spotVol {highfrequency}R Documentation

Spot volatility estimation

Description

Estimates a wide variety of spot volatility estimators.

Usage

spotVol(
  data,
  method = "detPer",
  alignBy = "minutes",
  alignPeriod = 5,
  marketOpen = "09:30:00",
  marketClose = "16:00:00",
  tz = "GMT",
  ...
)

Arguments

data

Can be one of two input types, xts or data.table. It is assumed that the input comprises prices in levels. Irregularly spaced observations are allowed. They will be aggregated to the level specified by parameters alignBy and alignPeriod.

method

specifies which method will be used to estimate the spot volatility. Valid options are "detPer", "stochPer" "kernel" "piecewise" "garch", "RM" ,"PARM" See ‘Details’ below for explanation and parameters to use in each of the methods.

alignBy

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

alignPeriod

positive integer, indicating the number of periods to aggregate over. For example, to aggregate an xts object to the 5-minute frequency, set alignPeriod = 5 and alignBy = "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

fallback time zone used in case we we are unable to identify the timezone of the data, by default: tz = NULL. We attempt to extract the timezone from the DT column (or index) of the data, which may fail. In case of failure we use tz if specified, and if it is not specified, we use "UTC"

...

method-specific parameters (see ‘Details’ below).

Details

The following estimation methods can be specified in method:

Deterministic periodicity method ("detPer")

Parameters:

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

Let there be T days of N equally-spaced log-returns r_{i,t}, i = 1, \dots, N and i = 1, \dots, T. In case of method = "detPer", the returns are modeled as

r_{i,t} = f_i s_t u_{i,t}

with independent u_{i,t} \sim \mathcal{N}(0,1). 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, see Taylor and Xu (1997) and Andersen and Bollerslev (1997). The jump robust versions by Boudt et al. (2011) have also been implemented.

If periodicVol = "SD", we have

\hat f_i^{SD} = \frac{SD_i}{\sqrt{\frac{1}{\lfloor{\lambda / \Delta}\rfloor} \sum_{j = 1}^N SD_j^2}}

with \Delta = 1 / N, cross-daily averages SD_i = \sqrt{1/T \sum_{i = t}^T r_{i,t}^2}, and \lambda being the length of the intraday time intervals.

If periodicVol = "WSD", we have another nonparametric estimator that is robust to jumps in contrast to periodicVol = "SD". The definition of this estimator can be found in Boudt et al. (2011, Eqs. 2.9-2.12).

The estimates when periodicVol = "OLS" and periodicVol = "TML" are based on the regression equation

\log \left| 1/T \sum_{t = 1}^T r_{i,t} \right| - c = \log f_i + \varepsilon_i

with i.i.d. zero-mean error term \varepsilon_i and c = -0.63518. periodicVol = "OLS" employs ordinary-least-squares estimation and periodicVol = "TML" truncated maximum-likelihood estimation (see Boudt et al., 2011, Section 2.2, for further details).

Stochastic periodicity method ("stochPer")

Parameters:

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

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.

The model for the intraday change in the return series is given by

r_{t,n} = \sigma_{t,n} \varepsilon_{t,n}, \ t = 1, \dots, T; \ n = 1, \dots, N,

where \sigma_{t,n} is the conditional standard deviation of the n-th interval of day t and \varepsilon_{t,n} is a i.i.d. mean-zero unit-variance process. The conditional standard deviations are modeled as

\sigma_{t,n} = \sigma \exp \left(\frac{\mu_{t,n} + h_{t,n} + c_{t,n}}{2} \right)

with \sigma being a scaling factor and \mu_{t,n} is the non-stationary volatility component

\mu_{t,n} = \mu_{t,n-1} + \xi_{t,n}

with independent \xi_{t,n} \sim \mathcal{N}(0,\sigma_\xi^2). h_{t,n} is the stochastic stationary acyclical volatility component

h_{t,n} = \phi h_{t,n-1} + \nu_{t,n}

with independent \eta_{t,n} \sim \mathcal{N}(0,\sigma_\eta^2) and | \phi | \leq 1. The cyclical component is separated in two components:

c_{t,n} = c_{1,t,n} + c_{2,t,n}

The first component is written in state-space form,

\left( \begin{array}{r} c_{1,t,n} \\ c_{1,t,n}^* \end{array}\right) = \rho \left(\begin{array}{rr} \cos \lambda & \sin \lambda \\ -\sin \lambda & \cos \lambda \end{array}\right) \left(\begin{array}{r} c_{1,t,n - 1} \\ c_{1,t,n-1}^* \end{array}\right) + \left(\begin{array}{r} \kappa_{1,t,n} \\ \kappa_{1,t,n}^* \end{array}\right)

with 0 \leq \rho \leq 1 and \kappa_{1,t,n}, \kappa_{1,t,n}^* are mutually independent zero-mean normal random variables with variance \sigma_\kappa^2. All other parameters and the process c_{1,t,n}^* in the state-space representation are only of instrumental use and are not part of the return value which is why we won't introduce them in detail in this vignette; see Beltratti and Morana (2001, pp. 208-209) for more information.

The second component is given by

c_{2,t,n} = \mu_1 n_1 + \mu_2 n_2 + \sum_{p = 2}^P (\delta_{cp} \cos(p\lambda) + \delta_{sp} \sin (p \lambda n))

with n_1 = 2n / (N+1) and n_2 = 6n^2 / (N+1) / (N+2).

Nonparametric filtering ("kernel")

Parameters:

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

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 appropriate.

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 \rightarrow \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.

A formal definition of the estimator is too extensive for the context of this vignette. Please refer to Kristensen (2010) for more detailed information. Our parameter names are aligned with this reference.

Piecewise constant volatility ("piecewise")

Parameters:

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

This nonparametric method by Fried (2012) is a two-step approach and 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. "MDa", "MDb" refer to different test statistics, see Section 2.2 in Fried (2012).

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:

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

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

In this model, daily returns r_t based on intraday observations r_{i,t}, i = 1, \dots, N are modeled as

r_t = \sum_{i = 1}^N r_{i,t} = \sigma_t \frac{1}{\sqrt{N}} \sum_{i = 1}^N s_i Z_{i,t}.

with \sigma_t > 0, intraday seasonality s_i > 0, and Z_{i,t} being a zero-mean unit-variance error term.

The overall approach is as in Appendix B of Andersen and Bollerslev (1997). This method generates the external regressors s_i needed to model the intraday seasonality with a flexible Fourier form (Andersen and Bollerslev, 1997, Eqs. A.1-A.4). The rugarch package is then employed to estimate the specified intraday GARCH(1,1) model on the residuals r_{i,t} / s_i.

Realized Measures ("RM")

This estimator takes trailing rolling window observations of intraday returns to estimate the spot volatility.

Parameters:

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

This method returns the estimates of the spot volatility, a string containing the realized measure used, and the lookBackPeriod.

(Non-overlapping) Pre-Averaged Realized Measures ("PARM")

This estimator takes rolling historical window observations of intraday returns to estimate the spot volatility as in the option "RM" but adds return pre-averaging of the realized measures. For a description of return pre-averaging see the details on spotDrift.

Parameters:

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

Value

A spotVol object, which is a list containing one or more of the following outputs, depending on the method used:

Author(s)

Jonathan Cornelissen, Kris Boudt, Onno Kleen, and Emil Sjoerup.

References

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

Beltratti, A. and Morana, C. (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, R. (2012). On the online estimation of local constant volatilities. Computational Statistics and Data Analysis, 56, 3080-3090.

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

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

Examples

## Not run: 
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 370 iterations
vol1 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)], method = "stochPer", init = init)
plot(vol1$spot[1:780])
legend("topright", c("stochPer"), col = c("black"), lty=1)
## End(Not run)

# Various kernel estimates
## Not run: 
h1 <- bw.nrd0((1:nrow(sampleOneMinuteData[, list(DT, PRICE = MARKET)]))*60)
vol2 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)],
                method = "kernel", h = h1)
vol3 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)], 
                method = "kernel", est = "quarticity")
vol4 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)],
                method = "kernel", est = "cv")
plot(cbind(vol2$spot, vol3$spot, vol4$spot))
xts::addLegend("topright", c("h = simple estimate", "h = quarticity corrected",
                     "h = crossvalidated"), col = 1:3, lty=1)

## End(Not run)

# Piecewise constant volatility
## Not run: 
vol5 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)], 
                method = "piecewise", m = 200, n  = 100, online = FALSE)
plot(vol5)
## End(Not run)

# Compare regular GARCH(1,1) model to eGARCH, both with external regressors
## Not run: 
vol6 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)], method = "garch", model = "sGARCH")
vol7 <- spotVol(sampleOneMinuteData[, list(DT, PRICE = MARKET)], method = "garch", model = "eGARCH")
plot(as.numeric(t(vol6$spot)), type = "l")
lines(as.numeric(t(vol7$spot)), col = "red")
legend("topleft", c("GARCH", "eGARCH"), col = c("black", "red"), lty = 1)

## End(Not run)

## Not run: 
# Compare realized measure spot vol estimation to pre-averaged version
vol8 <- spotVol(sampleTDataEurope[, list(DT, PRICE)], method = "RM", marketOpen = "09:00:00",
                marketClose = "17:30:00", tz = "UTC", alignPeriod = 1, alignBy = "mins",
                lookBackPeriod = 10)
vol9 <- spotVol(sampleTDataEurope[, list(DT, PRICE)], method = "PARM", marketOpen = "09:00:00",
                marketClose = "17:30:00", tz = "UTC", lookBackPeriod = 10)
plot(zoo::na.locf(cbind(vol8$spot, vol9$spot)))

## End(Not run)
[Package highfrequency version 1.0.1 Index]