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# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR Description. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
################################################################################
# FUNCTION: DESCRIPTION:
# maxddStats Expectation of Drawdowns for BM with drift
# .maxddStats Utility function called by "maxddStats"
# FUNCTION: DISTRIBUTION AND RANDOM VARIATES:
# dmaxdd Density function of mean Max-Drawdowns
# pmaxdd Probability function of mean Max-Drawdowns
# rmaxdd Random Variates of mean Max-Drawdowns
################################################################################
maxddStats <-
function(mean = 0, sd = 1, horizon = 1000)
{
# A function implemented by Diethelm Wuertz
# Description:
# Calculates Expectation Value E[D] of maximum Drawdowns of
# Brownian Motion for a given drift "mu", variance "sigma",
# and runtime "horizon" of the Brownian Motion process.
# Arguments:
# mu - Drift of Brownian Motion, a numeric vector.
# sigma - Standard Deviation of Brownian Motion, a numeric value.
# horizon - Runtime of the process, the time horizon of the
# investor, a numeric value.
# Details:
# Interpolates the data given in the table of Appendix B
# in Magdon-Ismail et al. [2003], "On the Maximum Drawdown
# of Brownian Motion". The interpolation is done with R's
# function spline().
# Notes.
# This computes for a vector of horizon values.
# FUNCTION:
# Statistics:
ans = NULL
for (i in 1:length(horizon)) {
ans = c(ans, .maxddStats(mu = mean, sigma = sd, horizon = horizon[i]))
}
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.maxddStats <-
function(mu = 0, sigma = 1, horizon = 1000)
{
# A function implemented by Diethelm Wuertz
# Description:
# Utility function called by "maxddStats"
# Arguments:
# see function "maxddStats
# FUNCTION:
# Internal Function - POSITIVE CASE: mu > 0
# Left Table from Appendix B:
QP = function(x) {
gamma = sqrt(pi/8)
vqn = t(matrix(c(
0.0005,0.019690, 0.0010,0.027694, 0.0015,0.033789, 0.0020,0.038896,
0.0025,0.043372, 0.0050,0.060721, 0.0075,0.073808, 0.0100,0.084693,
0.0125,0.094171, 0.0150,0.102651, 0.0175,0.110375, 0.0200,0.117503,
0.0225,0.124142, 0.0250,0.130374, 0.0275,0.136259, 0.0300,0.141842,
0.0325,0.147162, 0.0350,0.152249, 0.0375,0.157127, 0.0400,0.161817,
0.0425,0.166337, 0.0450,0.170702, 0.0500,0.179015, 0.0600,0.194248,
0.0700,0.207999, 0.0800,0.220581, 0.0900,0.232212, 0.1000,0.243050,
0.2000,0.325071, 0.3000,0.382016, 0.4000,0.426452, 0.5000,0.463159,
1.5000,0.668992, 2.5000,0.775976, 3.5000,0.849298, 4.5000,0.905305,
10.000,1.088998, 20.000,1.253794, 30.000,1.351794, 40.000,1.421860,
50.000,1.476457, 150.00,1.747485, 250.00,1.874323, 350.00,1.958037,
450.00,2.020630, 1000.0,2.219765, 2000.0,2.392826, 3000.0,2.494109,
4000.0,2.565985, 5000.0,2.621743), 2))
# Interpolation:
if (x < 0.0005) { y = gamma*sqrt(2*x) }
if (x >= 0.0005 & x <= 5000) {
y = spline(log(vqn[,1]), vqn[,2], n = 1, xmin = log(x),
xmax = log(x))$y
}
if (x > 5000) { y = 0.25*log(x) + 0.49088}
# Return Value:
y
}
# Internal Function - NEGATIVE CASE: mu < 0
# Right Table from Appendix B:
QN = function(x) {
gamma = sqrt(pi/8)
vqn = t(matrix(c(
0.0005,0.019965, 0.0010,0.028394, 0.0015,0.034874, 0.0020,0.040369,
0.0025,0.045256, 0.0050,0.064633, 0.0075,0.079746, 0.0100,0.092708,
0.0125,0.104259, 0.0150,0.114814, 0.0175,0.124608, 0.0200,0.133772,
0.0225,0.142429, 0.0250,0.150739, 0.0275,0.158565, 0.0300,0.166229,
0.0325,0.173756, 0.0350,0.180793, 0.0375,0.187739, 0.0400,0.194489,
0.0425,0.201094, 0.0450,0.207572, 0.0475,0.213877, 0.0500,0.220056,
0.0550,0.231797, 0.0600,0.243374, 0.0650,0.254585, 0.0700,0.265472,
0.0750,0.276070, 0.0800,0.286406, 0.0850,0.296507, 0.0900,0.306393,
0.0950,0.316066, 0.1000,0.325586, 0.1500,0.413136, 0.2000,0.491599,
0.2500,0.564333, 0.3000,0.633007, 0.3500,0.698849, 0.4000,0.762455,
0.5000,0.884593, 1.0000,1.445520, 1.5000,1.970740, 2.0000,2.483960,
2.5000,2.990940, 3.0000,3.492520, 3.5000,3.995190, 4.0000,4.492380,
4.5000,4.990430, 5.0000,5.498820), 2))
# Interpolation:
if (x < 0.0005) { y = gamma*sqrt(2*x) }
if (x >= 0.0005 & x <= 5) {
y = spline(vqn[,1], vqn[,2], n = 1, xmin = x, xmax = x)$y }
if (x > 5) { y = x + 1/2 }
# Return Value:
y
}
# Result:
ED = NULL
for (i in 1:length(mu)) {
if (mu[i] == 0) {
gamma = sqrt(pi/8)
ED[i] = 2 * gamma * sigma * sqrt(horizon)
} else {
x = mu[i]^2 * horizon / ( 2 * sigma^2 )
if (mu[i] > 0) { ED[i] = ( 2* sigma^2 / mu[i] ) * QP(x) }
if (mu[i] < 0) { ED[i] = - ( 2* sigma^2 / mu[i] ) * QN(x) }
}
}
# Return Value:
ED
}
################################################################################
dmaxdd <-
function(x, sd = 1, horizon = 100, N = 1000)
{
# Description:
# Calculates for a trendless Brownian process (mean=0) and
# standard deviation "sd" the density from the probability
# that the maximum drawdown "D" is larger or equal to "h"
# in the interval [0,T], where "T" denotes the time horizon
# of the investor.
# Arguments:
# x - Vector of Drawdowns
# sd - Standard Deviation
# horizon - Time horizon of the investor
# N - number of summands
# Notes:
# The drift dependent case is not yet implemented!
# FUNCTION:
# Use "h", "sigma" to be conform with Magdon-Ismael et al. [2003]
h = x
sigma = sd
# Settings:
n = 1:N
pn = 2 * sin((n-0.5)*pi) * sigma^2 * (n-0.5) * pi * horizon
en = sigma^2 * (n-0.5)^2 * pi^2 * horizon / 2
# Loop over all Drawdowns:
result = rep(NA, times = length(h))
for (i in 1:length(h)) {
if (h[i] == 0) {
result[i] = 0
} else {
g = pn * exp(-en/(h[i]^2)) / h[i]^3
result[i] = sum(g)
}
}
# Return Value:
result
}
# ------------------------------------------------------------------------------
pmaxdd <-
function(q, sd = 1, horizon = 100, N = 1000)
{
# Description:
# Calculates for a trendless Brownian process (mean=0)
# with standard deviation "sd" the probability that the
# maximum drawdown "D" is larger or equal to "h" in the
# interval [0,T], where "T" denotes the time horizon of
# the investor.
# Arguments:
# q - Vector of Drawdowns
# sd - Standard Deviation
# horizon - Time horizon of the investor
# N - number of summands
# Value:
# GD(h) - eqn(14) In Magdon-Ismail et al.
# Notes:
# The drift dependent case is not yet implemented!
# FUNCTION:
# Use "h", "sigma" to be conform with Magdon-Ismael et al. [2003]
h = q
sigma = sd
# Settings:
n = 1:N
pn = 2 * ( sin((n-0.5)*pi) / ((n-0.5)*pi) )
en = sigma^2 * (n-0.5)^2 * pi^2 * horizon / 2
# Loop over all Drawdowns:
result = rep(NA, times = length(h))
for (i in 1:length(h)) {
g = pn * ( 1 - exp(-en/(h[i]^2)) )
result[i] = sum(g)
}
# Return Value:
result
}
# ------------------------------------------------------------------------------
rmaxdd <-
function(n, mean = 0, sd = 1, horizon = 100)
{
# Description:
# Generates for a Brownian process with mean "mean" and
# standard deviation "sd" random variates of maximum
# Drawdowns.
# Arguments:
# n - Number of Drawdown rvs
# mean - Drift
# sd - Standard Deviation
# horizon - Time horizon of the investor
# FUNCTION:
# Simulation of "n" max Drawdowns "h":
result = NULL
for (i in 1:n) {
D = cumsum(rnorm(horizon, mean = mean, sd = sd))
result[i] = max(cummax(D)-D)
}
# Return Value:
result
}
################################################################################
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