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Skew Lognormal Cascade DistributionStephen Lihn / research archive

Quantitative finance
2008 to 2026

Simulation / interactive

Simulation and Demos

Set the three parameters and watch the probability density and cumulative probability respond. The market curve is a good place to start.

InteractiveThree parametersGNU Octave origin

Parameters

The browser version uses numerical quadrature. The original demonstration was computed with GNU Octave. The analytic method is described in the 2008 paper.

Probability density

Grid preview

Chart window: x equals negative 20 to 20.

Cumulative probability

Area normalized

Grid checks

StatisticGridAnalytic

The simulator accepts the same three parameters as the original demonstration: eta, beta and growth. Eta controls tail fatness, beta controls skew and growth shifts the center. Phi is fixed at one, as in the original demo.

Two practical notes. The tail gets fatter rapidly when eta is above 0.5. The chart window is fixed at x from negative 20 to 20, so a very large beta can push most of the mass outside the view.

Compare with market fitsSee the empirical series in the data atlas.Read the implementation notesThe original routines were written in GNU Octave.Study the N^3/2 lawHow the market level scaling follows from the cascade.

Source shelf

Data and further reading

11 entries
  1. Lihn 2008 on SSRNThe 2008 working paper on the scale invariant Brownian motion model.
  2. Lihn 2009 on SSRNThe working paper on higher order lognormal cascade distributions.
  3. Clark 1973Subordinated stochastic process model for speculative prices.
  4. Fat tailed distributionsA reference overview of heavy tailed probability distributions.
  5. Lognormal distributionDefinition and properties of the lognormal distribution.
  6. Stable distributionBackground on stable distributions and heavy tails.
  7. GNU OctaveThe GNU Octave scientific programming environment.
  8. MaximaThe Maxima computer algebra system.
  9. Fernholz 2002Stochastic portfolio theory and the diversity index.
  10. Mandelbrot 1963The variation of certain speculative prices.
  11. Probability density functionA reference overview of probability density.