SigmaEstimatorTwo.R

#' @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.
#' 
#' @details The difference between this estimator and the estimator defined in 
#' the calibrateSigma function is that this method makes use of overlapping 
#' windows of log price data. Whilst this does increase the number of 
#' observations and improve the accuracy of the estimator it is no longer 
#' unbiased. That said, Monte Carlo simulations indicate that this bias is 
#' negligable and, in fact, this estimator is more accurate than the one 
#' defined by calibrateSigma. 
#' 
#' @inheritParams calibrateSigma
#' @return sd.est double :: The estimated value of Sigma.
#' 
calibrateSigmaOverlapping <- function(X, q = 1, annualize = TRUE, 
                                      unbiased = TRUE) {
  # Get the estimate value for the drift component.
  mu.est <- calibrateMu(X, annualize = FALSE)
  
  # Ensure that the format of X is appropriate.
  X <- as.numeric(as.vector(X))
  
  # Calculate the number of times q goes into the length of X.
  n <- floor(length(X)/q)
  sd.est <- 0.0
  for (t in (q + 1):(n * q))
    sd.est <- sd.est + (X[t] - X[t - q] - (q * mu.est))^2
  
  # Calculate the average sigma using the unbiased or biased method.
  if (!unbiased) sd.est <- sd.est / (n * (q^2))
  else sd.est <- sd.est / ((q * ((n * q) - q + 1)) * (1 - (q / (n * q))))
  
  if (!annualize) return(sd.est)
  else return(sqrt((sd.est * 252)))
}