RND (version 1.2)

dmln.am: Density of Mixture Lognormal for American Options

Description

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

Usage

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

Value

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.

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

###
### 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.
###