# FMstable: Finite Moment Log Stable Distributions In FMStable: Finite Moment 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

 ```1 2 3 4 5``` ``` 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

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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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] ```

### Example output

```\$density
[1] 0.113639150 0.277743475 0.275637374 0.171134319 0.083715101 0.036120823
[7] 0.014582231 0.005692169 0.002190107 0.000840304

\$F
[1] 0.06046817 0.26235212 0.55443621 0.77969792 0.90368382 0.96072971
[7] 0.98456752 0.99403538 0.99770426 0.99911318

\$righttail
[1] 0.9395318258 0.7376478762 0.4455637942 0.2203020804 0.0963161839
[6] 0.0392702948 0.0154324778 0.0059646175 0.0022957398 0.0008868244

\$logdensity
[1] -2.174727 -1.281057 -1.288669 -1.765307 -2.480336 -3.320886 -4.227952
[8] -5.168664 -6.123805 -7.081747

\$logF
[1] -2.8056380987 -1.3380676938 -0.5898035269 -0.2488487169 -0.1012757407
[6] -0.0400621737 -0.0155527980 -0.0059824769 -0.0022983790 -0.0008872178

\$logrighttail
[1] -0.06237359 -0.30428870 -0.80841485 -1.51275558 -2.34011892 -3.23728690
[7] -4.17128104 -5.12191035 -6.07670014 -7.02786361

[1]  5.204170e-18 -4.163336e-17 -2.775558e-17  0.000000e+00  0.000000e+00
[6]  0.000000e+00
```

FMStable documentation built on May 29, 2017, 7:20 p.m.