BMRN - 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	3.12%	2.80%	2.71%	2.69%	2.43%	3.06%	62.54	65.66	61.04	67.27
Weekly	6.51%	7.06%	6.39%	6.13%	5.35%	6.73%	60.74	67.60	57.57	71.32
Monthly	13.84%	15.82%	13.77%	13.08%	11.14%	14.02%	57.32	71.63	51.28	80.08
Quarterly	20.68%	27.70%	22.66%	21.99%	19.30%	24.28%	52.83	77.72	43.56	94.27
Annual	—	—	—	—	38.60%	48.56%	43.56	94.27	29.61	138.69
Option Time Horizon	—	—	—	—	14.85%	18.68%	55.24	74.34	47.62	86.24

Calculation Notes: Computed at 2026-02-22T14:23 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $64.08 | Strike (K): $65.00 (first OTM call ≥ spot) | Observed Mid-Price: $3.5500 (Bid $2.7000 / Ask $4.4000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 38.60% (CRR binomial tree, American, n=20 steps) | BS IV: 38.77% (European Black-Scholes) | Yahoo IV (Annual): 48.56% → 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 0.64. Further, an Excess Kurtosis of 16.60 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.28%	45	0.67%	0.67%
-9.28% to -6.94%	58	0.87%	1.54%
-6.94% to -4.61%	171	2.56%	4.10%
-4.61% to -2.27%	703	10.52%	14.63%
-2.27% to 0.07%	2485	37.20%	51.83%
0.07% to 2.41%	2259	33.82%	85.64%
2.41% to 4.75%	636	9.52%	95.16%
4.75% to 7.09%	194	2.90%	98.07%
7.09% to 9.43%	72	1.08%	99.15%
Greater than 9.43%	57	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.05%	3.19%	81.11%	68.27%
±2σ	-6.16%	6.31%	95.21%	95.45%
±3σ	-9.28%	9.43%	98.47%	99.73%

Descriptive Statistics

Mean	0.072%
Standard Error	0.038%
Median	0.000%
Mode	0.000%
Standard Deviation	3.118%
Sample Variance	9.720%
Kurtosis	16.60
Skewness	0.64
Range	69.894%
Minimum	-35.279%
Maximum	34.615%
Count	6680

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.120%	3301	49.42%	1.048%
Negative	-1.991%	3273	49.00%	-0.976%
Zero	0.000%	106	1.59%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.41%
20	1 Month	3.08%
60	1 Quarter	3.06%
250	1 Year	3.00%
750	3 Years	2.62%
1,250	5 Years	2.69%
2,500	10 Years	2.96%
5,000	20 Years	3.31%
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 3.22. Further, an Excess Kurtosis of 18.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -4.51%	0	0.00%	0.00%
-4.51% to -2.41%	0	0.00%	0.00%
-2.41% to -0.31%	0	0.00%	0.00%
-0.31% to 1.79%	908	13.59%	13.59%
1.79% to 3.90%	3454	51.70%	65.29%
3.90% to 6.00%	1352	20.24%	85.53%
6.00% to 8.10%	523	7.83%	93.35%
8.10% to 10.20%	218	3.26%	96.62%
10.20% to 12.30%	100	1.50%	98.11%
Greater than 12.30%	126	1.89%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.09%	6.70%	87.79%	68.27%
±2σ	-1.71%	9.50%	95.73%	95.45%
±3σ	-4.51%	12.30%	98.11%	99.73%

Descriptive Statistics

Mean	3.895%
Standard Error	0.034%
Median	3.098%
Mode	4.000%
Standard Deviation	2.802%
Sample Variance	7.850%
Kurtosis	18.27
Skewness	3.22
Range	37.019%
Minimum	0.481%
Maximum	37.500%
Count	6681

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.895%	6681	100.00%	3.895%
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 0.34. Further, an Excess Kurtosis of 7.12 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.10%	43	0.64%	0.64%
-8.10% to -6.07%	79	1.18%	1.83%
-6.07% to -4.04%	218	3.26%	5.09%
-4.04% to -2.01%	754	11.29%	16.37%
-2.01% to 0.03%	2277	34.08%	50.46%
0.03% to 2.06%	2245	33.60%	84.06%
2.06% to 4.09%	733	10.97%	95.03%
4.09% to 6.12%	198	2.96%	97.99%
6.12% to 8.16%	74	1.11%	99.10%
Greater than 8.16%	60	0.90%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.68%	2.74%	78.55%	68.27%
±2σ	-5.39%	5.45%	94.96%	95.45%
±3σ	-8.10%	8.16%	98.46%	99.73%

Descriptive Statistics

Mean	0.026%
Standard Error	0.033%
Median	0.000%
Mode	0.000%
Standard Deviation	2.710%
Sample Variance	7.344%
Kurtosis	7.12
Skewness	0.34
Range	47.997%
Minimum	-23.944%
Maximum	24.053%
Count	6681

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.893%	3332	49.87%	0.944%
Negative	-1.885%	3254	48.70%	-0.918%
Zero	0.000%	95	1.42%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.24. Further, an Excess Kurtosis of 5.92 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.20%	9	0.65%	0.65%
-19.20% to -14.32%	9	0.65%	1.30%
-14.32% to -9.44%	46	3.32%	4.61%
-9.44% to -4.55%	170	12.26%	16.87%
-4.55% to 0.33%	478	34.46%	51.33%
0.33% to 5.21%	450	32.44%	83.78%
5.21% to 10.09%	149	10.74%	94.52%
10.09% to 14.98%	49	3.53%	98.05%
14.98% to 19.86%	15	1.08%	99.13%
Greater than 19.86%	12	0.87%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.18%	6.84%	77.51%	68.27%
±2σ	-12.69%	13.35%	95.10%	95.45%
±3σ	-19.21%	19.86%	98.49%	99.73%

Descriptive Statistics

Mean	0.328%
Standard Error	0.175%
Median	0.166%
Mode	0.000%
Standard Deviation	6.511%
Sample Variance	42.393%
Kurtosis	5.92
Skewness	0.24
Range	77.065%
Minimum	-38.907%
Maximum	38.158%
Count	1387

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.739%	711	51.26%	2.429%
Negative	-4.383%	665	47.95%	-2.101%
Zero	0.000%	11	0.79%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	5.86%
12	1 Quarter	7.30%
52	1 Year	6.45%
156	3 Years	5.89%
260	5 Years	6.13%
520	10 Years	6.64%
1,040	20 Years	7.56%
2,600	50 Years	9.08%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.81. 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 -11.59%	0	0.00%	0.00%
-11.59% to -6.30%	0	0.00%	0.00%
-6.30% to -1.01%	0	0.00%	0.00%
-1.01% to 4.29%	192	13.83%	13.83%
4.29% to 9.58%	721	51.95%	65.78%
9.58% to 14.87%	287	20.68%	86.46%
14.87% to 20.16%	91	6.56%	93.01%
20.16% to 25.46%	43	3.10%	96.11%
25.46% to 30.75%	23	1.66%	97.77%
Greater than 30.75%	31	2.23%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.52%	16.63%	88.62%	68.27%
±2σ	-4.54%	23.69%	95.17%	95.45%
±3σ	-11.59%	30.75%	97.77%	99.73%

Descriptive Statistics

Mean	9.578%
Standard Error	0.189%
Median	7.424%
Mode	3.895%
Standard Deviation	7.057%
Sample Variance	49.795%
Kurtosis	11.78
Skewness	2.81
Range	64.482%
Minimum	1.493%
Maximum	65.975%
Count	1388

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.578%	1388	100.00%	9.578%
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 1.52. Further, an Excess Kurtosis of 12.38 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.13%	6	0.43%	0.43%
-19.13% to -14.33%	8	0.58%	1.01%
-14.33% to -9.54%	50	3.60%	4.61%
-9.54% to -4.74%	173	12.46%	17.07%
-4.74% to 0.05%	497	35.81%	52.88%
0.05% to 4.85%	442	31.84%	84.73%
4.85% to 9.64%	144	10.37%	95.10%
9.64% to 14.44%	37	2.67%	97.77%
14.44% to 19.23%	19	1.37%	99.14%
Greater than 19.23%	12	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.34%	6.45%	78.17%	68.27%
±2σ	-12.73%	12.84%	96.11%	95.45%
±3σ	-19.13%	19.24%	98.70%	99.73%

Descriptive Statistics

Mean	0.053%
Standard Error	0.172%
Median	-0.170%
Mode	0.000%
Standard Deviation	6.394%
Sample Variance	40.877%
Kurtosis	12.38
Skewness	1.52
Range	90.824%
Minimum	-27.619%
Maximum	63.205%
Count	1388

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.653%	658	47.41%	2.206%
Negative	-4.150%	720	51.87%	-2.153%
Zero	0.000%	10	0.72%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.58. Further, an Excess Kurtosis of 14.85 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -40.09%	2	0.63%	0.63%
-40.09% to -29.71%	3	0.94%	1.57%
-29.71% to -19.33%	8	2.51%	4.08%
-19.33% to -8.95%	40	12.54%	16.61%
-8.95% to 1.43%	104	32.60%	49.22%
1.43% to 11.81%	113	35.42%	84.64%
11.81% to 22.19%	37	11.60%	96.24%
22.19% to 32.57%	8	2.51%	98.75%
32.57% to 42.95%	1	0.31%	99.06%
Greater than 42.95%	3	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.41%	15.27%	81.19%	68.27%
±2σ	-26.25%	29.11%	95.30%	95.45%
±3σ	-40.09%	42.95%	98.43%	99.73%

Descriptive Statistics

Mean	1.430%
Standard Error	0.775%
Median	1.493%
Mode	-49.824%
Standard Deviation	13.840%
Sample Variance	191.537%
Kurtosis	14.85
Skewness	1.58
Range	164.416%
Minimum	-49.824%
Maximum	114.592%
Count	319

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.779%	178	55.80%	5.457%
Negative	-9.174%	140	43.89%	-4.026%
Zero	0.000%	1	0.31%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	16.90%
12	1 Year	13.53%
36	3 Years	12.74%
60	5 Years	13.08%
120	10 Years	14.05%
240	20 Years	15.91%
600	50 Years	19.23%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.58. Further, an Excess Kurtosis of 9.05 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.30%	0	0.00%	0.00%
-25.30% to -13.43%	0	0.00%	0.00%
-13.43% to -1.57%	0	0.00%	0.00%
-1.57% to 10.30%	35	10.94%	10.94%
10.30% to 22.16%	178	55.62%	66.56%
22.16% to 34.03%	61	19.06%	85.62%
34.03% to 45.89%	24	7.50%	93.12%
45.89% to 57.76%	9	2.81%	95.94%
57.76% to 69.62%	4	1.25%	97.19%
Greater than 69.62%	9	2.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	6.34%	37.98%	87.50%	68.27%
±2σ	-9.48%	53.80%	95.62%	95.45%
±3σ	-25.30%	69.62%	97.19%	99.73%

Descriptive Statistics

Mean	22.161%
Standard Error	0.884%
Median	16.939%
Mode	13.502%
Standard Deviation	15.820%
Sample Variance	250.275%
Kurtosis	9.05
Skewness	2.58
Range	118.920%
Minimum	5.641%
Maximum	124.561%
Count	320

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.161%	320	100.00%	22.161%
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 1.39. Further, an Excess Kurtosis of 6.22 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -40.77%	1	0.31%	0.31%
-40.77% to -30.44%	1	0.31%	0.62%
-30.44% to -20.11%	8	2.50%	3.12%
-20.11% to -9.78%	45	14.06%	17.19%
-9.78% to 0.55%	128	40.00%	57.19%
0.55% to 10.88%	86	26.88%	84.06%
10.88% to 21.21%	36	11.25%	95.31%
21.21% to 31.54%	5	1.56%	96.88%
31.54% to 41.87%	3	0.94%	97.81%
Greater than 41.87%	7	2.19%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.22%	14.32%	81.56%	68.27%
±2σ	-27.00%	28.09%	95.31%	95.45%
±3σ	-40.77%	41.87%	97.50%	99.73%

Descriptive Statistics

Mean	0.548%
Standard Error	0.770%
Median	-1.324%
Mode	-9.845%
Standard Deviation	13.773%
Sample Variance	189.693%
Kurtosis	6.22
Skewness	1.39
Range	128.129%
Minimum	-54.000%
Maximum	74.129%
Count	320

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.402%	142	44.38%	5.060%
Negative	-8.156%	177	55.31%	-4.511%
Zero	0.000%	1	0.31%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.20. Further, an Excess Kurtosis of 0.69 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -58.35%	0	0.00%	0.00%
-58.35% to -42.84%	2	1.87%	1.87%
-42.84% to -27.32%	6	5.61%	7.48%
-27.32% to -11.81%	10	9.35%	16.82%
-11.81% to 3.70%	35	32.71%	49.53%
3.70% to 19.21%	34	31.78%	81.31%
19.21% to 34.73%	13	12.15%	93.46%
34.73% to 50.24%	5	4.67%	98.13%
50.24% to 65.75%	1	0.93%	99.07%
Greater than 65.75%	1	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.98%	24.38%	73.83%	68.27%
±2σ	-37.66%	45.07%	93.46%	95.45%
±3σ	-58.35%	65.75%	99.07%	99.73%

Descriptive Statistics

Mean	3.701%
Standard Error	2.000%
Median	4.421%
Mode	-50.493%
Standard Deviation	20.683%
Sample Variance	427.794%
Kurtosis	0.69
Skewness	0.20
Range	116.384%
Minimum	-50.493%
Maximum	65.891%
Count	107

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.024%	58	54.21%	9.770%
Negative	-13.252%	49	45.79%	-6.069%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	19.99%
12	3 Years	21.13%
20	5 Years	21.99%
40	10 Years	23.36%
80	20 Years	27.88%
200	50 Years	33.57%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.38. Further, an Excess Kurtosis of 8.19 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.66%	0	0.00%	0.00%
-41.66% to -20.88%	0	0.00%	0.00%
-20.88% to -0.10%	0	0.00%	0.00%
-0.10% to 20.68%	19	17.59%	17.59%
20.68% to 41.46%	51	47.22%	64.81%
41.46% to 62.24%	22	20.37%	85.19%
62.24% to 83.02%	8	7.41%	92.59%
83.02% to 103.80%	4	3.70%	96.30%
103.80% to 124.57%	1	0.93%	97.22%
Greater than 124.57%	3	2.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	13.75%	69.16%	89.81%	68.27%
±2σ	-13.95%	96.87%	95.37%	95.45%
±3σ	-41.66%	124.57%	97.22%	99.73%

Descriptive Statistics

Mean	41.459%
Standard Error	2.666%
Median	32.369%
Mode	14.202%
Standard Deviation	27.705%
Sample Variance	767.558%
Kurtosis	8.19
Skewness	2.38
Range	175.272%
Minimum	14.202%
Maximum	189.474%
Count	108

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	41.459%	108	100.00%	41.459%
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 4.14 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -67.04%	0	0.00%	0.00%
-67.04% to -50.05%	0	0.00%	0.00%
-50.05% to -33.05%	2	1.85%	1.85%
-33.05% to -16.05%	16	14.81%	16.67%
-16.05% to 0.94%	40	37.04%	53.70%
0.94% to 17.94%	34	31.48%	85.19%
17.94% to 34.93%	7	6.48%	91.67%
34.93% to 51.93%	4	3.70%	95.37%
51.93% to 68.92%	3	2.78%	98.15%
Greater than 68.92%	2	1.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.72%	23.60%	79.63%	68.27%
±2σ	-44.38%	46.26%	94.44%	95.45%
±3σ	-67.04%	68.92%	98.15%	99.73%

Descriptive Statistics

Mean	0.941%
Standard Error	2.181%
Median	-2.498%
Mode	-40.654%
Standard Deviation	22.661%
Sample Variance	513.535%
Kurtosis	4.14
Skewness	1.57
Range	141.317%
Minimum	-40.654%
Maximum	100.663%
Count	108

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.776%	51	47.22%	8.394%
Negative	-14.121%	57	52.78%	-7.453%
Zero	0.000%	0	0.00%	0.000%

BMRN Deep Analytical Report