FMstable | R Documentation |
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
.
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)
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
:
The probability density function.
The probability distribution function. i.e. the
probability of being less than or equal to x
.
The probability of being larger than x
.
The probability density function.
The logarithm of the probability of being less than
or equal to x
.
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
.
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]
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