| Title: | Brownian Semistationary Processes |
|---|---|
| Description: | Efficient simulation of Brownian semistationary (BSS) processes using the hybrid simulation scheme, as described in Bennedsen, Lunde, Pakkannen (2017) <arXiv:1507.03004v4>, as well as functions to fit BSS processes to data, and functions to estimate the stochastic volatility process of a BSS process. |
| Authors: | Phillip Murray [aut, cre] |
| Maintainer: | Phillip Murray <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.1.0 |
| Built: | 2024-12-19 06:27:25 UTC |
| Source: | CRAN |
bssAlphaFit uses the 'Change of Frequency' method to estimate the smoothness parameter, alpha,
of a BSS process. The COF method needs only minimal assumptions on the parametric form of the kernel,
therefore the estimate can be used in any kernel.
bssAlphaFit(Y, p = 2)bssAlphaFit(Y, p = 2)
Y |
a vector of observations of a BSS process at any frequency. |
p |
the power to be used in the change of frequency method. The default value is p = 2. |
The function returns a single value - an estimate for the smoothness parameter alpha.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss bssAlphaFit(y, p = 2)N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss bssAlphaFit(y, p = 2)
estimateAccumulatedVolatility estimates the pth power accumulated volatility process for a Brownian
semistationary process, using either parametric methods of model fitting first, or using a non-parametric
estimator for the scale factor.
estimateAccumulatedVolatility( Y, n, p = 2, method = "nonparametric", kernel = "gamma" )estimateAccumulatedVolatility( Y, n, p = 2, method = "nonparametric", kernel = "gamma" )
Y |
a vector of observations of a BSS process. |
n |
positive integer indicating the number of observations per unit of time. |
p |
the power to evaluate the accumulated power volatility process for. Defaults to 2, in order to estimate the accumulated squared volatility process. |
method |
text string representing the method used to estimate the accumulated volatility. Options are |
kernel |
text string representing the choice of kernel when fitting the model to estimate
the scale factor parametrically. Options are |
The function returns a vector of the same length as Y which is the estimate for the
accumulated volatility process, observed from time 0 to T, at intervals of T/n. Note that the values have been
divided by m_p in the output, so that the estimation is of the integral alone. If the non-parametric
estimator for tau_n is used then the values will be scaled by the expectation of the squared volatility, as
per the theory.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss estimateAccumulatedVolatility(y, n, p = 2, method = 'nonparametric', kernel = 'gamma') #'N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss estimateAccumulatedVolatility(y, n, p = 2, method = 'nonparametric', kernel = 'gamma') #'
estimateAccumulatedVolatility estimates a confidence interval for the pth power accumulated volatility process for a Brownian
semistationary process, using either parametric methods of model fitting first, or using a non-parametric
estimator for the scale factor.
estimateAccumulatedVolatilityCI( Y, n, p, method = "nonparametric", kernel = "gamma", confidence_level )estimateAccumulatedVolatilityCI( Y, n, p, method = "nonparametric", kernel = "gamma", confidence_level )
Y |
a vector of observations of a BSS process. |
n |
positive integer indicating the number of observations per unit of time. |
p |
the power to evaluate the accumulated power volatility process for. Defaults to 2, in order to estimate the accumulated squared volatility process. |
method |
text string representing the method used to estimate the accumulated volatility. Options are |
kernel |
text string representing the choice of kernel when fitting the model to estimate
the scale factor parametrically. Options are |
confidence_level |
the required level for the confidence interval, as a probability between 0 and 1. |
The function returns a list of two vectors of the same length as Y which are the estimates for the
lower and upper values for the confidence interval. Note that the values have been
divided by m_p in the output, so that the estimation is of the integral alone. If the non-parametric
estimator for tau_n is used then the values will be scaled by the expectation of the squared volatility, as
per the theory.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss estimateAccumulatedVolatility(y, n, p = 2, method = 'nonparametric', kernel = 'gamma') #'N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss estimateAccumulatedVolatility(y, n, p = 2, method = 'nonparametric', kernel = 'gamma') #'
exponentiatedOrnsteinUhlenbeck simulates an exponentiated Ornstein-Uhlenbeckprocess of the correct length to
be used as the volatility process within the hybrid scheme.
exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta)exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta)
N |
positive integer determining the number of terms in the Riemman sum element of the
hybrid scheme calculation. Should be of order at least |
n |
positive integer indicating the number of observations per unit of time. It represents the fineness or frequency of observations. |
T |
the time interval to simulate the BSS process over. |
theta |
positive number giving the mean reversion rate of the OU process. |
beta |
the factor in the exponential. |
The function returns a vector of length N + n*T + 1
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta)N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta)
gammaKernelBSS uses the Hybrid scheme to simulate a Brownian semistationary process from the
gamma kernel. It simulates a path where the volatility process is independent of the driving Brownian motion of the
BSS process.
gammaKernelBSS( N, n, T, kappa = 3, alpha, lambda, sigma = rep(1, N + n * T + 1) )gammaKernelBSS( N, n, T, kappa = 3, alpha, lambda, sigma = rep(1, N + n * T + 1) )
N |
positive integer determining the number of terms in the Riemman sum element of the
hybrid scheme calculation. Should be of order at least |
n |
positive integer indicating the number of observations per unit of time. It represents the fineness or frequency of observations. |
T |
the time interval to simulate the BSS process over. |
kappa |
positive integer giving the number of terms to use in the 'lower' sum of the hybrid scheme. Default set to 3. |
alpha |
the smoothness parameter of the BSS process to simulate. |
lambda |
the exponent parameter of the BSS process to simulate. |
sigma |
the volatility process used in the BSS simulation. This should be a vector of length |
The function returns a list of three objects, core gives the Gaussian core of the process
between 0 and T, at intervals of 1/n. bss gives the BSS sample path on the between 0 and T, at intervals of 1/n,
and vol gives the volatilty process over the same time period.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125
kappa <- 3 alpha <- -0.2 lambda <- 1.0
vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol)
gammaKernelBSSFit uses a method of moments to fit the parameters of a gamma kernel Brownian semistationary process
to a vector of observations. A least squares estimate of the parameters is obtained
by minimising the mean square error between the true gamma kernel autocorrelation function and the
empirical ACF of the data, using lags 0,...,H. The number of lags num_lags used can be adjusted.
The volatility process does not need to be specified.
gammaKernelBSSFit(Y, n, num_lags = 10)gammaKernelBSSFit(Y, n, num_lags = 10)
Y |
a vector of observations of a BSS process at frequency |
n |
positive integer indicating the number of observations per unit of time. |
num_lags |
the number of lags to be used in the regression. The default is to use the first 10 lags. |
The function returns a list containing the parameters alpha and lambda, and also the mean square
error mse of the least squares fit. This can be used to compare model fit when trying different kernels.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss gammaKernelBSSFit(y, n, num_lags = 10)N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta <- 0.125 kappa <- 3 alpha <- -0.2 lambda <- 1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, lambda, sigma = vol) y <- bss_simulation$bss gammaKernelBSSFit(y, n, num_lags = 10)
gammaKernelCorrelation calculates the value of the gamma kernel autocorrelation function
directly using the analytic expression.
gammaKernelCorrelation(alpha, lambda, h)gammaKernelCorrelation(alpha, lambda, h)
alpha |
the smoothness parameter, alpha, for the gamma kernel. |
lambda |
the exponent parameter, lambda, for the gamma kernel. |
h |
the lag to calculate the autocorrelation at. |
The function returns the autocorrelation for the gamma kernel with
parameters alpha and lambda at lag h.
gammaKernelTau evaluates the scale factor tau_n for the gamma kernel using the
exact expression derived from the covariance function.
gammaKernelTau(n, alpha, lambda)gammaKernelTau(n, alpha, lambda)
n |
a positive integer indicating the number of observations per unit of time. |
alpha |
the smoothness parameter, alpha, for the gamma kernel. |
lambda |
the exponent parameter, lambda, for the gamma kernel. |
The function returns the scale factor (tau_n) for the gamma kernel with
parameters alpha and lambda, observed at frequency n per unit of time.
Asymptotic scale factor for the gamma kernel
gammaKernelTauAsymptotic(n, alpha)gammaKernelTauAsymptotic(n, alpha)
n |
a positive integer indicating the number of observations per unit of time. |
alpha |
the smoothness parameter, alpha, for the gamma kernel. |
The function returns an approximation for the scale factor (tau_n) for the gamma kernel with
smoothness parameter alpha, observed at frequency n per unit of time, using the asymptotic
expression for the scale factor.
Generates the covariance matrix used in simulating Brownian semistationary processes by the hybrid scheme.
hybridSchemeCovarianceMatrix(kappa, n, alpha)hybridSchemeCovarianceMatrix(kappa, n, alpha)
kappa |
number of terms needed for the lower sum in the hybrid scheme. |
n |
number of observations per unit of time, n = 1/delta. |
alpha |
smoothness parameter used in the BSS simulation. |
Returns the covariance matrix for the lower sum in the hybrid scheme calculations. The dimensions of the covariance matrix will be (kappa + 1) by (kappa + 1).
kappa <- 3 n <- 100 alpha <- -0.2 hybridSchemeCovarianceMatrix(kappa, n, alpha)kappa <- 3 n <- 100 alpha <- -0.2 hybridSchemeCovarianceMatrix(kappa, n, alpha)
powerKernelBSS uses the Hybrid scheme to simulate a Brownian semistationary process from the
power law kernel. It simulates a path where the volatility process is independent of the driving Brownian motion of the
BSS process.
powerKernelBSS(N, n, T, kappa, alpha, beta, sigma = rep(1, N + n * T + 1))powerKernelBSS(N, n, T, kappa, alpha, beta, sigma = rep(1, N + n * T + 1))
N |
positive integer determining the number of terms in the Riemman sum element of the
hybrid scheme calculation. Should be of order at least |
n |
positive integer indicating the number of observations per unit of time. It represents the fineness or frequency of observations. |
T |
the time interval to simulate the BSS process over. |
kappa |
positive integer giving the number of terms to use in the 'lower' sum of the hybrid scheme. Default set to 3. |
alpha |
the smoothness parameter of the BSS process to simulate. |
beta |
the exponent parameter of the BSS process to simulate. |
sigma |
the volatility process used in the BSS simulation. This should be a vector of length |
The function returns a list of three objects, core gives the Gaussian core of the process
between 0 and T, at intervals of 1/n. bss gives the BSS sample path on the between 0 and T, at intervals of 1/n,
and vol gives the volatilty process over the same time period.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta_vol <- 0.125 kappa <- 3 alpha <- -0.2 beta_pwr <- -1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta_vol) bss_simulation <- powerKernelBSS(N, n, T, kappa, alpha, beta_pwr, sigma = vol)N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta_vol <- 0.125 kappa <- 3 alpha <- -0.2 beta_pwr <- -1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta_vol) bss_simulation <- powerKernelBSS(N, n, T, kappa, alpha, beta_pwr, sigma = vol)
powerKernelBSSFit uses a method of moments to fit the parameters of a power law kernel Brownian semistationary process
to a vector of observations. A least squares estimate of the parameters is obtained
by minimising the mean square error between the true power law kernel autocorrelation function (found by numerical intergration)
and the empirical ACF of the data, using lags 0,...,H. The number of lags num_lags used can be adjusted.
The volatility process does not need to be specified.
powerKernelBSSFit(Y, n, num_lags = 10)powerKernelBSSFit(Y, n, num_lags = 10)
Y |
a vector of observations of a BSS process at frequency |
n |
positive integer indicating the number of observations per unit of time. |
num_lags |
the number of lags to be used in the regression. The default is to use the first 10 lags. |
The function returns a list containing the parameters alpha and beta, and also the mean square
error mse of the least squares fit. This can be used to compare model fit when trying different kernels.
N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta_vol <- 0.125 kappa <- 3 alpha <- -0.2 beta_pwr <- -1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta_vol) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, beta_pwr, sigma = vol) y <- bss_simulation$bss powerKernelBSSFit(y, n, num_lags = 10)N <- 10000 n <- 100 T <- 1.0 theta <- 0.5 beta_vol <- 0.125 kappa <- 3 alpha <- -0.2 beta_pwr <- -1.0 vol <- exponentiatedOrnsteinUhlenbeck(N, n, T, theta, beta_vol) bss_simulation <- gammaKernelBSS(N, n, T, kappa, alpha, beta_pwr, sigma = vol) y <- bss_simulation$bss powerKernelBSSFit(y, n, num_lags = 10)
powerKernelCorrelation calculates the value of the power law kernel autocorrelation function
directly using numerical integration for the numerator (the covariance term) and the analytic expression
for the denominator (variance term).
powerKernelCorrelation(alpha, beta, h)powerKernelCorrelation(alpha, beta, h)
alpha |
the smoothness parameter, alpha, for the power law kernel. |
beta |
the exponent parameter, beta, for the power law kernel. |
h |
the lag to calculate the autocorrelation at. |
The function returns the autocorrelation for the power law kernel with
parameters alpha and beta at lag h.
powerKernelTau evaluates the scale factor tau_n for the power law kernel using
numerical integration for the covariance term, and exact evaluation for the variance term.
powerKernelTau(n, alpha, beta)powerKernelTau(n, alpha, beta)
n |
a positive integer indicating the number of observations per unit of time. |
alpha |
the smoothness parameter, alpha, for the power law kernel. |
beta |
the exponent parameter, beta, for the power law kernel. |
The function returns the scale factor (tau_n) for the power law kernel with
parameters alpha and beta, observed at frequency n per unit of time.
realisedPowerVariation calculates the realised power variation for a BSS process, which is then
used as a component of estimating the accumulated volatility process.
realisedPowerVariation(Y, p)realisedPowerVariation(Y, p)
Y |
a vector of observations of a BSS process. |
p |
the power variation to be calculated. |
The function returns the sum of the pth powers of the absolute first differences of the BSS process.
Non-parametric estimate of the scale factor
tauNonParametricEstimate(Y)tauNonParametricEstimate(Y)
Y |
a vector of observations of a BSS process. |
The function returns the non-parametric estimate for the scale factor. Note that this will be scaled by the expectation of the square of the volatitity.