R/G2-class.R
In ArnaudBu/esg2: Esg Based on a G2++ Model
Defines functions
g2
Documented in
g2
#### G2 Class ####
##################
#' G2 Model
#'
#' An S4 class to represent a G2 ++ model.
#'
#' @slot curve ZCCurve. Zero coupon curve as basis for projection.
#' @slot a Numeric. G2++ parameter.
#' @slot b Numeric. G2++ parameter.
#' @slot sigma Numeric. G2++ parameter.
#' @slot eta Numeric. G2++ parameter.
#' @slot rho Numeric. G2++ parameter.
#' @slot horizon Numeric. Projection horizon.
#' @slot nsimul Numeric. Number of simulations.
#' @slot x Matrix Projection of the first component.
#' @slot y Matrix. Projection of the second component.
#' @slot r Matrix. Short rate projection.
#' @slot Wx Matrix. Generated normal distribution for x projection.
#'
#' @include utils.R ZCCurve-class.R Swaptions-class.R
#' @exportClass G2
setClass(Class = "G2",
representation = representation(
curve = "ZCCurve",
a = "numeric",
b = "numeric",
sigma = "numeric",
eta = "numeric",
rho = "numeric",
horizon = "numeric",
nsimul = "numeric",
x = "matrix",
y = "matrix",
r = "matrix",
Wx = "matrix"
),
validity = check_g2
)
############################################
############################################
## Initializator for class G2
setMethod(
f = "initialize",
signature = "G2",
definition = function(.Object, curve, a, b, sigma, eta, rho, horizon, nsimul){
# Assignations
.Object@curve <- curve
.Object@a <- a
.Object@b <- b
.Object@sigma <- sigma
.Object@eta <- eta
.Object@rho <- rho
.Object@horizon <- horizon
.Object@nsimul <- nsimul
# Return object
validObject(.Object)
return(.Object)
}
)
############################################
############################################
## User constructor function
#' G2 ++ model: constructor
#'
#' \code{g2} creates a g2++ model object.
#'
#' @param curve ZCCurve. Zero coupon curve as basis for projection.
#' @param horizon Numeric. Projection horizon. Default to 50.
#' @param nsimul Numeric. Number of simulation. Default to 1000.
#' @param a Numeric. G2++ parameter. Optional.
#' @param b Numeric. G2++ parameter. Optional.
#' @param sigma Numeric. G2++ parameter. Optional.
#' @param eta Numeric. G2++ parameter. Optional.
#' @param rho Numeric. G2++ parameter. Optional.
#'
#' @return The model as a class
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,
#' 0.00684,0.00777,0.00861,0.00933,0.00989,0.0103,
#' 0.01061,0.01092,0.01127,0.0117,0.01222,0.01281,
#' 0.01345,0.01411,0.01478,0.01546,0.01613,0.01679,
#' 0.01743,0.01806,0.01867,0.01926,0.01983,0.02038,
#' 0.02092,0.02143,0.02192,0.02239,0.02285,0.02329,
#' 0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#'
#' @export
g2 <- function(curve, horizon = 50, nsimul = 1000, a = numeric(0), b = numeric(0), sigma = numeric(0), eta = numeric(0), rho = numeric(0)){
new(Class = "G2",
curve = curve,
a = a,
b = b,
sigma = sigma,
eta = eta,
rho = rho,
horizon = horizon,
nsimul = nsimul
)
}
############################################
############################################
## Plot method for class G2
#' @describeIn G2 plot method for G2
#' @param x the G2 object to plot
#' @param y classical plot parameters. Useless here.
#' @import graphics grDevices
#' @export
setMethod(
f = "plot",
signature = signature(x = "G2", y = "missing"),
definition = function(x, y){
tr <- x@r
tr2 <- tr
if(nrow(tr) > 100){
ord <- order(tr[, ncol(tr)])
sel <- ord[round(seq(1, nrow(tr), length.out = 100), 0)]
tr2 <- tr2[sel,]
}
matplot(colnames(tr),t(tr2), type = "l", xlab = "time", ylab = "Short Rate")
lines(colnames(tr),colMeans(tr), lwd = 3, col = rgb(99/255, 24/255, 66/255))
lines(colnames(tr),colMeans(tr) + apply(tr, 2, sd), lwd = 3, col = "#1de9b6")
lines(colnames(tr),colMeans(tr) - apply(tr, 2, sd), lwd = 3, col = "#1de9b6")
}
)
############################################
############################################
## Print method for class G2
#' @describeIn G2 print method for G2
#' @export
setMethod(
f = "print",
signature = "G2",
definition = function(x){
cat("\na: ", x@a, '\n \n')
cat("b: ", x@b, '\n \n')
cat("sigma: ", x@sigma, '\n \n')
cat("eta: ", x@eta, '\n \n')
cat("rho: ", x@rho, '\n \n')
cat("horizon: ", x@horizon, '\n \n')
cat("number of simulation: ", x@nsimul, '\n')
}
)
############################################
############################################
## Show method for class G2
#' @describeIn G2 show method for G2
#' @param object the G2 object to show
#' @export
setMethod(
f = "show",
signature = "G2",
definition = function(object){
cat("\na: ", object@a, '\n \n')
cat("b: ", object@b, '\n \n')
cat("sigma: ", object@sigma, '\n \n')
cat("eta: ", object@eta, '\n \n')
cat("rho: ", object@rho, '\n \n')
cat("horizon: ", object@horizon, '\n \n')
cat("number of simulation: ", object@nsimul, '\n')
}
)
############################################
############################################
## Projection function
#' G2 ++ model: projection
#'
#' \code{project} projects the rate model for the object G2.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param Wx Matrix. Matrix of dimension Number of simulations x Horizon representing the normal distribution for projection of the x component.
#' @param Wy Matrix. Matrix of dimension Number of simulations x Horizon representing the normal distribution for projection of the y component. Correlation rho with Wx.
#'
#' @return The updated object.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#'
#' @export
setGeneric(
name="project",
def = function(.Object, Wx = "auto", Wy = "auto")
{
standardGeneric("project")
}
)
setMethod(
f="project",
signature="G2",
definition=function(.Object, Wx = "auto", Wy = "auto")
{
# Extraction of useful parameters
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
# Construction of the normal laws if they do not exist
if(Wx[1] == "auto" || Wy[1] == "auto"){
correl <- cbind(c(1,rho), c(rho, 1))
V_norm=rnorm(.Object@nsimul * .Object@horizon * 2,mean=0,sd=1)
V_norm=array(V_norm,dim=c(.Object@nsimul , .Object@horizon, 2))
R=chol(correl)
V=array(0,dim=c(.Object@nsimul, .Object@horizon , 2))
for(i in 1:.Object@nsimul)
{
V[i,,]=V_norm[i,,] %*% R
}
Wx = V[,,1]
Wy = V[,,2]
}
# More parameters for projection
NP = ncol(Wx)
NS = nrow(Wx)
zcp = .Object@curve@zcp
f0 = .Object@curve@ifr
#Initialization x(0)=0 y(0)=0 r(0)=r0
x=array(0,dim=c(NS,NP))
y=array(0,dim=c(NS,NP))
r=array(0,dim=c(NS,NP))
colnames(x) <- 0:(NP-1)
colnames(y) <- 0:(NP-1)
colnames(r) <- 0:(NP-1)
r[,1] <- f0[1]
#Iterations
for(i in 2:(NP))
{
# Simulation of x factor
x[,i] = next_step_g2(x[,i-1], a, sigma, Wx[,i-1])
# Simulation of y factor
y[,i] = next_step_g2(y[,i-1], b, eta, Wy[,i-1])
# Short rate
r[,i]=x[,i]+y[,i]+phi_g2(f0[i], a, b, sigma, eta, rho, i-1)
}
# Assignation in object
.Object@x = x
.Object@y = y
.Object@r = r
.Object@Wx = Wx
# Return object
return(.Object)
}
)
############################################
############################################
## Zero coupon table
#' G2 ++ model: construction of a zero coupon prices table.
#'
#' \code{zctable} computes the zero coupon prices table at time t under the t-forward measure.
#'
#' @param .Object G2 object.
#' @param t Numeric. Time for computation.
#' @param length Numeric. Length of the curve, in time steps. If not given, the curve is given until the horizon of projection.
#'
#' @return The zero-coupon table at time t.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples zc10 <- zctable(g2model, 10)
#'
#' @export
setGeneric(
name="zctable",
def = function(.Object, t = 0, length = NULL)
{
standardGeneric("zctable")
}
)
setMethod(
f="zctable",
signature="G2",
definition=function(.Object, t = 0, length = NULL)
{
# Definition of the objects useful to computation
x = .Object@x
y = .Object@y
if(is.null(length)){
NP = .Object@horizon
} else {
NP = t + length
}
NS = .Object@nsimul
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
zcp = .Object@curve@zcp
p=array(1,dim=c(NS,NP+1 - t))
colnames(p) <- t:NP
#Iterations
for(i in (t+1):NP){
if(t == 0){
p[, i+1-t] <- zcp[i] * exp(A_g2(a, b, sigma, eta, rho, t, i+2, x[,1], y[,1]))
} else{
p[, i+1-t] <- zcp[i]/zcp[t] * exp(A_g2(a, b, sigma, eta, rho, t, i, x[,t+1], y[,t+1]))
}
}
# Return
return(p)
}
)
############################################
############################################
## Deflator table
#' G2 ++ model: deflator
#'
#' \code{deflator} computes the deflator table.
#'
#' @param .Object G2 object.
#'
#' @return The deflator table.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples def <- deflator(g2model)
#'
#' @export
setGeneric(
name="deflator",
def = function(.Object)
{
standardGeneric("deflator")
}
)
setMethod(
f="deflator",
signature="G2",
definition=function(.Object)
{
r = .Object@r
p <- exp(-t(apply(r, 1, cumsum)))
colnames(p) <- 1:.Object@horizon
return(p)
}
)
############################################
############################################
## Deflator test
#' Deflator test
#'
#' \code{test_deflator} compares the initial zero coupon rate curve with the one computed from the deflator.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples zc10 <- test_deflator(g2model)
#'
#' @export
setGeneric(
name="test_deflator",
def = function(.Object)
{
standardGeneric("test_deflator")
}
)
setMethod(
f="test_deflator",
signature="G2",
definition=function(.Object)
{
def_table <- colMeans(deflator(.Object))
r_table <- zcp2rate(1:length(def_table), def_table, method = .Object@curve@method)
plot(r_table, type = "l", lwd = 3, col = rgb(99/255, 24/255, 66/255), xlab = "maturity", ylab = "rate")
lines(.Object@curve@rates, lwd = 3, col = "#1de9b6")
cat("Differences in percent by maturity: \n")
dif <- paste0(round(abs(.Object@curve@rates[1:.Object@horizon] -r_table) *100, 2), "%")
names(dif) <- 1:length(r_table)
print(dif)
}
)
############################################
############################################
## Compute swaption prices
#' Swaptions prices
#'
#' \code{price_swaption} prices a swaption according to the g2++ model.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param freqS Numeric. Swaption frequence, as the number of payments per period.
#' @param matS Numeric. Maturity of the swaption.
#' @param tenorS Numeric. Tenor of the swaption.
#'
#' @return The price of the swaption
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta =0.010382461, rho =-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples price_swaption <- price_swaption(g2model, 2, 3, 3)
#'
#' @export
setGeneric(
name="price_swaption",
def = function(.Object, freqS, matS, tenorS)
{
standardGeneric("price_swaption")
}
)
setMethod(
f="price_swaption",
signature="G2",
definition=function(.Object, freqS, matS, tenorS)
{
# Define the relevant elements
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
L <- 1
mat <- 1:length(.Object@curve@zcp)
zcp <- .Object@curve@zcp
#Compute the price
times_payments <- seq(matS + 1/freqS, matS + tenorS, by = 1 / freqS)
amounts_payments <- approx(x = 1:length(zcp), y = zcp, xout = times_payments)$y
payments <- 1/freqS * sum(amounts_payments)
s0 <- (approx(x = 1:length(zcp), y = zcp, xout = matS)$y - approx(x = 1:length(zcp), y = zcp, xout = matS + tenorS)$y) / payments
strike <- s0
indices = seq(matS + 1 / freqS, matS + tenorS,by= 1 / freqS)
B_a=(1-exp(-a*(indices-matS)))/a
B_b=(1-exp(-b*(indices-matS)))/b
mux=-M_g2(a,b,sigma,eta ,rho,matS)
muy=-M_g2(b,a,eta,sigma,rho,matS)
sigmax=sigma*sqrt((1-exp(-2*a*matS))/(2*a))
sigmay=eta*sqrt((1-exp(-2*b*matS))/(2*b))
rhoxy=rho*sigma*eta*(1-exp(-(a+b)*matS))/((a+b)*sigmax*sigmay)
c=rep(strike * 1 / freqS, length(indices))
c[length(c)]=c[length(c)]+1
A = approx(x = mat, y = zcp, xout = indices)$y/approx(x = mat, y = zcp, xout = matS)$y * exp(A_g2(a, b, sigma, eta, rho, matS, indices, 0, 0))
g=function(x){ # function to integrate
y=uniroot(f=function(z){(sum(c*A*exp(-B_a*x-B_b*z))-1)},c(-100,100))$root
khi=-B_b*(muy-0.5*(1-rhoxy^2)*sigmay^2*B_b+rhoxy*sigmay*(x-mux)/sigmax)
lambda=c*A*exp(-B_a*x)
h1=((y-muy)/(sigmay*sqrt(1-rhoxy^2))) - (rhoxy*(x-mux)/(sigmax*sqrt(1-rhoxy^2)))
h2=h1+B_b*sigmay*sqrt(1-rhoxy^2)
density_x=exp(-0.5*((x-mux)/sigmax)^2)/(sigmax*sqrt(2*pi))
T1=pnorm(-h1)
T2=sum(lambda*exp(khi)*pnorm(-h2))
g=density_x * (T1-T2)
}
# Vector version of g
g_vect=function(z){
return(mapply(function(x){g(x)},x=z))
}
PrixExact=approx(x = mat, y = zcp, xout = matS)$y*integrate(g_vect,lower=mux-5*sigmax,upper=mux+5*sigmax)$value
# Return price
return(PrixExact)
}
)
############################################
############################################
## Calibration function
#' Rate model calibration
#'
#' \code{calibrate} calibrates the model on swaptions prices.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param swaptions Swaptions. List of swaptions under the format of the class Swaption.
#' @param maxIter Numeric. Max number of iterations for Nelder-Mead algorithm. Default to 1000.
#' @param input_param Data.frame. Input parameters for optimization. It must have 4 columns of 5 rows each:
#' #' \itemize{
#' \item{"Parameter": always equals to c("a", "b", "sigma", "eta", "rho").}
#' \item{"Initial.point": Initial point for optimization. Default to c(0.1, 0.25, 0.05, 0.025, 0).}
#' \item{"Min": Minimum admissible value for parameters. Default to c(0.001, 0.0001, 0.0001, 0.001, -1).}
#' \item{"Max": Maximimu admissible value for parameters. Default to c(0.2, 0.5, 0.1, 0.05, 1).}}
#' @param monitor Boolean. To display intermediary results. Default to TRUE.
#'
#' @return The calibrated model.
#'
#' @export
setGeneric(
name="calibrate",
def = function(.Object, swaptions, maxIter = 1000, input_param = "auto", monitor = TRUE)
{
standardGeneric("calibrate")
}
)
setMethod(
f="calibrate",
signature="G2",
definition=function(.Object, swaptions, maxIter = 1000, input_param = "auto", monitor = TRUE)
{
# Parameters for optimisation
if(all(input_param == "auto")){
input_param <- data.frame(Parameter = c("a", "b", "sigma", "eta", "rho"),
Initial.point = c(0.1, 0.25, 0.05, 0.025, 0),
Min = c(0.001, 0.0001, 0.0001, 0.001, -1),
Max = c(0.2, 0.5, 0.1, 0.05, 1))
}
freq = swaptions@freq
priceSwaptions <- data.frame(mat = swaptions@mat, tenor = swaptions@tenor, price = swaptions@price)
f0 <- .Object@curve@ifr[10]
nswap <- nrow(priceSwaptions)
# Probability of negative rate at 10 years horizon
ptn <- function(param){
a = param[1]
b = param[2]
sigma = param[3]
eta = param[4]
rho = param[5]
n <- f0 + 1/2*(sigma/a*(1-exp(-10*a)))^2 + 1/2*(eta/b*(1-exp(-10*b)))^2 + rho*sigma*eta/(a*b)*(1-exp(-10*a))*(1-exp(-10*b))
d <- sigma^2/(2*a)*(1-exp(-2*10*a)) + eta^2/(2*b)*(1-exp(-2*10*b)) + 2*sigma*rho*eta/(a+b)*(1-exp(-10*(a+b)))
return(pnorm(-n/sqrt(d)))
}
# Function to optimize
OptimFunction=function(param){
model <- .Object
model@a <- param[1]
model@b <- param[2]
model@sigma <- param[3]
model@eta <- param[4]
model@rho <- param[5]
if(param[1]<input_param[1,3]||param[2]<input_param[2,3]||param[3]<input_param[3,3]||param[4]<input_param[4,3]||param[5]<input_param[5,3]||param[1]>input_param[1,4]||param[2]>input_param[2,4]||param[3]>input_param[3,4]||param[4]>input_param[4,4]||param[5]>input_param[5,4])
{
ecartRelatif=100000
} else{
priceSwaptions$valueCalc <- tryCatch({mapply(function(x,y){price_swaption(model, freq, x, y)}, x = priceSwaptions$mat, y = priceSwaptions$tenor)}, error = function(e){1000})
ecartRelatif <- sum(abs((priceSwaptions$valueCalc - priceSwaptions$price) / priceSwaptions$price), na.rm = TRUE) + nswap * 0.3 *ptn(param)
}
return(ecartRelatif)
}
# Optimization loop
paramIni <- input_param$Initial.point
param = optim(paramIni ,function(param){OptimFunction(param)},
method="Nelder-Mead",
control = list(maxit = maxIter, trace = ifelse(monitor,1,0)))
param = optim(param$par ,function(param){OptimFunction(param)},
method="BFGS",
control = list(maxit = maxIter, trace = ifelse(monitor,1,0)))
# Save of the results
.Object@a <- param$par[1]
.Object@b <- param$par[2]
.Object@sigma <- param$par[3]
.Object@eta <- param$par[4]
.Object@rho <- param$par[5]
cat("\nRelative error: ", param$value / nrow(priceSwaptions) * 100, "% \n\n")
priceSwaptions$valueCalc <- tryCatch({mapply(function(x,y){price_swaption(.Object, freq, x, y)}, x = priceSwaptions$mat, y = priceSwaptions$tenor)}, error = function(e){1000})
priceSwaptions$diff <- abs(priceSwaptions$price - priceSwaptions$valueCalc)
priceSwaptions$relDiff <- paste0(round(priceSwaptions$diff / priceSwaptions$price * 100, 3), "%")
print(priceSwaptions)
return(.Object)
}
)
ArnaudBu/esg2 documentation built on Oct. 20, 2021, 10:01 a.m.
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Defines functions g2
Documented in g2
#### G2 Class ####
##################
#' G2 Model
#'
#' An S4 class to represent a G2 ++ model.
#'
#' @slot curve ZCCurve. Zero coupon curve as basis for projection.
#' @slot a Numeric. G2++ parameter.
#' @slot b Numeric. G2++ parameter.
#' @slot sigma Numeric. G2++ parameter.
#' @slot eta Numeric. G2++ parameter.
#' @slot rho Numeric. G2++ parameter.
#' @slot horizon Numeric. Projection horizon.
#' @slot nsimul Numeric. Number of simulations.
#' @slot x Matrix Projection of the first component.
#' @slot y Matrix. Projection of the second component.
#' @slot r Matrix. Short rate projection.
#' @slot Wx Matrix. Generated normal distribution for x projection.
#'
#' @include utils.R ZCCurve-class.R Swaptions-class.R
#' @exportClass G2
setClass(Class = "G2",
representation = representation(
curve = "ZCCurve",
a = "numeric",
b = "numeric",
sigma = "numeric",
eta = "numeric",
rho = "numeric",
horizon = "numeric",
nsimul = "numeric",
x = "matrix",
y = "matrix",
r = "matrix",
Wx = "matrix"
),
validity = check_g2
)
############################################
############################################
## Initializator for class G2
setMethod(
f = "initialize",
signature = "G2",
definition = function(.Object, curve, a, b, sigma, eta, rho, horizon, nsimul){
# Assignations
.Object@curve <- curve
.Object@a <- a
.Object@b <- b
.Object@sigma <- sigma
.Object@eta <- eta
.Object@rho <- rho
.Object@horizon <- horizon
.Object@nsimul <- nsimul
# Return object
validObject(.Object)
return(.Object)
}
)
############################################
############################################
## User constructor function
#' G2 ++ model: constructor
#'
#' \code{g2} creates a g2++ model object.
#'
#' @param curve ZCCurve. Zero coupon curve as basis for projection.
#' @param horizon Numeric. Projection horizon. Default to 50.
#' @param nsimul Numeric. Number of simulation. Default to 1000.
#' @param a Numeric. G2++ parameter. Optional.
#' @param b Numeric. G2++ parameter. Optional.
#' @param sigma Numeric. G2++ parameter. Optional.
#' @param eta Numeric. G2++ parameter. Optional.
#' @param rho Numeric. G2++ parameter. Optional.
#'
#' @return The model as a class
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,
#' 0.00684,0.00777,0.00861,0.00933,0.00989,0.0103,
#' 0.01061,0.01092,0.01127,0.0117,0.01222,0.01281,
#' 0.01345,0.01411,0.01478,0.01546,0.01613,0.01679,
#' 0.01743,0.01806,0.01867,0.01926,0.01983,0.02038,
#' 0.02092,0.02143,0.02192,0.02239,0.02285,0.02329,
#' 0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#'
#' @export
g2 <- function(curve, horizon = 50, nsimul = 1000, a = numeric(0), b = numeric(0), sigma = numeric(0), eta = numeric(0), rho = numeric(0)){
new(Class = "G2",
curve = curve,
a = a,
b = b,
sigma = sigma,
eta = eta,
rho = rho,
horizon = horizon,
nsimul = nsimul
)
}
############################################
############################################
## Plot method for class G2
#' @describeIn G2 plot method for G2
#' @param x the G2 object to plot
#' @param y classical plot parameters. Useless here.
#' @import graphics grDevices
#' @export
setMethod(
f = "plot",
signature = signature(x = "G2", y = "missing"),
definition = function(x, y){
tr <- x@r
tr2 <- tr
if(nrow(tr) > 100){
ord <- order(tr[, ncol(tr)])
sel <- ord[round(seq(1, nrow(tr), length.out = 100), 0)]
tr2 <- tr2[sel,]
}
matplot(colnames(tr),t(tr2), type = "l", xlab = "time", ylab = "Short Rate")
lines(colnames(tr),colMeans(tr), lwd = 3, col = rgb(99/255, 24/255, 66/255))
lines(colnames(tr),colMeans(tr) + apply(tr, 2, sd), lwd = 3, col = "#1de9b6")
lines(colnames(tr),colMeans(tr) - apply(tr, 2, sd), lwd = 3, col = "#1de9b6")
}
)
############################################
############################################
## Print method for class G2
#' @describeIn G2 print method for G2
#' @export
setMethod(
f = "print",
signature = "G2",
definition = function(x){
cat("\na: ", x@a, '\n \n')
cat("b: ", x@b, '\n \n')
cat("sigma: ", x@sigma, '\n \n')
cat("eta: ", x@eta, '\n \n')
cat("rho: ", x@rho, '\n \n')
cat("horizon: ", x@horizon, '\n \n')
cat("number of simulation: ", x@nsimul, '\n')
}
)
############################################
############################################
## Show method for class G2
#' @describeIn G2 show method for G2
#' @param object the G2 object to show
#' @export
setMethod(
f = "show",
signature = "G2",
definition = function(object){
cat("\na: ", object@a, '\n \n')
cat("b: ", object@b, '\n \n')
cat("sigma: ", object@sigma, '\n \n')
cat("eta: ", object@eta, '\n \n')
cat("rho: ", object@rho, '\n \n')
cat("horizon: ", object@horizon, '\n \n')
cat("number of simulation: ", object@nsimul, '\n')
}
)
############################################
############################################
## Projection function
#' G2 ++ model: projection
#'
#' \code{project} projects the rate model for the object G2.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param Wx Matrix. Matrix of dimension Number of simulations x Horizon representing the normal distribution for projection of the x component.
#' @param Wy Matrix. Matrix of dimension Number of simulations x Horizon representing the normal distribution for projection of the y component. Correlation rho with Wx.
#'
#' @return The updated object.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#'
#' @export
setGeneric(
name="project",
def = function(.Object, Wx = "auto", Wy = "auto")
{
standardGeneric("project")
}
)
setMethod(
f="project",
signature="G2",
definition=function(.Object, Wx = "auto", Wy = "auto")
{
# Extraction of useful parameters
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
# Construction of the normal laws if they do not exist
if(Wx[1] == "auto" || Wy[1] == "auto"){
correl <- cbind(c(1,rho), c(rho, 1))
V_norm=rnorm(.Object@nsimul * .Object@horizon * 2,mean=0,sd=1)
V_norm=array(V_norm,dim=c(.Object@nsimul , .Object@horizon, 2))
R=chol(correl)
V=array(0,dim=c(.Object@nsimul, .Object@horizon , 2))
for(i in 1:.Object@nsimul)
{
V[i,,]=V_norm[i,,] %*% R
}
Wx = V[,,1]
Wy = V[,,2]
}
# More parameters for projection
NP = ncol(Wx)
NS = nrow(Wx)
zcp = .Object@curve@zcp
f0 = .Object@curve@ifr
#Initialization x(0)=0 y(0)=0 r(0)=r0
x=array(0,dim=c(NS,NP))
y=array(0,dim=c(NS,NP))
r=array(0,dim=c(NS,NP))
colnames(x) <- 0:(NP-1)
colnames(y) <- 0:(NP-1)
colnames(r) <- 0:(NP-1)
r[,1] <- f0[1]
#Iterations
for(i in 2:(NP))
{
# Simulation of x factor
x[,i] = next_step_g2(x[,i-1], a, sigma, Wx[,i-1])
# Simulation of y factor
y[,i] = next_step_g2(y[,i-1], b, eta, Wy[,i-1])
# Short rate
r[,i]=x[,i]+y[,i]+phi_g2(f0[i], a, b, sigma, eta, rho, i-1)
}
# Assignation in object
.Object@x = x
.Object@y = y
.Object@r = r
.Object@Wx = Wx
# Return object
return(.Object)
}
)
############################################
############################################
## Zero coupon table
#' G2 ++ model: construction of a zero coupon prices table.
#'
#' \code{zctable} computes the zero coupon prices table at time t under the t-forward measure.
#'
#' @param .Object G2 object.
#' @param t Numeric. Time for computation.
#' @param length Numeric. Length of the curve, in time steps. If not given, the curve is given until the horizon of projection.
#'
#' @return The zero-coupon table at time t.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples zc10 <- zctable(g2model, 10)
#'
#' @export
setGeneric(
name="zctable",
def = function(.Object, t = 0, length = NULL)
{
standardGeneric("zctable")
}
)
setMethod(
f="zctable",
signature="G2",
definition=function(.Object, t = 0, length = NULL)
{
# Definition of the objects useful to computation
x = .Object@x
y = .Object@y
if(is.null(length)){
NP = .Object@horizon
} else {
NP = t + length
}
NS = .Object@nsimul
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
zcp = .Object@curve@zcp
p=array(1,dim=c(NS,NP+1 - t))
colnames(p) <- t:NP
#Iterations
for(i in (t+1):NP){
if(t == 0){
p[, i+1-t] <- zcp[i] * exp(A_g2(a, b, sigma, eta, rho, t, i+2, x[,1], y[,1]))
} else{
p[, i+1-t] <- zcp[i]/zcp[t] * exp(A_g2(a, b, sigma, eta, rho, t, i, x[,t+1], y[,t+1]))
}
}
# Return
return(p)
}
)
############################################
############################################
## Deflator table
#' G2 ++ model: deflator
#'
#' \code{deflator} computes the deflator table.
#'
#' @param .Object G2 object.
#'
#' @return The deflator table.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples def <- deflator(g2model)
#'
#' @export
setGeneric(
name="deflator",
def = function(.Object)
{
standardGeneric("deflator")
}
)
setMethod(
f="deflator",
signature="G2",
definition=function(.Object)
{
r = .Object@r
p <- exp(-t(apply(r, 1, cumsum)))
colnames(p) <- 1:.Object@horizon
return(p)
}
)
############################################
############################################
## Deflator test
#' Deflator test
#'
#' \code{test_deflator} compares the initial zero coupon rate curve with the one computed from the deflator.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta=0.010382461, rho=-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples zc10 <- test_deflator(g2model)
#'
#' @export
setGeneric(
name="test_deflator",
def = function(.Object)
{
standardGeneric("test_deflator")
}
)
setMethod(
f="test_deflator",
signature="G2",
definition=function(.Object)
{
def_table <- colMeans(deflator(.Object))
r_table <- zcp2rate(1:length(def_table), def_table, method = .Object@curve@method)
plot(r_table, type = "l", lwd = 3, col = rgb(99/255, 24/255, 66/255), xlab = "maturity", ylab = "rate")
lines(.Object@curve@rates, lwd = 3, col = "#1de9b6")
cat("Differences in percent by maturity: \n")
dif <- paste0(round(abs(.Object@curve@rates[1:.Object@horizon] -r_table) *100, 2), "%")
names(dif) <- 1:length(r_table)
print(dif)
}
)
############################################
############################################
## Compute swaption prices
#' Swaptions prices
#'
#' \code{price_swaption} prices a swaption according to the g2++ model.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param freqS Numeric. Swaption frequence, as the number of payments per period.
#' @param matS Numeric. Maturity of the swaption.
#' @param tenorS Numeric. Tenor of the swaption.
#'
#' @return The price of the swaption
#'
#' @examples rates <- c(-0.00316,-0.00269,-0.00203,-0.00122,
#' -0.00022,0.00092,0.00215,0.00342,0.00465,0.00581,0.00684,
#' 0.00777,0.00861,0.00933,0.00989,0.0103,0.01061,0.01092,
#' 0.01127,0.0117,0.01222,0.01281,0.01345,0.01411,0.01478,
#' 0.01546,0.01613,0.01679,0.01743,0.01806,0.01867,0.01926,
#' 0.01983,0.02038,0.02092,0.02143,0.02192,0.02239,0.02285,
#' 0.02329,0.02371,0.02411,0.0245,0.02488,0.02524,0.02558,
#' 0.02592,0.02624,0.02655,0.02685)
#' @examples curve <-curvezc(rates, "continuous")
#' @examples g2model <- g2(curve, a=0.773511777, b=0.082013014,
#' sigma=0.022284644, eta =0.010382461, rho =-0.701985206)
#' @examples g2model <- project(g2model)
#' @examples price_swaption <- price_swaption(g2model, 2, 3, 3)
#'
#' @export
setGeneric(
name="price_swaption",
def = function(.Object, freqS, matS, tenorS)
{
standardGeneric("price_swaption")
}
)
setMethod(
f="price_swaption",
signature="G2",
definition=function(.Object, freqS, matS, tenorS)
{
# Define the relevant elements
a = .Object@a
b = .Object@b
sigma = .Object@sigma
eta = .Object@eta
rho = .Object@rho
L <- 1
mat <- 1:length(.Object@curve@zcp)
zcp <- .Object@curve@zcp
#Compute the price
times_payments <- seq(matS + 1/freqS, matS + tenorS, by = 1 / freqS)
amounts_payments <- approx(x = 1:length(zcp), y = zcp, xout = times_payments)$y
payments <- 1/freqS * sum(amounts_payments)
s0 <- (approx(x = 1:length(zcp), y = zcp, xout = matS)$y - approx(x = 1:length(zcp), y = zcp, xout = matS + tenorS)$y) / payments
strike <- s0
indices = seq(matS + 1 / freqS, matS + tenorS,by= 1 / freqS)
B_a=(1-exp(-a*(indices-matS)))/a
B_b=(1-exp(-b*(indices-matS)))/b
mux=-M_g2(a,b,sigma,eta ,rho,matS)
muy=-M_g2(b,a,eta,sigma,rho,matS)
sigmax=sigma*sqrt((1-exp(-2*a*matS))/(2*a))
sigmay=eta*sqrt((1-exp(-2*b*matS))/(2*b))
rhoxy=rho*sigma*eta*(1-exp(-(a+b)*matS))/((a+b)*sigmax*sigmay)
c=rep(strike * 1 / freqS, length(indices))
c[length(c)]=c[length(c)]+1
A = approx(x = mat, y = zcp, xout = indices)$y/approx(x = mat, y = zcp, xout = matS)$y * exp(A_g2(a, b, sigma, eta, rho, matS, indices, 0, 0))
g=function(x){ # function to integrate
y=uniroot(f=function(z){(sum(c*A*exp(-B_a*x-B_b*z))-1)},c(-100,100))$root
khi=-B_b*(muy-0.5*(1-rhoxy^2)*sigmay^2*B_b+rhoxy*sigmay*(x-mux)/sigmax)
lambda=c*A*exp(-B_a*x)
h1=((y-muy)/(sigmay*sqrt(1-rhoxy^2))) - (rhoxy*(x-mux)/(sigmax*sqrt(1-rhoxy^2)))
h2=h1+B_b*sigmay*sqrt(1-rhoxy^2)
density_x=exp(-0.5*((x-mux)/sigmax)^2)/(sigmax*sqrt(2*pi))
T1=pnorm(-h1)
T2=sum(lambda*exp(khi)*pnorm(-h2))
g=density_x * (T1-T2)
}
# Vector version of g
g_vect=function(z){
return(mapply(function(x){g(x)},x=z))
}
PrixExact=approx(x = mat, y = zcp, xout = matS)$y*integrate(g_vect,lower=mux-5*sigmax,upper=mux+5*sigmax)$value
# Return price
return(PrixExact)
}
)
############################################
############################################
## Calibration function
#' Rate model calibration
#'
#' \code{calibrate} calibrates the model on swaptions prices.
#'
#' @param .Object G2 object. The object whose model is to be projected.
#' @param swaptions Swaptions. List of swaptions under the format of the class Swaption.
#' @param maxIter Numeric. Max number of iterations for Nelder-Mead algorithm. Default to 1000.
#' @param input_param Data.frame. Input parameters for optimization. It must have 4 columns of 5 rows each:
#' #' \itemize{
#' \item{"Parameter": always equals to c("a", "b", "sigma", "eta", "rho").}
#' \item{"Initial.point": Initial point for optimization. Default to c(0.1, 0.25, 0.05, 0.025, 0).}
#' \item{"Min": Minimum admissible value for parameters. Default to c(0.001, 0.0001, 0.0001, 0.001, -1).}
#' \item{"Max": Maximimu admissible value for parameters. Default to c(0.2, 0.5, 0.1, 0.05, 1).}}
#' @param monitor Boolean. To display intermediary results. Default to TRUE.
#'
#' @return The calibrated model.
#'
#' @export
setGeneric(
name="calibrate",
def = function(.Object, swaptions, maxIter = 1000, input_param = "auto", monitor = TRUE)
{
standardGeneric("calibrate")
}
)
setMethod(
f="calibrate",
signature="G2",
definition=function(.Object, swaptions, maxIter = 1000, input_param = "auto", monitor = TRUE)
{
# Parameters for optimisation
if(all(input_param == "auto")){
input_param <- data.frame(Parameter = c("a", "b", "sigma", "eta", "rho"),
Initial.point = c(0.1, 0.25, 0.05, 0.025, 0),
Min = c(0.001, 0.0001, 0.0001, 0.001, -1),
Max = c(0.2, 0.5, 0.1, 0.05, 1))
}
freq = swaptions@freq
priceSwaptions <- data.frame(mat = swaptions@mat, tenor = swaptions@tenor, price = swaptions@price)
f0 <- .Object@curve@ifr[10]
nswap <- nrow(priceSwaptions)
# Probability of negative rate at 10 years horizon
ptn <- function(param){
a = param[1]
b = param[2]
sigma = param[3]
eta = param[4]
rho = param[5]
n <- f0 + 1/2*(sigma/a*(1-exp(-10*a)))^2 + 1/2*(eta/b*(1-exp(-10*b)))^2 + rho*sigma*eta/(a*b)*(1-exp(-10*a))*(1-exp(-10*b))
d <- sigma^2/(2*a)*(1-exp(-2*10*a)) + eta^2/(2*b)*(1-exp(-2*10*b)) + 2*sigma*rho*eta/(a+b)*(1-exp(-10*(a+b)))
return(pnorm(-n/sqrt(d)))
}
# Function to optimize
OptimFunction=function(param){
model <- .Object
model@a <- param[1]
model@b <- param[2]
model@sigma <- param[3]
model@eta <- param[4]
model@rho <- param[5]
if(param[1]<input_param[1,3]||param[2]<input_param[2,3]||param[3]<input_param[3,3]||param[4]<input_param[4,3]||param[5]<input_param[5,3]||param[1]>input_param[1,4]||param[2]>input_param[2,4]||param[3]>input_param[3,4]||param[4]>input_param[4,4]||param[5]>input_param[5,4])
{
ecartRelatif=100000
} else{
priceSwaptions$valueCalc <- tryCatch({mapply(function(x,y){price_swaption(model, freq, x, y)}, x = priceSwaptions$mat, y = priceSwaptions$tenor)}, error = function(e){1000})
ecartRelatif <- sum(abs((priceSwaptions$valueCalc - priceSwaptions$price) / priceSwaptions$price), na.rm = TRUE) + nswap * 0.3 *ptn(param)
}
return(ecartRelatif)
}
# Optimization loop
paramIni <- input_param$Initial.point
param = optim(paramIni ,function(param){OptimFunction(param)},
method="Nelder-Mead",
control = list(maxit = maxIter, trace = ifelse(monitor,1,0)))
param = optim(param$par ,function(param){OptimFunction(param)},
method="BFGS",
control = list(maxit = maxIter, trace = ifelse(monitor,1,0)))
# Save of the results
.Object@a <- param$par[1]
.Object@b <- param$par[2]
.Object@sigma <- param$par[3]
.Object@eta <- param$par[4]
.Object@rho <- param$par[5]
cat("\nRelative error: ", param$value / nrow(priceSwaptions) * 100, "% \n\n")
priceSwaptions$valueCalc <- tryCatch({mapply(function(x,y){price_swaption(.Object, freq, x, y)}, x = priceSwaptions$mat, y = priceSwaptions$tenor)}, error = function(e){1000})
priceSwaptions$diff <- abs(priceSwaptions$price - priceSwaptions$valueCalc)
priceSwaptions$relDiff <- paste0(round(priceSwaptions$diff / priceSwaptions$price * 100, 3), "%")
print(priceSwaptions)
return(.Object)
}
)
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