StuartGordonReid

# Plot fifteen asset price paths without stochastic volatility.
charts.PerformanceSummary(returnProcesses(15, t = (252*5), 
                                          stochastic.volatility = FALSE), 
                          colorset = seq(1,15))

# Plot fifteen asset price paths with stochastic volatility.
charts.PerformanceSummary(returnProcesses(15, t = (252*5), 
                                          stochastic.volatility = TRUE), 
                          colorset = seq(1,15))

StuartGordonReid

testRobustness <- function(t = (252 * 7)) {
  # Generate an underlying signal.
  signal <- sin(seq(1, t)) / 50
  signal <- signal - mean(signal)
  
  # For different noise levels
  sds <- seq(0.0, 0.020, 0.0005)
  cratios <- c()
  for (s in sds) {
    # Generate a noisy signal

StuartGordonReid

#' @title Given a log price process, X, compute the Z-score which can be used 
#' to accept or reject the hypothesis that the process evolved according to a 
#' Brownian Motion model with drift and stochastic volatility.
#' 
#' @description Given a log price process, X, and a sampling interval, q, this 
#' method returns a Z score indicating the confidence we have that X evolved 
#' according to a Brownian Motion mode with drift and stochastic volatility. This
#' heteroskedasticity-consistent variance ratio test essentially checks to see 
#' whether or not the observed Mr statistic for the number of observations, is 
#' within or out of the limiting distribution defined by the Asymptotic Variance.

StuartGordonReid

#' @title Estimator for the value of the asymptotic variance of the Mr statistic. 
#' This is equivalent to a weighted sum of the asymptotic variances for each of 
#' the autocorrelation co-efficients under the null hypothesis.
#' 
#' @details Given a log price process, X, and a sampling interval, q, this 
#' method is used to estimate the asymptoticvariance of the Mr statistic in the 
#' presence of stochastic volatility. In other words, it is a heteroskedasticity 
#' consistent estimator of the variance of the Mr statistic. This parameter is 
#' used to estimate the probability that the given log price process was 
#' generated by a Brownian Motion model with drift and stochastic volatility. 

StuartGordonReid

#' @title Compute the Mr statistic.
#' 
#' @description Compute the Mr statistic. The Md statistic is the ratio between 
#' two estimate values for Sigma computed using the calibrateSigma function for 
#' sampling intervals 1 and the estimate value for Sima computed using the
#' CalibrateSigmaOverlapping function for sampling inveral q minus 1. 
#' This statistic should converge to zero.
#' 
#' @inheritParams calibrateSigma
#' @return Mr double :: The Mr statistic defined by Lo and MacKinlay.

StuartGordonReid

#' @title Compute the Md statistic.
#' 
#' @description Compute the Md statistic. The Md statistic is the difference 
#' between two estimate values for Sigma computed using the calibrateSigma 
#' function for sampling intervals 1 and the estimate value for Sima computed 
#' using the CalibrateSigmaOverlapping function for sampling inveral q. 
#' This statistic should converge to zero.
#' 
#' @inheritParams calibrateSigma
#' @return Md double :: The Md statistic defined by Lo and MacKinlay.

StuartGordonReid

#' @title A more efficient estimator for the value of sigma. Sigma is the 
#' standard deviation of the random component of returns in the Geometric 
#' Brownian Motion model. This estimate can be calculated in an unbiased manner.
#' 
#' @description Given a log price process and a parameter, q, which specifies 
#' the sampling intervel this function estimates the value of Sigma. Sigma 
#' represents the standarddeviation of the random disturbance component of 
#' daily  returns. This estimate can be annualized and can be computed in 
#' a biased or unbiased manner.
#' 

StuartGordonReid

#' @title Estimator for the value of sigma. Sigma is the standard deviation of 
#' the random component of returns in the Geometric Brownian Motion model. This 
#' estimate can be calculated in a biased or unbiased manner.
#' 
#' @description Given a log price process and a parameter, q, which specifies 
#' the sampling intervel this function estimates the value of Sigma. Sigma 
#' represents the standarddeviation of the random disturbance component of daily 
#' returns. This estimate can be annualized and can be computed in a biased or 
#' unbiased manner.
#' 

StuartGordonReid

#' @title Estimator for the value of mu. Mu is the drift component in the 
#' Geometric Brownian Motion model.
#' 
#' @description Given a log price process, this function estimates the value of 
#' mu. Mu is the daily component of the returns which is attributable to upward, 
#' or downward, drift. This estimate can be annualized.
#' 
#' @param X vector :: A log price process.
#' @param annualize logical :: Annualize the parameter estimate.
#' @return mu.est double :: The estimated value of mu.

StuartGordonReid

# Load the required packages.
library(PerformanceAnalytics)
library(xts)


#' @title Sample a random disturbance from either a normal distribution with a 
#' constant standard deviation (Geometric Brownian Motion model) or from a 
#' distribution with a stochastic standard deviation (Stochastic Volatility GBM)
#' 
#' @description Given a long run random disturbance mean, mu, and a standard