Description Usage Arguments Details Value References See Also Examples

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`

.

1 2 3 4 5 |

`x` |
Vector of quantiles. |

`stableParamObj` |
An object of class |

`mean` |
Mean of logstable distribution. |

`sd` |
Standard deviation of logstable distribution. |

`p` |
Vector of tail probabilities. |

`log` |
Logical; if |

`lower.tail` |
Logical; if |

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

`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`

.

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`

.

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]
``` |

```
$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
```

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