Probability Analysis:
ETHA ETF Price Risk Assessment

A comprehensive quantitative analysis of the probability that ETHA ETF will decline to the $20-$24 range within a 3-month timeframe, utilizing log-normal distribution modeling and Black-Scholes framework.

17.26%
Log-Normal Probability
15.61%
Normal Distribution

Key Parameters

Current Price: $28.20
Annual Volatility: 44%
Time Horizon: 3 Months
Target Range: $20 - $24

Model Insights

The analysis employs both log-normal and normal distribution approaches, revealing the impact of distributional assumptions on probability outcomes in financial risk assessment.

Executive Summary

This comprehensive analysis calculates the probability of ETHA ETF's price declining to the $20-$24 range within a 3-month period. Using established financial mathematics frameworks, we present two distinct modeling approaches with their respective outcomes.

Log-Normal Distribution (Recommended)

Theoretically sound approach preventing negative prices

17.26%

Normal Distribution (User Request)

Direct application of normal returns assumption

15.61%

Problem Definition

Objective

Calculate the probability that ETHA ETF price will decline to the $20-$24 range within a 3-month timeframe under specified volatility conditions and distributional assumptions.

Given Parameters

  • • Current Price: $28.20
  • • Annual Volatility: 44%
  • • Time Horizon: 3 months
  • • Target Range: $20-$24

Key Assumptions

  • • Log-normal price distribution
  • • Constant 44% volatility
  • • 95% confidence framework
  • • No dividend payments

Modeling Approach

Log-Normal Distribution Framework

Theoretical Foundation

The log-normal distribution is preferred for asset pricing models because it prevents negative prices and better captures the compounding nature of returns [116], [137].

ln(ST) ~ N(ln(S₀), σ²T)

Black-Scholes Integration

The model assumes Geometric Brownian Motion, where future prices follow a log-normal distribution with normally distributed log-returns [148], [186].

ST = S₀ exp{(μ - σ²/2)T + σ√T Z}

Modeling Framework Flow

graph TD A["Current Price
$28.20"] --> B["Log Transformation
ln28.20 = 3.3393"] B --> C["Normal Distribution
μ = 3.3393, σ = 0.22"] C --> D["Price Bounds
$20-$24"] D --> E["Log Bounds
ln20 = 2.9957
ln24 = 3.1781"] E --> F["Z-Score Calculation
Z_lower = -1.5618
Z_upper = -0.7327"] F --> G["CDF Application
ΦZ_upper - ΦZ_lower"] G --> H["Final Probability
17.26%"] style A fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#0d47a1 style B fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px,color:#4a148c style C fill:#e8f5e8,stroke:#388e3c,stroke-width:3px,color:#1b5e20 style D fill:#fff3e0,stroke:#f57c00,stroke-width:3px,color:#e65100 style E fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px,color:#4a148c style F fill:#e8f5e8,stroke:#388e3c,stroke-width:3px,color:#1b5e20 style G fill:#e3f2fd,stroke:#1976d2,stroke-width:3px,color:#0d47a1 style H fill:#ffebee,stroke:#d32f2f,stroke-width:4px,color:#b71c1c

Volatility Calculation

Annual to 3-Month Conversion

Volatility scales with the square root of time, following the principle that returns are independently and identically distributed [162].

σ3-month = σannual × √T
σ3-month = 0.44 × √0.25
σ3-month = 0.44 × 0.5 = 0.22

Volatility Parameters

Annual Volatility: 44%
Time Period: 0.25 years
3-Month Volatility: 22%

Probability Calculation

Z-Score Determination

Log-Price Transformation

ln(20) ≈ 2.9957
ln(24) ≈ 3.1781
ln(28.20) ≈ 3.3393

Conversion of price bounds to log scale enables normal distribution probability calculations.

Z-Score Calculation

Z = (ln(X) - ln(S₀)) / (σ√T)
Zlower = (2.9957 - 3.3393) / 0.22 ≈ -1.5618
Zupper = (3.1781 - 3.3393) / 0.22 ≈ -0.7327

Cumulative Distribution Function Application

Standard Normal CDF

The probability is calculated as the difference between the CDF values at the upper and lower Z-scores [173].

P = Φ(Zupper) - Φ(Zlower)
P = Φ(-0.7327) - Φ(-1.5618)

Probability Result

17.26%

Probability of ETHA ETF price being between $20 and $24 in 3 months

Final Results

Parameter Log-Normal Model Normal Distribution
Distribution Type Log-Normal Prices Normal Returns
Theoretical Foundation Prevents negative prices Allows negative prices
Z-Score Lower -1.5618 -1.32173
Z-Score Upper -0.7327 -0.67698
Final Probability 17.26% 15.61%

95% Certainty Interpretation

The "95% certainty" refers to the confidence level of the statistical model used, not a modifier for the calculated probability. This indicates high reliability in the modeling framework rather than adjusting the 17.26% probability figure.

Alternative Approach: Normal Distribution

User-Specified Normal Distribution Method

Model Framework

This approach models simple arithmetic returns as normally distributed, directly following the user's initial assumption.

ST = S₀ × (1 + R)
R ~ N(0, 0.22²)
ST ~ N(28.20, 6.204²)

Calculation Process

Zlower: (20 - 28.20) / 6.204 ≈ -1.32173
Zupper: (24 - 28.20) / 6.204 ≈ -0.67698
Probability: Φ(-0.67698) - Φ(-1.32173) ≈ 15.61%

Theoretical Limitation

The normal distribution approach theoretically allows for negative prices (when R < -1), which is economically impossible for asset prices. This limitation makes the log-normal approach theoretically superior despite the user's initial specification.

Confidence Interval Analysis

95% Confidence Interval for Future Price

Log-Normal CI Calculation

Using the properties of normal distribution for log-prices, we construct a 95% confidence interval [134].

CI for ln(ST): [ln(S₀) - 1.96σ√T, ln(S₀) + 1.96σ√T]
[3.3393 - 1.96×0.22, 3.3393 + 1.96×0.22]
[2.9081, 3.7705]

95% Price Confidence Interval

$18.33 - $43.40

There is 95% probability that ETHA ETF price will be within this range after 3 months

Comparison Analysis

Log-Normal Advantages

  • Prevents negative asset prices
  • Better captures compounding returns
  • Consistent with Black-Scholes framework
  • Empirically validated for asset prices

Normal Distribution Limitations

  • Allows theoretically impossible negative prices
  • Does not account for compounding effects
  • Less accurate for longer time horizons
  • Inconsistent with option pricing theory

Key Insight

The 1.65 percentage point difference between models (17.26% vs 15.61%) demonstrates the material impact of distributional assumptions in financial risk modeling. The log-normal approach provides a more theoretically sound foundation while maintaining computational tractability.

Conclusions

Primary Findings

  • 17.26% probability of ETHA ETF reaching $20-$24 range using log-normal distribution
  • 15.61% probability using normal distribution of returns
  • 95% confidence interval of $18.33-$43.40 for 3-month price
  • 22% 3-month volatility derived from 44% annual figure

Recommendations

  • Prefer log-normal distribution for theoretical soundness
  • Consider sensitivity to volatility assumptions
  • Monitor actual volatility for model validation
  • Use confidence intervals for risk management

This analysis provides a quantitative foundation for risk assessment and investment decision-making, demonstrating the importance of appropriate statistical modeling in financial analysis.