FMstable {FMStable}

Finite Moment Log Stable Distributions

Description

Density function, distribution function, and quantile function for a log stable distribution with location, scale and shape parameters. For such families of distributions all moments are finite. Carr and Wu (2003) refer to such distributions as “finite moment log stable processes”.

The finite moment log stable distribution is well-defined for \alpha=0, when the distribution is discrete with probability concentrated at x=0 and at one other point. The distribution function may be computed by pFMstable.alpha0.

Usage

  dFMstable(x, stableParamObj, log=FALSE) 
  pFMstable(x, stableParamObj, log=FALSE, lower.tail=TRUE)
  pFMstable.alpha0(x, mean=1, sd=1, lower.tail=TRUE)
  qFMstable(p, stableParamObj, lower.tail=TRUE)
  tailsFMstable(x, stableParamObj)

Arguments

x

Vector of quantiles.

stableParamObj

An object of class stableParameters which describes a maximally skew stable distribution. It may, for instance, have been created by setMomentsFMstable or fitGivenQuantile.

mean

Mean of logstable distribution.

sd

Standard deviation of logstable distribution.

p

Vector of tail probabilities.

log

Logical; if TRUE, the log density or log tail probability is returned by functions dFMstable and pFMstable; and logarithms of probabilities are input to function qFMstable.

lower.tail

Logical; if TRUE, the lower tail probability is returned. Otherwise, the upper tail probability.

Details

The values are worked out by interpolation, with several different interpolation formulae in various regions.

Value

dFMstable gives the density function; pFMstable gives the distribution function or its complement; qFMstable gives quantiles; tailsFMstable returns a list with the following components which are all the same length as x:

density

The probability density function.

F

The probability distribution function. i.e. the probability of being less than or equal to x.

righttail

The probability of being larger than x.

logdensity

The probability density function.

logF

The logarithm of the probability of being less than or equal to x.

logrighttail

The logarithm of the probability of being larger than x.

References

Carr, P. and Wu, L. (2003). The Finite Moment Log Stable Process and Option Pricing. Journal of Finance, American Finance Association, vol. 58(2), pages 753-778

See Also

If a random variable X has a finite moment stable distribution then \log(X) has the corresponding extremal stable distribution. The density of \log(X) can be found using dEstable. Option prices can be found using callFMstable and putFMstable.

Examples

tailsFMstable(1:10, setMomentsFMstable(3, 1.5, alpha=1.7))

x <- c(-1, 0, 1.e-5, .001, .01, .03, seq(from=.1, to=4.5, length=100))
plot(x, pFMstable(x, setMomentsFMstable(1, 1.5, 2)), type="l" ,xlim=c(0, 4.3),
  ylim=c(0,1), ylab="Distribution function")
for (alpha in c(.03, 1:19/10)) lines(x, pFMstable(x,
  setMomentsFMstable(1, 1.5, alpha)), col=2)
lines(x, pFMstable.alpha0(x, mean=1, sd=1.5), col=3)

p <- c(1.e-10, .01, .1, .2, .5, .99, 1-1.e-10)
obj <- setMomentsFMstable(alpha=1.95)
result <- qFMstable(p, obj)
OK <- result > 0
pFMstable(result[OK], obj) - p[OK]