I am using the ODE function In R in order to solve this equation:
library(deSolve)
FluidH <- function(t,state,parameters) {
with(as.list(c(state,parameters)),
dh <- Qin/A - ((5073.3*h^2+6430.1*h)/(60*A))
list(c(dh))
})
}
parameters <- c(Qin =10, A=6200)
state<- c(h=0.35)
time <- seq(0,2000,by=1)
out <- ode(y= state, func = FluidH, parms = parameters, times = time)
I might be missing something with math, but when I try to calculate h by myself by assigning the initial state I don't get the same numbers as the output of the function!
for example to calculate h at time 1 : h=h0+ dh*dt -> h= 0.35 + 10/6200 - ((5073.3*h^2+6430.1*h)/(60*6200))=0.3438924348
and the output of ode gives 0.343973044412394
Can anyone tell what am I missing?
You computed the Euler step with step size dt=1. The solver uses a higher order method with (usually) a smaller step size that is adapted to meet the default error tolerances of 1e-6 for relative and absolute error. The step-size 1 that you give only determines where the numerical solution is sampled for the output, internally the solver may use many more or sometimes even less steps (interpolating the output values).
Related
I would like to solve a differential equation in R (with deSolve?) for which I do not have the initial condition, but only the final condition of the state variable. How can this be done?
The typical code is: ode(times, y, parameters, function ...) where y is the initial condition and function defines the differential equation.
Are your equations time reversible, that is, can you change your differential equations so they run backward in time? Most typically this will just mean reversing the sign of the gradient. For example, for a simple exponential growth model with rate r (gradient of x = r*x) then flipping the sign makes the gradient -r*x and generates exponential decay rather than exponential growth.
If so, all you have to do is use your final condition(s) as your initial condition(s), change the signs of the gradients, and you're done.
As suggested by #LutzLehmann, there's an even easier answer: ode can handle negative time steps, so just enter your time vector as (t_end, 0). Here's an example, using f'(x) = r*x (i.e. exponential growth). If f(1) = 3, r=1, and we want the value at t=0, analytically we would say:
x(T) = x(0) * exp(r*T)
x(0) = x(T) * exp(-r*T)
= 3 * exp(-1*1)
= 1.103638
Now let's try it in R:
library(deSolve)
g <- function(t, y, parms) { list(parms*y) }
res <- ode(3, times = c(1, 0), func = g, parms = 1)
print(res)
## time 1
## 1 1 3.000000
## 2 0 1.103639
I initially misread your question as stating that you knew both the initial and final conditions. This type of problem is called a boundary value problem and requires a separate class of numerical algorithms from standard (more elementary) initial-value problems.
library(sos)
findFn("{boundary value problem}")
tells us that there are several R packages on CRAN (bvpSolve looks the most promising) for solving these kinds of problems.
Given a differential equation
y'(t) = F(t,y(t))
over the interval [t0,tf] where y(tf)=yf is given as initial condition, one can transform this into the standard form by considering
x(s) = y(tf - s)
==> x'(s) = - y'(tf-s) = - F( tf-s, y(tf-s) )
x'(s) = - F( tf-s, x(s) )
now with
x(0) = x0 = yf.
This should be easy to code using wrapper functions and in the end some list reversal to get from x to y.
Some ODE solvers also allow negative step sizes, so that one can simply give the times for the construction of y in the descending order tf to t0 without using some intermediary x.
I am trying to use DEoptim to optimize the parameters of the Heston pricing model (NMOF package). My goal is to minimize the difference between the real option price and the heston price. However, when running my code, DEoptim does not save the best result but always displays the value that is obtained by using the initial parameters, not the optimized ones. Unfortunately, I'm totally new to R (and any kind of programming) and thus I cannot seem to fix the problem.
My data, for one exemplary subset of an option looks like this.
#Load data
#Real option price
C0116_P=as.vector(c(1328.700000, 1316.050000, 1333.050000, 1337.900000, 1344.800000))
#Strike price
C0116_K=as.vector(c(500, 500, 500, 500, 500))
#Time to maturity in years
C0116_T_t=as.vector(c(1.660274, 1.657534, 1.654795, 1.652055, 1.649315))
#Interest rate percentage
C0116_r=as.vector(c(0.080000, 0.080000, 0.090000, 0.090000, 0.090000))
#Dividend yield percentage
C0116_DY=as.vector(c(2.070000, 2.090000, 2.070000, 2.070000,2.060000))
#Price underlying
C0116_SP_500_P=as.vector(c(1885.08, 1872.83, 1888.03, 1892.49, 1900.53))
In the next step, I want to define the function I want to minimize (difference between real and heston price) and set some initial parameters. To optimize, I am running a loop which unfortunately at the end only returns the difference between the real option price and the heston price using the initial parameters as a best value and not the actual parameters that minimize the difference.
#Load packages
require(NMOF)
require(DEoptim)
#Initial parameters
v0=0.2
vT=0.2
rho=0.2
k=0.2
sigma=0.2
#Define function
error_heston<-function(x)
{error<-P-callHestoncf(S, X, tau, r, q, v0, vT, rho, k, sigma)
return(error)}
#Run optimization
outDEoptim<-matrix()
for (i in 1:5)
{
#I only want the parameters v0, vT, rho, k and sigma to change. That is why I kept the others constant
lower<-c(C0116_P[i],C0116_SP_500_P[i],C0116_K[i],C0116_T_t[i],C0116_r[i]/100,C0116_DY[i]/100,0.0001,0.0001,-1,0.0001,0.0001)
upper<-c(C0116_P[i],C0116_SP_500_P[i],C0116_K[i],C0116_T_t[i],C0116_r[i]/100,C0116_DY[i]/100,10,10,1,10,10)
outDEoptim<-(DEoptim(error_heston, lower, upper, DEoptim.control(VTR=0,itermax=100)))
print(outDEoptim$opti$bestval)
i=i+1
}
Any help is much appreciated!
One of the first problems is that your objective function only has one argument (the parameters to optimize), so all the others objects used inside the function must be looked up. It's better practice to pass them explicitly.
Plus, many of the necessary values aren't defined in your example (e.g. S, X, etc). All the parameters you want to optimize will be passed to your objective function via the first argument. It can help clarify things if you explicitly assign each element inside your objective function. So a more robust objective function definition is:
# Define objective function
error_heston <- function(x, P, S, K, tau, r, q) {
v0 <- x[1]
vT <- x[2]
rho <- x[3]
k <- x[4]
sigma <- x[5]
error <- abs(P - callHestoncf(S, K, tau, r, q, v0, vT, rho, k, sigma))
return(error)
}
Also note that I took the absolute error. DEoptim is going to minimize the objective function, so it would try to make P - callHestoncf() as negative as possible, when you want it to be close to zero instead.
You specified the box constraints upper and lower even for the parameters that don't vary. It's best to only have DEoptim generate a population for the parameters that do vary, so I removed the non-varying parameters from the box constraints. I also defined them outside the for loop.
# Only need to set bounds for varying parameters
lower <- c(1e-4, 1e-4, -1, 1e-4, 1e-4)
upper <- c( 10, 10, 1, 10, 10)
Now to the actual DEoptim call. Here is where you will pass the values for all the non-varying parameters. You set them as named arguments to the DEoptim call, as I've done below.
i <- 1
outDEoptim <- DEoptim(error_heston, lower, upper,
DEoptim.control(VTR=0, itermax=100), P = C0116_P[i], S = C0116_SP_500_P[i],
K = C0116_K[i], tau = C0116_T_t[i], r = C0116_r[i], q = C0116_DY[i])
I only ran one iteration of the for loop, because the callHestoncf() function frequently throws an error because the numerical integration routine fails. This stops the optimization. You should look into the cause of that, and ask a new question if you have trouble.
I also noticed you specified one of the non-varying inputs incorrectly. Your dividend yield percentages are 100 times too large. Your non-varying inputs should be:
# Real option price
C0116_P <- c(1328.70, 1316.05, 1333.05, 1337.90, 1344.80)
# Strike price
C0116_K <- c(500, 500, 500, 500, 500)
# Time to maturity in years
C0116_T_t <- c(1.660274, 1.657534, 1.654795, 1.652055, 1.649315)
# Interest rate percentage
C0116_r <- c(0.08, 0.08, 0.09, 0.09, 0.09)
# Dividend yield percentage
C0116_DY <- c(2.07, 2.09, 2.07, 2.07, 2.06) / 100
# Price underlying
C0116_SP_500_P <- c(1885.08, 1872.83, 1888.03, 1892.49, 1900.53)
As an aside, you should take a little time to format your code better. It makes it more readable, which should help you avoid typo-like errors.
I would like to solve a differential equation in R (with deSolve?) for which I do not have the initial condition, but only the final condition of the state variable. How can this be done?
The typical code is: ode(times, y, parameters, function ...) where y is the initial condition and function defines the differential equation.
Are your equations time reversible, that is, can you change your differential equations so they run backward in time? Most typically this will just mean reversing the sign of the gradient. For example, for a simple exponential growth model with rate r (gradient of x = r*x) then flipping the sign makes the gradient -r*x and generates exponential decay rather than exponential growth.
If so, all you have to do is use your final condition(s) as your initial condition(s), change the signs of the gradients, and you're done.
As suggested by #LutzLehmann, there's an even easier answer: ode can handle negative time steps, so just enter your time vector as (t_end, 0). Here's an example, using f'(x) = r*x (i.e. exponential growth). If f(1) = 3, r=1, and we want the value at t=0, analytically we would say:
x(T) = x(0) * exp(r*T)
x(0) = x(T) * exp(-r*T)
= 3 * exp(-1*1)
= 1.103638
Now let's try it in R:
library(deSolve)
g <- function(t, y, parms) { list(parms*y) }
res <- ode(3, times = c(1, 0), func = g, parms = 1)
print(res)
## time 1
## 1 1 3.000000
## 2 0 1.103639
I initially misread your question as stating that you knew both the initial and final conditions. This type of problem is called a boundary value problem and requires a separate class of numerical algorithms from standard (more elementary) initial-value problems.
library(sos)
findFn("{boundary value problem}")
tells us that there are several R packages on CRAN (bvpSolve looks the most promising) for solving these kinds of problems.
Given a differential equation
y'(t) = F(t,y(t))
over the interval [t0,tf] where y(tf)=yf is given as initial condition, one can transform this into the standard form by considering
x(s) = y(tf - s)
==> x'(s) = - y'(tf-s) = - F( tf-s, y(tf-s) )
x'(s) = - F( tf-s, x(s) )
now with
x(0) = x0 = yf.
This should be easy to code using wrapper functions and in the end some list reversal to get from x to y.
Some ODE solvers also allow negative step sizes, so that one can simply give the times for the construction of y in the descending order tf to t0 without using some intermediary x.
I am trying to replicate results in R from Excel's "Solver" add-in. I don't know about the inner workings of optimization (mathematically), hence my confusion at most post results as well as the error messages I am receiving. I tried using the optimx package, but apparently that doesn't allow for too much control over the constraints in the optimization, so now I'm trying out the nloptr package.
Basically, what I'm trying to do is replicate an optimum portfolio calculation (financial). Below is a sample of my code:
ret.cov <- cov(as.matrix(ret.p[,1:30]))
wts <- rep(1/portfolioSize, times = portfolioSize)
sharpe <- function(wts) {
mean.p <- sum(colMeans(ret.p[,1:30])*wts)
var.p <- t(wts) %*% (ret.cov %*% (wts))
sd.p <- sqrt(var.p)
SR <- (mean.p - Rf)/sd.p
return(as.numeric(SR))
}
fun.eq <- function(wts) {
sum(wts) == 1
}
optim.p <- nloptr(x0 = wts, eval_f = sharpe, lb = 0, ub = 1, eval_g_eq = fun.eq)
sharpe(as.numeric(optim.p$solution))
Calculates the covariance matrix of 30 stocks and their returns
Initializes the weights of those stocks to optimize (equally weighted to start)
Sets up a function to maximize which calculates the portfolio's Sharpe Ratio
Tries (???) to specify the equality function for nloptr that states that the sum of the wts vector must be equal to 1.
Tries to maximize the function (though I think it's minimizing by default, and I don't know how to change that to maximize instead).
Checks the resulting, maximized Sharpe Ratio
The Sharpe calculation function works fine, when I try it outside of the nloptr function. The issues are various, from needing to specify the proper algorithm to use, to the function not accepting the equality function I supplied.
So, the questions I have are:
How do you change the nloptr to maximize instead of minimize?
How would one write an equality function to specify that the sum of the input vector (weights) must be equal to 1?
What is the proper algorithm to specify using opts = list() here? Excel uses something called "GRG Nonlinear".
Thank you in advance!
Hope it's still relevant...
You don't supply data so I can't run it but I'll try to help.
1) In order to maximize just minimize the -sharpe
2) eval_g_eq needs to be in format of h(x)=0, meaning that you need on fun.eq to change sum(wts) == 1 to sum(wts) - 1.
3) There are a lot of decent options. I use NLOPT_LN_COBYLA
I am teaching myself how to run some Markov models in R, by working through the textbook "Hidden Markov Models for Time Series: An Introduction using R". I am a bit stuck on how to go about implementing something that is mentioned in the text.
So, I have the following function:
f <- function(samples,lambda,delta) -sum(log(outer(samples,lambda,dpois)%*%delta))
Which I can optimize with respect to, say, lambda using:
optim(par, fn=f, samples=x, delta=d)
where "par" is the initial guess for lambda, for some x and d.
Which works perfectly fine. However, in the part of the text corresponding to the example I am trying to reproduce, they note: "The parameters delta and lambda are constrained by sum(delta_i)=1 for i=1,...m, delta_i>0, and lambda_i>0. It is therefore necessary to reparametrize if one wishes to use an unconstrained optimizer such as nlm". One possibility is to maximize the likelihood with respect to the 2m-1 unconstrained parameters".
The unconstrained parameters are given by
eta<-log(lambda)
tau<-log(delta/(1-sum(delta))
I don't entirely understand how to go about implementing this. How would I write a function to optimize over this transformed parameter space?
When using optim() without parmater transfromations like so:
simpleFun <- function(x)
(x-3)^2
out = optim(par=5,
fn=simpleFun)
the set of parmaters estimates would be obtained via out$par which is 3 in
the case, as you might expect. Alternatively, you can wrap your function
f in a transformation the parameters like so:
out = optim(par=5,
fn=function(x)
# apply the transformation x -> x^3
simpleFun(x^3))
and now the trick to get the correct set of optimal parmeters to your
function you need to apply the same transfromation to the parameter
estimates as in:
(out$par)^3
#> 2.99741
(and yes, the parameter estimate is slightly different. For this contrived
example, you could set method="BFGS" for a slightly better estimate. Anyhow, this goes to show that the choice of transformation does matter in
some cases, but that's for another discussion...)
To complete the answer, It sounds like you a want to use a wrapper like so
# the function to be optimized
f <- function(samples,lambda,delta)
-sum(log(outer(samples,lambda,dpois)%*%delta))
out <- optim(# par it now a 2m vector
par = c(eta1 = 1,
eta2 = 1,
eta3 = 1,
tau1 = 1,
tau2 = 1,
tau3 = 1),
# a wrapper that applies the constraints
fn=function(x,samples){
# exp() guarantees that the values of lambda are > 0
lambda = exp(x[1:3])
# delta is also > 0
delta = exp(x[4:6])
# and now it sums to 1
delta = delta / sum(delta)
f(samples,lambda,delta)
},
samples=samples)
The above guarantees that the the parameters passed to f()have the correct constraints, and as long as you apply the same transformation to out$par, optim() will estimate an optimal set of parameters for f().