________________________________
Date: Tue, 19 Oct 2010 09:03:55 +1100
From: stephen at organicfoodmarkets.com.au
To: r-sig-finance at stat.math.ethz.ch
CC: bjorn.skogtro at gmail.com
Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck
Hi
I am still trying to sort this one out. Any comments from anyone would
be most welcome.
Stephen Choularton Ph.D., FIoD
On 14/10/2010 7:29 AM, Stephen Choularton wrote:
Thanks for this help.
Trying to make sense of it so I have added some notes to the code. I
have marked them #?#
Delighted if you can tell me if I am write or wrong, add any comments,
answers.
#?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck
#?# process work' particularly via dcOU
#?# I have noted in several places that I am after:
#?# 'the half-life of the decay equals ln(2)/?'
#?# 'The half-life is given as log(2)/mean-reversion speed.'
#?# and I see theta appearing at a number of points in the code.
#?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do?
#?# eg is one of these the theta I am after?
# ex3.01.R
OU.lik <- function(theta1, theta2, theta3){
n <- length(X)
dt <- deltat(X)
-sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE))
}
require(stats4)
require(sde)
#?# random numer generation seed
set.seed(123)
#?# creation of a data set
X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1)
#?# If I Look at X its like this:
#?# Time Series:
#?# Start = 0
#?# End = 1000
#?# Frequency = 1
#?# [1] 1.00000000 etc
#?# What sort of data object is it and how would I coerce an object with one
#?# column from a read.csv into it?
mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1),
method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit
summary(fit)
#?# This gives:
#?# Maximum likelihood estimation
#?# Call:
#?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5,
#?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0))
#?# Coefficients:
#?# Estimate Std. Error
#?# theta1 3.355322 0.28159504
#?# theta2 1.106107 0.09010627
#?# theta3 2.052815 0.07624441
#?# -2 log L: 3366.389
#?# What's this telling me?
# ex3.01.R (cont.)
prof <- profile(fit)
par(mfrow=c(1,3))
plot(prof)
par(mfrow=c(1,1))
vcov(fit)
confint(fit)
#?# This provides me with this output using 'fit' from before:
#?# > vcov(fit)
#?# theta1 theta2 theta3
#?# theta1 0.07929576 0.024620718 0.016634557
#?# theta2 0.02462072 0.008119141 0.005485549
#?# theta3 0.01663456 0.005485549 0.005813209
#?# > confint(fit)
#?# Profiling...
#?# 2.5 % 97.5 %
#?# theta1 2.8448980 3.960982
#?# theta2 0.9433338 1.300629
#?# theta3 1.9147136 2.216113
#?# and 'fit' is:
#?# Call:
#?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5,
#?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0))
#?# Coefficients:
#?# theta1 theta2 theta3
#?# 3.355322 1.106107 2.052815
#?# plus some graphic output
#?# Again, what's this telling me.
#?# This looks like a further example?
# ex3.01.R (cont.)
set.seed(123)
X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3)
mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1),
method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2
summary(fit2)
Please excuse the length of this email (and my lack of understanding)
Hope you can help and thanks.
Stephen Choularton Ph.D., FIoD
On 13/10/2010 2:41 AM, stefano iacus wrote:
just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM.
sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function.
This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU
# ex3.01.R
OU.lik <- function(theta1, theta2, theta3){
n <- length(X)
dt <- deltat(X)
-sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE))
}
require(stats4)
require(sde)
set.seed(123)
X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1)
mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1),
method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit
summary(fit)
# ex3.01.R (cont.)
prof <- profile(fit)
par(mfrow=c(1,3))
plot(prof)
par(mfrow=c(1,1))
vcov(fit)
confint(fit)
# ex3.01.R (cont.)
set.seed(123)
X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3)
mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1),
method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2
summary(fit2)
I hope this helps out
stefano
On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote:
Hi Stephen,
You could take a look at
http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model
for the linear regression method, or take a look at the package "sde" which
contains some examples using GMM (not for the Ornstein-Uhlenbeck process,
though, only the CIR).
The half-life is given as log(2)/mean-reversion speed.
Do keep an eye on the partition of the time-axis, e.g. what frequency you
are using (daily, yearly) for interpreting the half-life.
BR,
Bj?rn
------------------------------
Message: 2
Date: Tue, 12 Oct 2010 05:43:32 -0400
From: Sarbo
To: r-sig-finance at stat.math.ethz.ch
Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck
Message-ID:
Content-Type: text/plain; charset="utf-8"
By half-life, do you mean the speed of mean-reversion?
If so, there's a bit of algebraic tomfoolery that's required to
discretise the equation and then fit the data to it. I don't have the
time right now to go into all the details but it's not hard- you can
parameterise the process using simple linear regression. If you need
help with that I'll try and get back to you tonight about it.
On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote:
Hi
Wonder if anyone could point me how I use this method to discover the
half life of a mean reverting process.
I am looking into pair trading and the time it takes for a
cointegrated pair to revert to the norm.
--
Stephen Choularton Ph.D., FIoD
9999 2226
0413 545 182
for insurance go to www.netinsure.com.au
for markets go to www.organicfoodmarkets.com.au
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-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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