Modeling Framework
Probability Distributions
Laplace
The Laplace (Double Exponential) function is a probability distribution with a higher peak and fatter tails than the Gaussian (Normal) distribution—two properties observed empirically in daily and monthly stock return data.
Normal
The Gaussian (Normal) function is the most common probability distribution used to model portfolio returns in professional finance software. Rooted in academic tradition, this approach assumes that returns conform to the classic bell curve.
Student
The Student’s t-distribution, like the normal distribution, is symmetrical and bell-shaped, but with a shorter peak and much heavier tails. Modeling returns with the Student distribution produces highly volatile and extreme results.
Time Series Models & Other Calibrations
Random Walk
The Random Walk Theory suggests that changes in stock prices move in random (or unpredictable) paths that render it impossible to accurately and reliably predict returns over the long-run. Simulations generating investment returns as a random walk will produce a greater dispersion of portfolio results (higher highs and lower lows).
Mean Reversion
Some research suggests that stock prices move in cycles or trends, and tend to gravitate toward a long-term average. For example, high performing periods (bull markets) are often followed by low performing periods (bear markets), and vice versa. Simulations generating investment returns with mean-reverting tendencies produce a narrower range of long-term outcomes.
Volatility Clustering
Stock prices have been observed to exhibit volatility clustering (or momentum) over shorter periods (days, weeks, or months) due to economic sentiment, reactions to world events, psychological phenomena, or other forces. Simulations calibrated with a momentum factor will exhibit occasional spikes in volatility.
Fat Tails & Black Swans
Stochastic modeling techniques can be added to probability models to accommodate unusually large and random movements in asset prices occasionally exhibited in markets. Honest Math allows users to calibrate the “fatness” of the tails, and to specify extreme portfolio shocks by dictating a large drop in portfolio value across all simulation trials.
