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	3.11%	2.80%	2.71%	2.71%	3.33%	3.36%	52.88	56.52	51.15	58.44
Weekly	6.51%	7.05%	6.39%	6.19%	7.33%	7.39%	50.81	58.83	47.21	63.30
Monthly	13.83%	15.80%	13.76%	13.20%	15.26%	15.39%	46.93	63.68	40.29	74.18
Quarterly	20.67%	27.64%	22.63%	22.39%	26.43%	26.66%	41.97	71.21	32.22	92.76
Annual	—	—	—	—	52.87%	53.32%	32.22	92.76	18.99	157.37
Option Time Horizon	—	—	—	—	19.95%	20.13%	44.78	66.74	36.68	81.48

Calculation Notes: Computed at 2026-03-23T16:39 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $54.67 | Strike (K): $55.00 (first OTM call ≥ spot) | Observed Mid-Price: $4.3000 (Bid $3.2000 / Ask $5.4000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 52.87% (CRR binomial tree, American, n=20 steps) | BS IV: 52.54% (European Black-Scholes) | Yahoo IV (Annual): 53.32% → 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 16.63 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.27%	45	0.67%	0.67%
-9.27% to -6.94%	59	0.88%	1.55%
-6.94% to -4.60%	171	2.55%	4.10%
-4.60% to -2.27%	704	10.51%	14.61%
-2.27% to 0.07%	2495	37.23%	51.84%
0.07% to 2.40%	2266	33.82%	85.66%
2.40% to 4.74%	638	9.52%	95.18%
4.74% to 7.08%	193	2.88%	98.06%
7.08% to 9.41%	73	1.09%	99.15%
Greater than 9.41%	57	0.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.04%	3.18%	81.09%	68.27%
±2σ	-6.16%	6.30%	95.21%	95.45%
±3σ	-9.27%	9.41%	98.48%	99.73%

Descriptive Statistics

Mean	0.070%
Standard Error	0.038%
Median	0.000%
Mode	0.000%
Standard Deviation	3.114%
Sample Variance	9.698%
Kurtosis	16.63
Skewness	0.64
Range	69.894%
Minimum	-35.279%
Maximum	34.615%
Count	6701

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.118%	3307	49.35%	1.045%
Negative	-1.988%	3288	49.07%	-0.975%
Zero	0.000%	106	1.58%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.01%
20	1 Month	3.77%
60	1 Quarter	3.20%
250	1 Year	3.11%
750	3 Years	2.64%
1,250	5 Years	2.71%
2,500	10 Years	2.95%
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 18.29 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -4.50%	0	0.00%	0.00%
-4.50% to -2.40%	0	0.00%	0.00%
-2.40% to -0.30%	0	0.00%	0.00%
-0.30% to 1.80%	913	13.62%	13.62%
1.80% to 3.90%	3462	51.66%	65.28%
3.90% to 5.99%	1358	20.26%	85.54%
5.99% to 8.09%	524	7.82%	93.36%
8.09% to 10.19%	218	3.25%	96.61%
10.19% to 12.29%	101	1.51%	98.12%
Greater than 12.29%	126	1.88%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.10%	6.69%	87.78%	68.27%
±2σ	-1.70%	9.49%	95.72%	95.45%
±3σ	-4.50%	12.29%	98.12%	99.73%

Descriptive Statistics

Mean	3.895%
Standard Error	0.034%
Median	3.101%
Mode	4.000%
Standard Deviation	2.799%
Sample Variance	7.837%
Kurtosis	18.29
Skewness	3.22
Range	37.019%
Minimum	0.481%
Maximum	37.500%
Count	6702

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.895%	6702	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 7.13 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%	1.82%
-6.07% to -4.03%	220	3.28%	5.10%
-4.03% to -2.00%	754	11.25%	16.35%
-2.00% to 0.03%	2283	34.06%	50.42%
0.03% to 2.06%	2255	33.65%	84.06%
2.06% to 4.09%	736	10.98%	95.05%
4.09% to 6.12%	197	2.94%	97.99%
6.12% to 8.15%	74	1.10%	99.09%
Greater than 8.15%	61	0.91%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.68%	2.73%	78.57%	68.27%
±2σ	-5.39%	5.44%	94.97%	95.45%
±3σ	-8.09%	8.15%	98.43%	99.73%

Descriptive Statistics

Mean	0.026%
Standard Error	0.033%
Median	0.000%
Mode	0.000%
Standard Deviation	2.707%
Sample Variance	7.329%
Kurtosis	7.13
Skewness	0.34
Range	47.997%
Minimum	-23.944%
Maximum	24.053%
Count	6702

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	1.890%	3345	49.91%	0.943%
Negative	-1.884%	3262	48.67%	-0.917%
Zero	0.000%	95	1.42%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.25. 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%	1.29%
-14.32% to -9.44%	46	3.31%	4.60%
-9.44% to -4.56%	173	12.44%	17.04%
-4.56% to 0.32%	478	34.36%	51.40%
0.32% to 5.20%	452	32.49%	83.90%
5.20% to 10.07%	148	10.64%	94.54%
10.07% to 14.96%	49	3.52%	98.06%
14.96% to 19.83%	15	1.08%	99.14%
Greater than 19.83%	12	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.19%	6.82%	77.64%	68.27%
±2σ	-12.70%	13.33%	95.11%	95.45%
±3σ	-19.20%	19.83%	98.49%	99.73%

Descriptive Statistics

Mean	0.316%
Standard Error	0.174%
Median	0.165%
Mode	0.000%
Standard Deviation	6.506%
Sample Variance	42.323%
Kurtosis	5.92
Skewness	0.25
Range	77.065%
Minimum	-38.907%
Maximum	38.158%
Count	1391

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.732%	712	51.19%	2.422%
Negative	-4.386%	668	48.02%	-2.106%
Zero	0.000%	11	0.79%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	8.84%
12	1 Quarter	7.19%
52	1 Year	6.84%
156	3 Years	5.96%
260	5 Years	6.19%
520	10 Years	6.63%
1,040	20 Years	7.57%
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 11.81 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -11.56%	0	0.00%	0.00%
-11.56% to -6.28%	0	0.00%	0.00%
-6.28% to -0.99%	0	0.00%	0.00%
-0.99% to 4.29%	193	13.86%	13.86%
4.29% to 9.58%	721	51.80%	65.66%
9.58% to 14.87%	290	20.83%	86.49%
14.87% to 20.15%	91	6.54%	93.03%
20.15% to 25.44%	43	3.09%	96.12%
25.44% to 30.72%	23	1.65%	97.77%
Greater than 30.72%	31	2.23%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.53%	16.63%	88.65%	68.27%
±2σ	-4.52%	23.68%	95.19%	95.45%
±3σ	-11.56%	30.72%	97.77%	99.73%

Descriptive Statistics

Mean	9.579%
Standard Error	0.189%
Median	7.428%
Mode	3.895%
Standard Deviation	7.048%
Sample Variance	49.668%
Kurtosis	11.81
Skewness	2.81
Range	64.482%
Minimum	1.493%
Maximum	65.975%
Count	1392

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.579%	1392	100.00%	9.579%
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 12.39 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.10%	6	0.43%	0.43%
-19.10% to -14.31%	8	0.57%	1.01%
-14.31% to -9.52%	50	3.59%	4.60%
-9.52% to -4.73%	172	12.36%	16.95%
-4.73% to 0.06%	502	36.06%	53.02%
0.06% to 4.85%	440	31.61%	84.63%
4.85% to 9.64%	146	10.49%	95.11%
9.64% to 14.43%	37	2.66%	97.77%
14.43% to 19.22%	19	1.36%	99.14%
Greater than 19.22%	12	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.32%	6.45%	78.16%	68.27%
±2σ	-12.71%	12.84%	96.05%	95.45%
±3σ	-19.10%	19.22%	98.71%	99.73%

Descriptive Statistics

Mean	0.062%
Standard Error	0.171%
Median	-0.163%
Mode	0.000%
Standard Deviation	6.387%
Sample Variance	40.795%
Kurtosis	12.39
Skewness	1.52
Range	90.824%
Minimum	-27.619%
Maximum	63.205%
Count	1392

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.651%	661	47.49%	2.209%
Negative	-4.144%	721	51.80%	-2.146%
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 1.59. Further, an Excess Kurtosis of 14.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -40.10%	2	0.62%	0.62%
-40.10% to -29.73%	3	0.94%	1.56%
-29.73% to -19.36%	8	2.50%	4.06%
-19.36% to -8.99%	41	12.81%	16.88%
-8.99% to 1.38%	104	32.50%	49.38%
1.38% to 11.75%	114	35.62%	85.00%
11.75% to 22.12%	36	11.25%	96.25%
22.12% to 32.49%	8	2.50%	98.75%
32.49% to 42.86%	1	0.31%	99.06%
Greater than 42.86%	3	0.94%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.45%	15.20%	80.62%	68.27%
±2σ	-26.28%	29.03%	95.00%	95.45%
±3σ	-40.10%	42.86%	98.44%	99.73%

Descriptive Statistics

Mean	1.377%
Standard Error	0.773%
Median	1.468%
Mode	-49.824%
Standard Deviation	13.827%
Sample Variance	191.199%
Kurtosis	14.88
Skewness	1.59
Range	164.416%
Minimum	-49.824%
Maximum	114.592%
Count	320

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.755%	178	55.62%	5.426%
Negative	-9.190%	141	44.06%	-4.049%
Zero	0.000%	1	0.31%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	14.81%
12	1 Year	14.36%
36	3 Years	12.78%
60	5 Years	13.20%
120	10 Years	14.03%
240	20 Years	15.87%
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 2.59. Further, an Excess Kurtosis of 9.09 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -25.24%	0	0.00%	0.00%
-25.24% to -13.39%	0	0.00%	0.00%
-13.39% to -1.54%	0	0.00%	0.00%
-1.54% to 10.30%	35	10.90%	10.90%
10.30% to 22.15%	179	55.76%	66.67%
22.15% to 34.00%	61	19.00%	85.67%
34.00% to 45.84%	24	7.48%	93.15%
45.84% to 57.69%	9	2.80%	95.95%
57.69% to 69.54%	4	1.25%	97.20%
Greater than 69.54%	9	2.80%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	6.36%	37.95%	87.23%	68.27%
±2σ	-9.44%	53.74%	95.64%	95.45%
±3σ	-25.24%	69.54%	97.20%	99.73%

Descriptive Statistics

Mean	22.151%
Standard Error	0.882%
Median	17.029%
Mode	13.502%
Standard Deviation	15.796%
Sample Variance	249.502%
Kurtosis	9.09
Skewness	2.59
Range	118.920%
Minimum	5.641%
Maximum	124.561%
Count	321

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.151%	321	100.00%	22.151%
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 -40.68%	1	0.31%	0.31%
-40.68% to -30.36%	1	0.31%	0.62%
-30.36% to -20.04%	8	2.49%	3.12%
-20.04% to -9.73%	44	13.71%	16.82%
-9.73% to 0.59%	130	40.50%	57.32%
0.59% to 10.91%	85	26.48%	83.80%
10.91% to 21.23%	37	11.53%	95.33%
21.23% to 31.54%	5	1.56%	96.88%
31.54% to 41.86%	3	0.93%	97.82%
Greater than 41.86%	7	2.18%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.17%	14.35%	81.62%	68.27%
±2σ	-26.92%	28.10%	95.33%	95.45%
±3σ	-40.68%	41.86%	97.51%	99.73%

Descriptive Statistics

Mean	0.591%
Standard Error	0.768%
Median	-1.321%
Mode	-9.845%
Standard Deviation	13.756%
Sample Variance	189.218%
Kurtosis	6.22
Skewness	1.39
Range	128.129%
Minimum	-54.000%
Maximum	74.129%
Count	321

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.400%	143	44.55%	5.078%
Negative	-8.137%	177	55.14%	-4.487%
Zero	0.000%	1	0.31%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.22. Further, an Excess Kurtosis of 0.71 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -58.47%	0	0.00%	0.00%
-58.47% to -42.97%	2	1.87%	1.87%
-42.97% to -27.46%	6	5.61%	7.48%
-27.46% to -11.96%	10	9.35%	16.82%
-11.96% to 3.54%	36	33.64%	50.47%
3.54% to 19.05%	33	30.84%	81.31%
19.05% to 34.55%	13	12.15%	93.46%
34.55% to 50.06%	5	4.67%	98.13%
50.06% to 65.56%	1	0.93%	99.07%
Greater than 65.56%	1	0.93%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-17.13%	24.22%	73.83%	68.27%
±2σ	-37.80%	44.89%	92.52%	95.45%
±3σ	-58.47%	65.56%	99.07%	99.73%

Descriptive Statistics

Mean	3.545%
Standard Error	1.999%
Median	3.258%
Mode	-50.493%
Standard Deviation	20.673%
Sample Variance	427.361%
Kurtosis	0.71
Skewness	0.22
Range	116.384%
Minimum	-50.493%
Maximum	65.891%
Count	107

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.106%	57	53.27%	9.645%
Negative	-13.054%	50	46.73%	-6.100%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	22.00%
12	3 Years	21.80%
20	5 Years	22.39%
40	10 Years	23.56%
80	20 Years	27.98%
200	50 Years	33.64%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.40. Further, an Excess Kurtosis of 8.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.36%	0	0.00%	0.00%
-41.36% to -20.64%	0	0.00%	0.00%
-20.64% to 0.09%	0	0.00%	0.00%
0.09% to 20.82%	19	17.59%	17.59%
20.82% to 41.54%	51	47.22%	64.81%
41.54% to 62.27%	22	20.37%	85.19%
62.27% to 83.00%	8	7.41%	92.59%
83.00% to 103.72%	4	3.70%	96.30%
103.72% to 124.45%	1	0.93%	97.22%
Greater than 124.45%	3	2.78%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	13.91%	69.18%	89.81%	68.27%
±2σ	-13.73%	96.81%	95.37%	95.45%
±3σ	-41.36%	124.45%	97.22%	99.73%

Descriptive Statistics

Mean	41.544%
Standard Error	2.659%
Median	32.369%
Mode	14.960%
Standard Deviation	27.635%
Sample Variance	763.693%
Kurtosis	8.27
Skewness	2.40
Range	174.514%
Minimum	14.960%
Maximum	189.474%
Count	108

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	41.544%	108	100.00%	41.544%
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 1.56. 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 -66.80%	0	0.00%	0.00%
-66.80% to -49.83%	0	0.00%	0.00%
-49.83% to -32.86%	2	1.85%	1.85%
-32.86% to -15.89%	17	15.74%	17.59%
-15.89% to 1.08%	38	35.19%	52.78%
1.08% to 18.05%	36	33.33%	86.11%
18.05% to 35.02%	6	5.56%	91.67%
35.02% to 51.99%	4	3.70%	95.37%
51.99% to 68.96%	3	2.78%	98.15%
Greater than 68.96%	2	1.85%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-21.54%	23.71%	77.78%	68.27%
±2σ	-44.17%	46.34%	94.44%	95.45%
±3σ	-66.80%	68.96%	98.15%	99.73%

Descriptive Statistics

Mean	1.082%
Standard Error	2.177%
Median	-1.293%
Mode	-40.654%
Standard Deviation	22.627%
Sample Variance	512.001%
Kurtosis	4.14
Skewness	1.56
Range	141.317%
Minimum	-40.654%
Maximum	100.663%
Count	108

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	17.495%	52	48.15%	8.423%
Negative	-14.160%	56	51.85%	-7.342%
Zero	0.000%	0	0.00%	0.000%

BMRN Deep Analytical Report