LogNormalDistribution

Generate a set of random numbers that are lognormally distributed:

Compare the histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness grows exponentially with standard deviation :

Limiting values:

Kurtosis grows exponentially with standard deviation :

Limiting values:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

GammaDistribution data can be approximated by a lognormal distribution:

Comparing log-likelihoods with estimation by gamma distribution:

Lognormal distribution was traditionally used to analyze the fractional stock price changes from the previous closing price. Find the estimated distribution for the daily fractional price changes of the S&P 500 index from January 1, 2000 to January 1, 2009:

To fit lognormal distribution, all data must be positive:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the probability of the fractional price change being greater than 0.5%:

Find the mean fractional price change:

Simulate fractional price changes for 30 days:

Show that using LogisticDistribution provides better fit than using lognormal distribution:

Lognormal distribution can be used to model stock prices:

Fit the distribution to the data:

Compare the histogram to the PDF:

Find the probability that the price is above $500:

Find the mean price:

Simulate the price for the consecutive 30 days:

Lognormal distribution can be used to approximate wind speeds:

Find the estimated distribution:

Compare the PDF to the histogram of the wind data:

Find the probability of a day with wind speed greater than 30 km/h:

Find the mean wind speed:

Simulate wind speeds for a month:

The fractional change of stock price at time (in years) is assumed lognormally distributed with parameters and :

Compute expected stock price at epoch :

Assuming an investor can invest money for a year at a continuously compounded yearly rate risk-free, the risk-neutral pricing condition requires:

Solve for parameter :

Consider an option to buy this stock a year from now, at a fixed price . The value of such an option is:

The risk-neutral price of the option is determined as the present value of the expected option value:

Assuming rate of 5%, volatility parameter of 0.087, an initial price of $200 per share of stock, and a strike price of $190 per share, the Black-Scholes option price is:

Parameter influence on the CDF for each :

Lognormal distribution is closed under scaling by a positive factor:

Power of a LogNormalDistribution follows a lognormal distribution:

In particular, a reciprocal of a lognormal distribution follows a lognormal distribution:

The product of two independent lognormally distributed variates follows lognormal distribution:

Quotient of two independent lognormally distributed variates follows lognormal distribution:

Geometric mean of independent identically lognormally distributed variates follows lognormal distribution:

Relationships to other distributions:

NormalDistribution is exponentially related to LogNormalDistribution:

Reverse transformation:

Lognormal distribution is a special case of SL JohnsonDistribution:

SuzukiDistribution can be obtained from lognormal distribution and RayleighDistribution:

LogNormalDistribution is not defined when is not a real number:

LogNormalDistribution is not defined when is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

LogNormalDistribution is not uniquely determined by its sequence of moments:

Compute the sequence of moments:

Compare it to the sequence of moments of the LogNormalDistribution:

Plot distribution densities: