highfrequency (version 0.4)

spotvol: Spot volatility estimation

Description

The spotvol function offers several methods to estimate spot volatility and its intraday seasonality, using high-frequency data. It returns an object of class spotvol, which can contain various outputs, depending on the method used. See 'Details' for a description of each method. In any case, the output will contain the spot volatility estimates.

The input can consist of price data or return data, either tick by tick or sampled at set intervals. The data will be converted to equispaced high-frequency returns $r_{t,i}$ (read: the $i$th return on day $t$).

Usage

spotvol(data, method = "detper", ..., on = "minutes", k = 5, marketopen = "09:30:00", marketclose = "16:00:00", tz = "GMT")

Arguments

data

Either an xts object, containing price data, or a matrix containing returns. For price data, irregularly spaced observations are allowed. They will be aggregated to the level specified by parameters on and k. For return data, the observations are assumed to be equispaced, with the time between them specified by on and k. Return data should be in matrix form, where each row corresponds to a day, and each column to an intraday period. The output will be in the same form as the input (xts or matrix/numeric).

method

specifies which method will be used to estimate the spot volatility. Options include "detper" and "stochper". See 'Details'.

on

string indicating the time scale in which k is expressed. Possible values are: "secs", "seconds", "mins", "minutes", "hours".

k

positive integer, indicating the number of periods to aggregate over. E.g. to aggregate an xts object to the 5 minute frequency, set k = 5 and on = "minutes".

marketopen

the market opening time. This should be in the time zone specified by tz. By default, marketopen = "09:30:00".

marketclose

the market closing time. This should be in the time zone specified by tz. By default, marketclose = "16:00:00".

tz

string specifying the time zone to which the times in data and/or marketopen/ marketclose belong. Default = "GMT".

...

method-specific parameters (see 'Details').

Value

A spotvol object, which is a list containing one or more of the following outputs, depending on the method used:spotAn xts or matrix object (depending on the input) containing spot volatility estimates $\sigma_{t,i}$, reported for each interval $i$ between marketopen and marketclose for every day $t$ in data. The length of the intervals is specifiedby k and on. Methods that provide this output: All.dailyAn xts or numeric object (depending on the input) containing estimates of the daily volatility levels for each day $t$ in data, if the used method decomposed spot volatility into a daily and an intraday component. Methods that provide this output: "detper".periodicAn xts or numeric object (depending on the input) containing estimates of the intraday periodicity factor for each day interval $i$ between marketopen and marketclose, if the spot volatility was decomposed into a daily and an intraday component. If the output is in xts format, this periodicity factor will be dated to the first day of the input data, but it is identical for each day in the sample. Methods that provide this output: "detper".parA named list containing parameter estimates, for methods that estimate one or more parameters. Methods that provide this output: "stochper", "kernel".cpA vector containing the change points in the volatility, i.e. the observation indices after which the volatility level changed, according to the applied tests. The vector starts with a 0. Methods that provide this output: "piecewise".ugarchfitA ugarchfit object, as used by the rugarch package, containing all output from fitting the GARCH model to the data. Methods that provide this output: "garch".

Details

The following estimation methods can be specified in method:

Deterministic periodicity method ("detper")

Parameters:

dailyvol

A string specifying the estimation method for the daily component $s_t$. Possible values are "bipower", "rv", "medrv". Default = "bipower".

periodicvol

A string specifying the estimation method for the component of intraday volatility, that depends in a deterministic way on the intraday time at which the return is observed. Possible values are "TML", "SD", "WSD", "OLS". See Boudt et al. (2011) for details. Default = "TML".

P1

A positive integer corresponding to the number of cosinus terms used in the flexible Fourier specification of the periodicity function, see Andersen et al. (1997) for details. Default = 5.

P2

Same as P1, but for the sinus terms. Default = 5.

Outputs (see 'Value' for a full description of each component):

• spot
• daily
• periodic

The spot volatility is decomposed into a deterministic periodic factor $f_{i}$ (identical for every day in the sample) and a daily factor $s_{t}$ (identical for all observations within a day). Both components are then estimated separately. For more details, see Taylor and Xu (1997) and Andersen and Bollerslev (1997). The jump robust versions by Boudt et al. (2011) have also been implemented.

Stochastic periodicity method ("stochper")

Parameters:

P1

A positive integer corresponding to the number of cosinus terms used in the flexible Fourier specification of the periodicity function. Default = 5.

P2

Same as P1, but for the sinus terms. Default = 5.

init

A named list of initial values to be used in the optimization routine ("BFGS" in optim). Default = list(sigma = 0.03, sigma_mu = 0.005, sigma_h = 0.005, sigma_k = 0.05, phi = 0.2, rho = 0.98, mu = c(2, -0.5), delta_c = rep(0, max(1,P1)), delta_s = rep(0, max(1,P2))). See Beltratti & Morana (2001) for a definition of each parameter. init can contain any number of these parameters. For parameters not specified in init, the default initial value will be used.

Outputs (see 'Value' for a full description of each component):

• spot
• par

This method by Beltratti and Morana (2001) assumes the periodicity factor to be stochastic. The spot volatility estimation is split into four components: a random walk, an autoregressive process, a stochastic cyclical process and a deterministic cyclical process. The model is estimated using a quasi-maximum likelihood method based on the Kalman Filter. The package FKF is used to apply the Kalman filter. In addition to the spot volatility estimates, all parameter estimates are returned.

Nonparametric filtering ("kernel")

Parameters:

type

String specifying the type of kernel to be used. Options include "gaussian", "epanechnikov", "beta". Default = "gaussian".

h

Scalar or vector specifying bandwidth(s) to be used in kernel. If h is a scalar, it will be assumed equal throughout the sample. If it is a vector, it should contain bandwidths for each day. If left empty, it will be estimated. Default = NULL.

est

String specifiying the bandwidth estimation method. Possible values include "cv", "quarticity". Method "cv" equals cross-validation, which chooses the bandwidth that minimizes the Integrated Square Error. "quarticity" multiplies the simple plug-in estimator by a factor based on the daily quarticity of the returns. est is obsolete if h has already been specified by the user. Default = "cv".

lower

Lower bound to be used in bandwidth optimization routine, when using cross-validation method. Default is $0.1n^{-0.2}$.

upper

Upper bound to be used in bandwidth optimization routine, when using cross-validation method. Default is $n^{-0.2}$.

Outputs (see 'Value' for a full description of each component):

• spot
• par

This method by Kristensen (2010) filters the spot volatility in a nonparametric way by applying kernel weights to the standard realized volatility estimator. Different kernels and bandwidths can be used to focus on specific characteristics of the volatility process.

Estimation results heavily depend on the bandwidth parameter $h$, so it is important that this parameter is well chosen. However, it is difficult to come up with a method that determines the optimal bandwidth for any kind of data or kernel that can be used. Although some estimation methods are provided, it is advised that you specify $h$ yourself, or make sure that the estimation results are appropiate.

One way to estimate $h$, is by using cross-validation. For each day in the sample, $h$ is chosen as to minimize the Integrated Square Error, which is a function of $h$. However, this function often has multiple local minima, or no minima at all ($h -> \infty$). To ensure a reasonable optimum is reached, strict boundaries have to be imposed on $h$. These can be specified by lower and upper, which by default are $0.1n^{-0.2}$ and $n^{-0.2}$ respectively, where $n$ is the number of observations in a day.

When using the method "kernel", in addition to the spot volatility estimates, all used values of the bandwidth $h$ are returned.

Piecewise constant volatility ("piecewise")

Parameters:

type

String specifying the type of test to be used. Options include "MDa", "MDb", "DM". See Fried (2012) for details. Default = "MDa".

m

Number of observations to include in reference window. Default = 40.

n

Number of observations to include in test window. Default = 20.

alpha

Significance level to be used in tests. Note that the test will be executed many times (roughly equal to the total number of observations), so it is advised to use a small value for alpha, to avoid a lot of false positives. Default = 0.005.

volest

String specifying the realized volatility estimator to be used in local windows. Possible values are "bipower", "rv", "medrv". Default = "bipower".

online

Boolean indicating whether estimations at a certain point $t$ should be done online (using only information available at $t-1$), or ex post (using all observations between two change points). Default = TRUE.

Outputs (see 'Value' for a full description of each component):

• spot
• cp

This nonparametric method by Fried (2012) assumes the volatility to be piecewise constant over local windows. Robust two-sample tests are applied to detect changes in variability between subsequent windows. The spot volatility can then be estimated by evaluating regular realized volatility estimators within each local window.

Along with the spot volatility estimates, this method will return the detected change points in the volatility level. When plotting a spotvol object containing cp, these change points will be visualized.

GARCH models with intraday seasonality ("garch")

Parameters:

model

String specifying the type of test to be used. Options include "sGARCH", "eGARCH". See ugarchspec in the rugarch package. Default = "eGARCH".

garchorder

Numeric value of length 2, containing the order of the GARCH model to be estimated. Default = c(1,1).

dist

String specifying the distribution to be assumed on the innovations. See distribution.model in ugarchspec for possible options. Default = "norm".

solver.control

List containing solver options. See ugarchfit for possible values. Default = list().

P1

A positive integer corresponding to the number of cosinus terms used in the flexible Fourier specification of the periodicity function. Default = 5.

P2

Same as P1, but for the sinus terms. Default = 5.

Outputs (see 'Value' for a full description of each component):

• spot
• ugarchfit

This method generates the external regressors needed to model the intraday seasonality with a Flexible Fourier form. The rugarch package is then employed to estimate the specified GARCH(1,1) model.

Along with the spot volatility estimates, this method will return the ugarchfit object used by the rugarch package.

References

Andersen, T. G. and T. Bollerslev (1997). Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance 4, 115-158.

Beltratti, A. and C. Morana (2001). Deterministic and stochastic methods for estimation of intraday seasonal components with high frequency data. Economic Notes 30, 205-234.

Boudt K., Croux C. and Laurent S. (2011). Robust estimation of intraweek periodicity in volatility and jump detection. Journal of Empirical Finance 18, 353-367.

Fried, Roland (2012). On the online estimation of local constant volatilities. Computational Statistics and Data Analysis 56, 3080-3090.

Kristensen, Dennis (2010). Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory 26, 60-93.

Taylor, S. J. and X. Xu (1997). The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance 4, 317-340.

Examples

# Load sample data

data(sample_real5minprices)
data(sample_returns_5min)

# Default method, deterministic periodicity

vol1 <- spotvol(sample_real5minprices)
plot(vol1)

# Compare to stochastic periodicity

init = list(sigma = 0.03, sigma_mu = 0.005, sigma_h = 0.007,
            sigma_k = 0.06, phi = 0.194, rho = 0.986, mu = c(1.87,-0.42),
            delta_c = c(0.25, -0.05, -0.2, 0.13, 0.02), delta_s = c(-1.2,
            0.11, 0.26, -0.03, 0.08))
# next method will take around 110 iterations
vol2 <- spotvol(sample_real5minprices, method = "stochper", init = init)
plot(as.numeric(vol1$spot[1:780]), type="l")
lines(as.numeric(vol2$spot[1:780]), col="red")
legend("topright", c("detper", "stochper"), col = c("black", "red"), lty=1)

# Various kernel estimates

h1 = bw.nrd0((1:nrow(sample_returns_5min))*(5*60))
vol3 <- spotvol(sample_returns_5min, method = "kernel", h = h1)
vol4 <- spotvol(sample_returns_5min, method = "kernel", est = "quarticity")
vol5 <- spotvol(sample_returns_5min, method = "kernel", est = "cv")
plot(vol3, length = 2880)
lines(as.numeric(t(vol4$spot))[1:2880], col="red")
lines(as.numeric(t(vol5$spot))[1:2880], col="blue")
legend("topright", c("h = simple estimate", "h = quarticity corrected",
       "h = crossvalidated"), col = c("black", "red", "blue"), lty=1)

# Piecewise constant volatility, using an example from Fried (2012)

simdata <- matrix(sqrt(5/3)*rt(3000, df = 5), ncol = 500, byrow = TRUE)
simdata <- c(1, 1, 1.5, 1.5, 2, 1)*simdata
# the volatility of the simulated now changes at 1000, 2000 and 2500
vol6 <- spotvol(simdata, method = "piecewise", m = 200, n  = 100,
                online = FALSE)
plot(vol6)

# Compare regular GARCH(1,1) model to eGARCH, both with external regressors

vol7 <- spotvol(sample_returns_5min, method = "garch", model = "sGARCH")
vol8 <- spotvol(sample_returns_5min, method = "garch", model = "eGARCH")
plot(as.numeric(t(vol7$spot)), type = "l")
lines(as.numeric(t(vol8$spot)), col = "red")
legend("topleft", c("GARCH", "eGARCH"), col = c("black", "red"), lty=1)