Package 'stochcorr' reference manual

Title:	Stochastic Correlation Modelling via Circular Diffusion
Description:	Performs simulation and inference of diffusion processes on circle. Stochastic correlation models based on circular diffusion models are provided. For details see Majumdar, S. and Laha, A.K. (2024) "Diffusion on the circle and a stochastic correlation model" <doi:10.48550/arXiv.2412.06343>.
Authors:	Sourav Majumdar [aut, cre]
Maintainer:	Sourav Majumdar <[email protected]>
License:	GPL (>= 3)
Version:	0.0.1
Built:	2026-05-16 07:18:35 UTC
Source:	https://github.com/cran/stochcorr

Bootstrap for circdiff

Description

stoch.bootstrap returns the Bootstrap confidence interval for estimated parameters from circdiff.

Usage

circ.bootstrap(est, theta, boot_iter=500, p=0.05, seed = NULL)
circ.bootstrap(est, theta, boot_iter=500, p=0.05, seed = NULL)

Arguments

est

object of class cbm or vmp

theta

data of the discretely observed diffusion

boot_iter

number of bootstrap iteration (Default is 500)

p

1-p% confidence interval (Default is 0.05)

seed

(optional) seed value

Details

This function returns a (1-p)% confidence interval for estimated parameters from circdiff using parametric bootstrap. See section 4 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

Returns a matrix of the bootstrap (1-p)% confidence interval for the parameters. The first row is the lower bound and the second row is the upper bound.

Examples


library(stochcorr)


data(wind)

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(wind, aes(x = Date, y = Dir)) +
  geom_line() +
  labs(title = "Sotavento Wind Farm",
       x = "Date",
       y = "Wind Direction") +
  scale_x_datetime(date_labels = "%d-%b", date_breaks = "2 day") +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )
}

a<-circdiff(wind$Dir,10/1440,"vmp")
a

b<-circdiff(wind$Dir,10/1440,"cbm")
b



estimates_vmp<-circ.bootstrap(a,wind$Dir, seed = 100)
estimates_vmp

estimates_cbm<-circ.bootstrap(b,wind$Dir, seed = 100)
estimates_cbm
library(stochcorr)


data(wind)

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(wind, aes(x = Date, y = Dir)) +
  geom_line() +
  labs(title = "Sotavento Wind Farm",
       x = "Date",
       y = "Wind Direction") +
  scale_x_datetime(date_labels = "%d-%b", date_breaks = "2 day") +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )
}

a<-circdiff(wind$Dir,10/1440,"vmp")
a

b<-circdiff(wind$Dir,10/1440,"cbm")
b



estimates_vmp<-circ.bootstrap(a,wind$Dir, seed = 100)
estimates_vmp

estimates_cbm<-circ.bootstrap(b,wind$Dir, seed = 100)
estimates_cbm

Estimation of circular diffusion models

Description

circdiff returns the estimates of parameters of a discretely observed von-Mises process or Circular Brownian motion

Usage

circdiff(theta, dt, corr_process, iter=2000, lambda=0, lambda_1=0, lambda_2=0)
circdiff(theta, dt, corr_process, iter=2000, lambda=0, lambda_1=0, lambda_2=0)

Arguments

theta

data of the discretely observed diffusion

dt

time step $\Delta t$

corr_process

vmp for von-Mises process; cbm for circular brownian motion

iter

number of iterations (Default is 2000)

lambda

regularization parameter for cbm (Default is 0)

lambda_1

regularization parameter for vmp (Default is 0)

lambda_2

regularization parameter for vmp (Default is 0)

Details

Let $\theta_0,\theta_{\Delta t},\theta_{2\Delta t}, \ldots, \theta_{n\Delta t}$ be a discretely observed circular diffusion at time step $\Delta t$ . The circular diffusion could either be von-Mises process,

$d\theta_t=-\lambda\sin(\theta_t-\mu)dt+\sigma dW_t$

or the circular brownian motion,

$d\theta_t=\sigma dW_t$

under periodic boundary condition. The function returns the MLE of $\lambda,\sigma,\mu$ for von-Mises process or $\sigma$ for circular brownian motion.

We provide an option to perform penalised MLE using an iterative optimization procedure as following, we maximise for von Mises process, where $n$ is the number of observations,

$\text{Log-lik}-n\lambda_1\sum(1-cos(\theta_{i+1}-\theta_{i}))-\lambda_2\frac{2\lambda}{\sigma^2}$

For Circular Brownian motion we maximise,

$\text{Log-lik}-n\lambda\sum(1-cos(\theta_{i+1}-\theta_{i}))$

See section 3 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

A list containing the estimates of the model if corr_process=vmp then it returns

dt time step $\Delta t$
lambda_vm the drift parameter of the von Mises process
sigma_vm the volatility parameter of the von Mises process
mu_vm the mean direction of the von Mises process
lambda_1, lambda_2 value of the regularization parameters used for estimation

else if corr_process=cbm then it returns

dt time step $\Delta t$
sigma_cbm the volatility parameter of the circular Brownian motion
lambda value of the regularization parameter used for estimation

Examples


library(stochcorr)

data(wind)
if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(wind, aes(x = Date, y = Dir)) +
  geom_line() +
  labs(title = "Sotavento Wind Farm",
       x = "Date",
       y = "Wind Direction") +
  scale_x_datetime(date_labels = "%d-%b", date_breaks = "2 day") +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )
 }

a<-circdiff(wind$Dir,10/1440,"vmp")
a

b<-circdiff(wind$Dir,10/1440,"cbm")
b
library(stochcorr)

data(wind)
if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(wind, aes(x = Date, y = Dir)) +
  geom_line() +
  labs(title = "Sotavento Wind Farm",
       x = "Date",
       y = "Wind Direction") +
  scale_x_datetime(date_labels = "%d-%b", date_breaks = "2 day") +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  )
 }

a<-circdiff(wind$Dir,10/1440,"vmp")
a

b<-circdiff(wind$Dir,10/1440,"cbm")
b

Probability transition density function for Circular Brownian Motion

Description

dcbm evaluates the transition density for Circular Brownian Motion

Usage

dcbm(theta, t, theta_0, sigma)
dcbm(theta, t, theta_0, sigma)

Arguments

theta

vector of data

t

time step $\Delta t$

theta_0

initial value

sigma

sigma

Details

See section 2 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

dcbm gives the transition density function of the circular Brownian motion

Probability transition density function for von Mises process

Description

dvmp evaluates the transition density for von Mises process

Usage

dvmp(theta, t, theta_0, lambda, sigma, mu)
dvmp(theta, t, theta_0, lambda, sigma, mu)

Arguments

theta

vector of data

t

time step $\Delta t$

theta_0

initial value

lambda

lambda

sigma

sigma

mu

Details

See section 2 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

dvmp gives the transition density function of the von Mises process

2020 USD/GBP and FTSE250 data

Description

End of the day closing price for USD/GBP and FTSE250 in the year 2020.

Usage

data("ftse2020")
data("ftse2020")

Format

A data frame with 224 rows and 3 columns:

USD/GBP: EOD USD/GBP rate
FTSE250: EOD FTSE250
Date: Date

Source

Yahoo Finance

2020 USD/JPY and Nikkei data

Description

End of the day closing price for USD/JPY and Nikkei in the year 2020.

Usage

data("nikkei2020")
data("nikkei2020")

Format

A data frame with 213 rows and 3 columns:

USD/JPY: EOD USD/JPY rate
Nikkei: EOD Nikkei
Date: Date

Source

Yahoo Finance

2020 USD/INR and NIFTY data

Description

End of the day closing price for USD/INR and NIFTY in the year 2020.

Usage

data("nse2020")
data("nse2020")

Format

A data frame with 250 rows and 3 columns:

USD/INR: EOD USD/INR rate
Nifty: EOD Nifty
Date: Date

Source

Yahoo Finance

Simulate circular Brownian motion

Description

rtraj.cbm returns a simulated path of a circular Brownian motion for given parameters

Usage

rtraj.cbm(n, theta_0, dt, sigma, burnin=1000)
rtraj.cbm(n, theta_0, dt, sigma, burnin=1000)

Arguments

n

number of steps in the simulated path

theta_0

initial point

dt

Time step

sigma

volatility parameter

burnin

number of initial samples to be rejected (Default is 1000)

Details

Let $\theta_t$ evolve according to a circular Brownian motion given by,

$d\theta_t=\sigma dW_t$

We simulate $\theta_t$ by simulating from its transition density.

Value

A vector of length n of the simulated path from circular Brownian motion

Simulate von Mises process

Description

rtraj.vmp returns a simulated path of a von Mises process for given parameters

Usage

rtraj.vmp(n, theta_0, dt, mu, lambda, sigma)
rtraj.vmp(n, theta_0, dt, mu, lambda, sigma)

Arguments

n

number of steps in the simulated path

theta_0

initial point

dt

Time step

mu

mean parameter

lambda

drift parameter

sigma

volatility parameter

Details

Let $\theta_t$ evolve according to a von Mises process given by,

$d\theta_t=-\lambda\sin(\theta_t-\mu)dt+\sigma dW_t$

We simulate $\theta_t$ by the Euler-Maruyama discretization of the above SDE.

Value

A vector of length n of the simulated path from von Mises process

2020 EUR/USD and S&P500 data

Description

End of the day closing price for EUR/USD and S&P500 in the year 2020.

Usage

data("s&p2020")
data("s&p2020")

Format

A data frame with 224 rows and 3 columns:

EUR/USD: EOD EUR/USD rate
S&P500: EOD S&P500
Date: Date

Source

Yahoo Finance

Bootstrap for stochcorr

Description

stoch.bootstrap returns the Bootstrap confidence interval for estimated rho from stochcorr.

Usage

stoch.bootstrap(est, S_1, S_2, boot_iter=500, p=0.05, seed = NULL)
stoch.bootstrap(est, S_1, S_2, boot_iter=500, p=0.05, seed = NULL)

Arguments

est

object of class cbm or vmp

S_1

historical price of the first asset

S_2

historical price of the second asset

boot_iter

number of bootstrap iteration (Default is 500)

p

1-p% confidence interval (Default is 0.05)

seed

(optional) seed value

Details

This function returns a p% confidence interval for estimated rho from stochcorr using parametric bootstrap. See section 4 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

Returns a matrix for the bootstrap (1-p)% confidence interval for rho. The first row of the matrix is the lower bound and the second row is the upper bound.

Examples


library(stochcorr)

data("nse2020")

## using von Mises process as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "vmp")
b <- stoch.bootstrap(a, nse2020$`USD/INR`, nse2020$Nifty, seed = 100)

rho_data <- as.data.frame(cbind(a$rho, nse2020$Date))
rho_data[, 2] <- as.Date(rho_data[, 2], origin = "1970-01-01")
colnames(rho_data) <- c("Correlation", "Time")

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(rho_data, aes(x = Time, y = Correlation)) +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  ) +
  geom_line() +
  geom_ribbon(aes(ymin = b[1, ], ymax = b[2, ]), fill = "blue", alpha = 0.15) +
  scale_y_continuous(breaks = round(seq(-1, 1, by = 0.05), 1)) +
  scale_x_date(breaks = "1 month", date_labels = "%B %Y")
  }

## using Circular Brownian Motions as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "cbm")
b <- stoch.bootstrap(a, nse2020$`USD/INR`, nse2020$Nifty, seed = 100)

rho_data <- as.data.frame(cbind(a$rho, nse2020$Date))
rho_data[, 2] <- as.Date(rho_data[, 2], origin = "1970-01-01")
colnames(rho_data) <- c("Correlation", "Time")

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(rho_data, aes(x = Time, y = Correlation)) +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  ) +
  geom_line() +
  geom_ribbon(aes(ymin = b[1, ], ymax = b[2, ]), fill = "blue", alpha = 0.15) +
  scale_y_continuous(breaks = round(seq(-1, 1, by = 0.05), 1)) +
  scale_x_date(breaks = "1 month", date_labels = "%B %Y")}
  
library(stochcorr)

data("nse2020")

## using von Mises process as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "vmp")
b <- stoch.bootstrap(a, nse2020$`USD/INR`, nse2020$Nifty, seed = 100)

rho_data <- as.data.frame(cbind(a$rho, nse2020$Date))
rho_data[, 2] <- as.Date(rho_data[, 2], origin = "1970-01-01")
colnames(rho_data) <- c("Correlation", "Time")

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(rho_data, aes(x = Time, y = Correlation)) +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  ) +
  geom_line() +
  geom_ribbon(aes(ymin = b[1, ], ymax = b[2, ]), fill = "blue", alpha = 0.15) +
  scale_y_continuous(breaks = round(seq(-1, 1, by = 0.05), 1)) +
  scale_x_date(breaks = "1 month", date_labels = "%B %Y")
  }

## using Circular Brownian Motions as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "cbm")
b <- stoch.bootstrap(a, nse2020$`USD/INR`, nse2020$Nifty, seed = 100)

rho_data <- as.data.frame(cbind(a$rho, nse2020$Date))
rho_data[, 2] <- as.Date(rho_data[, 2], origin = "1970-01-01")
colnames(rho_data) <- c("Correlation", "Time")

if(requireNamespace("ggplot2")){
library(ggplot2)
ggplot2::ggplot(rho_data, aes(x = Time, y = Correlation)) +
  theme_test() +
  theme(
    text = element_text(size = 15),
    axis.text.x = element_text(angle = 90, hjust = 1)
  ) +
  geom_line() +
  geom_ribbon(aes(ymin = b[1, ], ymax = b[2, ]), fill = "blue", alpha = 0.15) +
  scale_y_continuous(breaks = round(seq(-1, 1, by = 0.05), 1)) +
  scale_x_date(breaks = "1 month", date_labels = "%B %Y")}

Estimate a stochastic correlation model

Description

stochcorr returns the estimates of the instantaneous correlation and other model parameters.

Usage

stochcorr(S_1, S_2, dt, corr_process, iter=2000, lambda=4, lambda_1=10, lambda_2=0)
stochcorr(S_1, S_2, dt, corr_process, iter=2000, lambda=4, lambda_1=10, lambda_2=0)

Arguments

S_1

Historical price of the first asset

S_2

Historical price of the second asset

dt

Time step

corr_process

specify the correlation process, vmp for von Mises process or cbm for Circular Brownian Motion

iter

Number of iteration (Default is 2000)

lambda

regularization parameter for circular Brownian motion (Default is 4)

lambda_1

(Default is 10)

lambda_2

0 (Default)

Details

Let $S^1_t$ and $S^2_t$ be two discretely observed geometric Brownian motions observed at a time step of dt.

$dS^1_t=\mu_1S^1_tdt+\sigma_1dW_t^1\\ dS^2_t=\mu_2S^2_tdt+\sigma_2(\rho_tdW_t^1+\sqrt{1-\rho_t^2}dW_t^2)$

where $\rho_t=\cos\theta_t$ , with $\theta_t$ being specified by the von Mises Process ( $d\theta_t=\lambda\sin(\theta_t-\mu)+\sigma dW_t^3$ ) or the Circular Brownian Motion,

$d\theta=\sigma dW_t^3$

with periodic boundary conditions. Here $W_t^1,W_t^2,W_t^3$ are mutually independent Brownian Motions.

We estimate the model by maximising penalized MLE, using an iterative optimization procedure. In case of the von Mises process as the correlation process we maximise,

$\text{Log-lik}-\lambda_1\sum(\rho_{i+1}-\rho_{i})^2-\lambda_2\frac{2\lambda}{\sigma^2}$

For Circular Brownian motion as the correlation process we maximise,

$\text{Log-lik}-\lambda\sum(\rho_{i+1}-\rho_{i})^2$

See section 4 of Majumdar and Laha (2024) https://doi.org/10.48550/arXiv.2412.06343.

Value

A list containing the estimates of the model including the asset price parameters, instantaneous correlations and estimates of the correlation process

rho Estimated instantaneous correlations
mu_1 Drift of the first asset
mu_2 Drift of the second asset
sigma_1 Volatility of the first asset
sigma_2 Volatility of the second asset

if corr_process=vmp then it additionally returns

lambda_vm the drift parameter of the von Mises process
sigma_vm the volatility parameter of the von Mises process
mu_vm the mean direction of the von Mises process
lambda_1, lambda_2 value of the regularization parameters used for estimation

else if corr_process=cbm then it returns

sigma_cbm the volatility parameter of the circular Brownian motion
lambda value of the regularization parameter used for estimation

Examples


data("nse2020")

## using von Mises process as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "vmp")

## using Circular Brownian Motions as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "cbm")
data("nse2020")

## using von Mises process as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "vmp")

## using Circular Brownian Motions as the correlation process

a <- stochcorr(nse2020$`USD/INR`, nse2020$Nifty, 1 / 250, corr_process = "cbm")

Simulate stochastic correlation model

Description

stochcorr.sim returns the paths of stock price under a stochastic correlation model

Usage

stochcorr.sim(m=500, n, dt, S1_0, S2_0, mu1, sigma1, mu2, sigma2,
mu, lambda, sigma, corr_process)
stochcorr.sim(m=500, n, dt, S1_0, S2_0, mu1, sigma1, mu2, sigma2,
mu, lambda, sigma, corr_process)

Arguments

m

number of paths (Default is 500)

n

number of steps in each simulated path

dt

time step

S1_0

initial price of the first asset

S2_0

initial price of the second asset

mu1

drift of the first asset

sigma1

volatility of the first asset

mu2

drift of the second asset

sigma2

volatility of the second asset

mu

mean direction of the correlation process (if corr_process=vmp)

lambda

drift of the correlation process (if corr_process=vmp)

sigma

volatility of the correlation process (if corr_process=vmp or corr_process=cbm)

corr_process

specify the correlation process, vmp for von Mises process or cbm for Circular Brownian Motion

Details

This function returns the simulated paths of two stock prices following a stochastic correlation model. See stochcorr() details of the stochastic correlation model

Value

Returns a list with prices of two assets S1 and S2 under the stochastic correlation model

Examples

library(stochcorr)
# Generate 500 paths of two geometric Brownian motions, S1 and S2, of length 100 each
# following the von Mises process with mu=pi/2, lambda=1 and sigma =1

a<-stochcorr.sim(m=500,100,0.01,100,100,0.05,0.05,0.06,0.1,pi/2,1,1,"vmp")
t<-seq(0,100*0.01-0.01,0.01)

# Plot the first realization of S1 and S2

plot(t,a$S1[1,], ylim=c(min(a$S1[1,],a$S2[1,]),max(a$S1[1,],a$S2[1,])),type="l")
lines(t,a$S2[1,], col="red",type="l")
legend(0.01,max(a$S1[1,],a$S2[1,]), legend = c("S1","S2"), col = c("black", "red"), lty=1)

library(stochcorr)
# Generate 500 paths of two geometric Brownian motions, S1 and S2, of length 100 each
# following the von Mises process with mu=pi/2, lambda=1 and sigma =1

a<-stochcorr.sim(m=500,100,0.01,100,100,0.05,0.05,0.06,0.1,pi/2,1,1,"vmp")
t<-seq(0,100*0.01-0.01,0.01)

# Plot the first realization of S1 and S2

plot(t,a$S1[1,], ylim=c(min(a$S1[1,],a$S2[1,]),max(a$S1[1,],a$S2[1,])),type="l")
lines(t,a$S2[1,], col="red",type="l")
legend(0.01,max(a$S1[1,],a$S2[1,]), legend = c("S1","S2"), col = c("black", "red"), lty=1)

Wind direction data

Description

Wind direction data from Sotavento Wind farm from 1 November 2024 to 30 November 2024 at a 10 minute frequency.

Usage

data("wind")
data("wind")

Format

A data frame with 4320 rows and 2 columns:

Dir: Wind Direction data
Date: Date

Source

https://www.sotaventogalicia.com/en/technical-area/real-time-data/historical/

Package 'stochcorr'

Help Index

Bootstrap for circdiff

Description

Usage

Arguments

Details

Value

See Also

Examples

Estimation of circular diffusion models

Description

Usage

Arguments

Details

Value

See Also

Examples

Probability transition density function for Circular Brownian Motion

Description

Usage

Arguments

Details

Value

Probability transition density function for von Mises process

Description

Usage

Arguments

Details

Value

2020 USD/GBP and FTSE250 data

Description

Usage

Format

Source

2020 USD/JPY and Nikkei data

Description

Usage

Format

Source

2020 USD/INR and NIFTY data

Description

Usage

Format

Source

Simulate circular Brownian motion

Description

Usage

Arguments

Details

Value

Simulate von Mises process

Description

Usage

Arguments

Details

Value

2020 EUR/USD and S&P500 data

Description

Usage

Format

Source

Bootstrap for stochcorr

Description

Usage

Arguments

Details

Value

See Also

Examples

Estimate a stochastic correlation model

Description

Usage

Arguments

Details

Value

See Also

Examples

Simulate stochastic correlation model

Description