yuima (version 1.15.27)

mpv: Realized Multipower Variation

Description

The function returns the realized MultiPower Variation (mpv), defined in Barndorff-Nielsen and Shephard (2004), for each component.

Usage

mpv(yuima, r = 2, normalize = TRUE)

Value

A numeric vector with the same length as the zoo.data of yuima

Arguments

yuima

an object of yuima-class or yuima.data-class.

r

a vector of non-negative numbers or a list of vectors of non-negative numbers.

normalize

logical. See `Details'.

Author

Yuta Koike with YUIMA Project Team

Details

Let d be the number of the components of the zoo.data of yuima.

Let \(X^i_{t_0},X^i_{t_1},\dots,X^i_{t_n}\) be the observation data of the \(i\)-th component (i.e. the \(i\)-th component of the zoo.data of yuima).

When \(r\) is a \(k\)-dimensional vector of non-negative numbers, mpv(yuima,r,normalize=TRUE) is defined as the d-dimensional vector with i-th element equal to $$\mu_{r[1]}^{-1}\cdots\mu_{r[k]}^{-1}n^{\frac{r[1]+\cdots+r[k]}{2}-1}\sum_{j=1}^{n-k+1}|\Delta X^i_{t_{j}}|^{r[1]}|\Delta X^i_{t_{j+1}}|^{r[2]}\cdots|\Delta X^i_{t_{j+k-1}}|^{r[k]},$$ where \(\mu_p\) is the p-th absolute moment of the standard normal distribution and \(\Delta X^i_{t_{j}}=X^i_{t_j}-X^i_{t_{j-1}}\). If normalize is FALSE the result is not multiplied by \(\mu_{r[1]}^{-1}\cdots\mu_{r[k]}^{-1}\).

When \(r\) is a list of vectors of non-negative numbers, mpv(yuima,r,normalize=TRUE) is defined as the d-dimensional vector with i-th element equal to $$\mu_{r^i_1}^{-1}\cdots\mu_{r^i_{k_i}}^{-1}n^{\frac{r^i_1+\cdots+r^i_{k_i}}{2}-1}\sum_{j=1}^{n-k_i+1}|\Delta X^i_{t_{j}}|^{r^i_1}|\Delta X^i_{t_{j+1}}|^{r^i_2}\cdots|\Delta X^i_{t_{j+k_i-1}}|^{r^i_{k_i}},$$ where \(r^i_1,\dots,r^i_{k_i}\) is the i-th component of r. If normalize is FALSE the result is not multiplied by \(\mu_{r^i_1}^{-1}\cdots\mu_{r^i_{k_i}}^{-1}\).

References

Barndorff-Nielsen, O. E. and Shephard, N. (2004) Power and bipower variation with stochastic volatility and jumps, Journal of Financial Econometrics, 2, no. 1, 1--37.

Barndorff-Nielsen, O. E. , Graversen, S. E. , Jacod, J. , Podolskij M. and Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales, in: Kabanov, Y. , Lipster, R. , Stoyanov J. (Eds.), From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, Springer-Verlag, Berlin, pp. 33--68.

See Also

setData, cce, minrv, medrv

Examples

if (FALSE) {
set.seed(123)

# One-dimensional case
## Model: dXt=t*dWt+t*dzt, 
## where zt is a compound Poisson process with intensity 5 and jump sizes distribution N(0,0.1). 

model <- setModel(drift=0,diffusion="t",jump.coeff="t",measure.type="CP",
                  measure=list(intensity=5,df=list("dnorm(z,0,sqrt(0.1))")),
                  time.variable="t")

yuima.samp <- setSampling(Terminal = 1, n = 390) 
yuima <- setYuima(model = model, sampling = yuima.samp) 
yuima <- simulate(yuima)
plot(yuima)

mpv(yuima) # true value is 1/3
mpv(yuima,1) # true value is 1/2
mpv(yuima,rep(2/3,3)) # true value is 1/3

# Multi-dimensional case
## Model: dXkt=t*dWk_t (k=1,2,3).

diff.matrix <- diag(3)
diag(diff.matrix) <- c("t","t","t")
model <- setModel(drift=c(0,0,0),diffusion=diff.matrix,time.variable="t",
                  solve.variable=c("x1","x2","x3"))

yuima.samp <- setSampling(Terminal = 1, n = 390) 
yuima <- setYuima(model = model, sampling = yuima.samp) 
yuima <- simulate(yuima)
plot(yuima)

mpv(yuima,list(c(1,1),1,rep(2/3,3))) # true varue is c(1/3,1/2,1/3)

}