# dmln.am: Density of Mixture Lognormal for American Options In RND: Risk Neutral Density Extraction Package

## Description

`mln.am` is the probability density function of a mixture of three lognormal densities.

## Usage

 `1` ```dmln.am(x, u.1, u.2, u.3, sigma.1, sigma.2, sigma.3, p.1, p.2) ```

## Arguments

 `x` value at which the denisty is to be evaluated `u.1` log mean of the first lognormal `u.2` log mean of the second lognormal `u.3` log mean of the third lognormal `sigma.1` log standard deviation of the first lognormal `sigma.2` log standard deviation of the second lognormal `sigma.3` log standard deviation of the third lognormal `p.1` weight assigned to the first density `p.2` weight assigned to the second density

## Details

mln is density f(x) = p.1 * f1(x) + p.2 * f2(x) + (1 - p.1 - p.2) * f3(x), where f1, f2, and f3 are lognormal densities with log means u.1,u.2, and u.3 and standard deviations sigma.1, sigma.2, and sigma.3 respectively.

## Value

 `out ` density value at x

Kam Hamidieh

## References

Melick, W. R. and Thomas, C. P. (1997). Recovering an asset's implied pdf from option prices: An application to crude oil during the gulf crisis. Journal of Financial and Quantitative Analysis, 32(1), 91-115.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```### ### Just look at a generic density and see if it integrates to 1. ### u.1 = 4.2 u.2 = 4.5 u.3 = 4.8 sigma.1 = 0.30 sigma.2 = 0.20 sigma.3 = 0.15 p.1 = 0.25 p.2 = 0.45 x = seq(from = 0, to = 250, by = 0.01) y = dmln.am(x = x, u.1 = u.1, u.2 = u.2, u.3 = u.3, sigma.1 = sigma.1, sigma.2 = sigma.2, sigma.3 = sigma.3, p.1 = p.1, p.2 = p.2) plot(y ~ x, type="l") sum(y * 0.01) ### ### Yes, the sum is near 1. ### ```

### Example output ``` 0.9999984
```

RND documentation built on May 1, 2019, 10:52 p.m.