.615%
Count	135

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	14.593%	79	58.52%	8.540%
Negative	-11.880%	56	41.48%	-4.928%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	18.92%
12	3 Years	18.52%
20	5 Years	17.08%
40	10 Years	18.31%
80	20 Years	21.94%
200	50 Years	26.72%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.21. Further, an Excess Kurtosis of 6.66 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -29.98%	0	0.00%	0.00%
-29.98% to -14.37%	0	0.00%	0.00%
-14.37% to 1.24%	0	0.00%	0.00%
1.24% to 16.84%	23	16.91%	16.91%
16.84% to 32.45%	64	47.06%	63.97%
32.45% to 48.06%	29	21.32%	85.29%
48.06% to 63.67%	10	7.35%	92.65%
63.67% to 79.28%	4	2.94%	95.59%
79.28% to 94.88%	4	2.94%	98.53%
Greater than 94.88%	2	1.47%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	11.64%	53.26%	84.56%	68.27%
±2σ	-9.17%	74.07%	93.38%	95.45%
±3σ	-29.98%	94.88%	98.53%	99.73%

Descriptive Statistics

Mean	32.451%
Standard Error	1.785%
Median	26.524%
Mode	22.727%
Standard Deviation	20.811%
Sample Variance	433.117%
Kurtosis	6.66
Skewness	2.21
Range	130.514%
Minimum	8.632%
Maximum	139.146%
Count	136

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	32.451%	136	100.00%	32.451%
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 1.57. Further, an Excess Kurtosis of 5.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -58.43%	0	0.00%	0.00%
-58.43% to -43.90%	0	0.00%	0.00%
-43.90% to -29.36%	6	4.41%	4.41%
-29.36% to -14.83%	18	13.24%	17.65%
-14.83% to -0.29%	53	38.97%	56.62%
-0.29% to 14.24%	39	28.68%	85.29%
14.24% to 28.77%	10	7.35%	92.65%
28.77% to 43.31%	6	4.41%	97.06%
43.31% to 57.84%	3	2.21%	99.26%
Greater than 57.84%	1	0.74%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-19.67%	19.08%	77.21%	68.27%
±2σ	-39.05%	38.46%	94.12%	95.45%
±3σ	-58.43%	57.84%	99.26%	99.73%

Descriptive Statistics

Mean	-0.295%
Standard Error	1.662%
Median	-3.661%
Mode	-43.025%
Standard Deviation	19.379%
Sample Variance	375.530%
Kurtosis	5.27
Skewness	1.57
Range	141.167%
Minimum	-43.025%
Maximum	98.142%
Count	136

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	15.046%	59	43.38%	6.527%
Negative	-12.050%	77	56.62%	-6.822%
Zero	0.000%	0	0.00%	0.000%

BSX Deep Analytical Report