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Diffstat (limited to 'R/tranche_functions.R')
| -rw-r--r-- | R/tranche_functions.R | 886 |
1 files changed, 0 insertions, 886 deletions
diff --git a/R/tranche_functions.R b/R/tranche_functions.R deleted file mode 100644 index c969c1fd..00000000 --- a/R/tranche_functions.R +++ /dev/null @@ -1,886 +0,0 @@ -## todo:
-## -investigate other ways to interpolate the random severities on the grid
-## I'm thinking that at eah severity that we add to the distribution, round it down
-## and keep track of the missing mass: namely if X_i=S_i w.p p_i, then add
-## X_i=lu*floor(S_i/lu) with probability p_i and propagate
-## X_{i+1}=S_{i+1}+(S_i-lu*floor(S_i/lu)) with the right probability
-## - investigate truncated distributions more (need to compute loss and recov distribution
-## separately, for the 0-10 equity tranche, we need the loss on the 0-0.1 support and
-## recovery with 0.1-1 support, so it's not clear that there is a big gain.
-## - do the correlation adjustments when computing the deltas since it seems to be
-## the market standard
-hostname <- system("hostname", intern=TRUE)
-
-checkSymbol <- function(name){
- if(!is.loaded(name)){
- if(.Platform$OS.type == "unix"){
- root.dir <- "/home/share/CorpCDOs"
- dyn.load(file.path(root.dir, "code", "R", paste0("lossdistrib",
- hostname,
- .Platform$dynlib.ext)))
- }else{
- root.dir <- "//WDSENTINEL/share/CorpCDOs"
- dyn.load(file.path(root.dir, "code", "R", paste0("lossdistrib",
- .Platform$dynlib.ext)))
- }
- }
-}
-lossdistrib <- function(p){
- ## basic recursive algorithm of Andersen, Sidenius and Basu
- n <- length(p)
- q <- rep(0, (n+1))
- q[1] <- 1
- for(i in 1:n){
- q[-1] <- p[i]*q[-(n+1)]+(1-p[i])*q[-1]
- q[1] <- (1-p[i])*q[1]
- }
- return(q)
-}
-
-lossdistrib.fft <- function(p){
- ## computes loss distribution using the fft
- ## complexity is of order O(n*m)+O(m*log(m))
- ## where m is the size of the grid and n the number of probabilities.
- ## this is slower than the recursive algorithm
- theta <- 2*pi*1i*(0:n)/(n+1)
- Phi <- 1 - p + p%o%exp(theta)
- v <- apply(Phi, 2, prod)
- return(1/(n+1)*Re(fft(v)))
-}
-
-lossdistrib2 <- function(p, w, S, N, defaultflag=FALSE){
- ## recursive algorithm with first order correction
- ## p vector of default probabilities
- ## w vector of weigths
- ## S vector of severities
- ## N number of ticks in the grid
- ## defaultflag if true computes the default distribution
- n <- length(p)
- lu <- 1/(N-1)
- q <- rep(0, N)
- q[1] <- 1
- for(i in 1:n){
- if(defaultflag){
- d <- w[i] /lu
- }else{
- d <- S[i] * w[i] / lu
- }
- d1 <- floor(d)
- d2 <- ceiling(d)
- p1 <- p[i]*(d2-d)
- p2 <- p[i] - p1
- q1 <- c(rep(0,d1), p1*q[1:(N-d1)])
- q2 <- c(rep(0,d2), p2*q[1:(N-d2)])
- q <- q1 + q2 + (1-p[i])*q
- }
- q[length(q)] <- q[length(q)]+1-sum(q)
- return(q)
-}
-
-lossdistrib2.truncated <- function(p, w, S, N, cutoff=N){
- ## recursive algorithm with first order correction
- ## p vector of default probabilities
- ## w vector of weigths
- ## S vector of severities
- ## N number of ticks in the grid (for best accuracy should
- ## be a multiple of the number of issuers)
- ## cutoff where to stop computing the exact probabilities
- ## (useful for tranche computations)
-
- ## this is actually slower than lossdistrib2. But in C this is
- ## twice as fast.
- ## for high severities, M can become bigger than N, and there is
- ## some probability mass escaping.
- n <- length(p)
- lu <- 1/(N-1)
- q <- rep(0, truncated)
- q[1] <- 1
- M <- 1
- for(i in 1:n){
- d <- S[i] * w[i] / lu
- d1 <- floor(d)
- d2 <- ceiling(d)
- p1 <- p[i]*(d2-d)
- p2 <- p[i] - p1
- q1 <- p1*q[1:min(M, cutoff-d1)]
- q2 <- p2*q[1:min(M, cutoff-d2)]
- q[1:min(M, cutoff)] <- (1-p[i])*q[1:min(M, cutoff)]
- q[(d1+1):min(M+d1, cutoff)] <- q[(d1+1):min(M+d1, cutoff)]+q1
- q[(d2+1):min(M+d2, cutoff)] <- q[(d2+1):min(M+d2, cutoff)]+q2
- M <- M+d2
- }
- return(q)
-}
-
-recovdist <- function(dp, pp, w, S, N){
- ## computes the recovery distribution for a sum of independent variables
- ## R=\sum_{i=1}^n X_i
- ## where X_i = 0 w.p 1-dp[i]-pp[i]
- ## = w[i]*(1-S[i]) w.p dp[i]
- ## = w[i] w.p pp[i]
- ## each non zero value v is interpolated on the grid as
- ## the pair of values floor(v/lu) and ceiling(v/lu) so that
- ## X_i has four non zero values
- n <- length(dp)
- q <- rep(0, N)
- q[1] <- 1
- lu <- 1/(N-1)
- for(i in 1:n){
- d1 <- w[i]*(1-S[i])/lu
- d1l <- floor(d1)
- d1u <- ceiling(d1)
- d2 <- w[i] / lu
- d2l <- floor(d2)
- d2u <- ceiling(d2)
- dp1 <- dp[i] * (d1u-d1)
- dp2 <- dp[i] - dp1
- pp1 <- pp[i] * (d2u - d2)
- pp2 <- pp[i] - pp1
- q1 <- c(rep(0, d1l), dp1 * q[1:(N-d1l)])
- q2 <- c(rep(0, d1u), dp2 * q[1:(N-d1u)])
- q3 <- c(rep(0, d2l), pp1 * q[1:(N-d2l)])
- q4 <- c(rep(0, d2u), pp2 *q[1:(N-d2u)])
- q <- q1+q2+q3+q4+(1-dp[i]-pp[i])*q
- }
- return(q)
-}
-
-lossdist.joint <- function(p, w, S, N, defaultflag=FALSE){
- ## recursive algorithm with first order correction
- ## to compute the joint probability distribution of the loss and recovery
- ## inputs:
- ## p: vector of default probabilities
- ## w: vector of issuer weights
- ## S: vector of severities
- ## N: number of tick sizes on the grid
- ## defaultflag: if true computes the default distribution
- ## output:
- ## q: matrix of joint loss, recovery probability
- ## colSums(q) is the recovery distribution marginal
- ## rowSums(q) is the loss distribution marginal
- n <- length(p)
- lu <- 1/(N-1)
- q <- matrix(0, N, N)
- q[1,1] <- 1
- for(k in 1:n){
- if(defaultflag){
- x <- w[k] / lu
- }else{
- x <- S[k] * w[k]/lu
- }
- y <- (1-S[k]) * w[k]/lu
- i <- floor(x)
- j <- floor(y)
- weights <- c((i+1-x)*(j+1-y), (i+1-x)*(y-j), (x-i)*(y-j), (j+1-y)*(x-i))
- psplit <- p[k] * weights
- qtemp <- matrix(0, N, N)
- qtemp[(i+1):N,(j+1):N] <- qtemp[(i+1):N,(j+1):N] + psplit[1] * q[1:(N-i),1:(N-j)]
- qtemp[(i+1):N,(j+2):N] <- qtemp[(i+1):N,(j+2):N] + psplit[2] * q[1:(N-i), 1:(N-j-1)]
- qtemp[(i+2):N,(j+2):N] <- qtemp[(i+2):N,(j+2):N] + psplit[3] * q[1:(N-i-1), 1:(N-j-1)]
- qtemp[(i+2):N, (j+1):N] <- qtemp[(i+2):N, (j+1):N] + psplit[4] * q[1:(N-i-1), 1:(N-j)]
- q <- qtemp + (1-p[k])*q
- }
- q[length(q)] <- q[length(q)]+1-sum(q)
- return(q)
-}
-
-lossdist.prepay.joint <- function(dp, pp, w, S, N, defaultflag=FALSE){
- ## recursive algorithm with first order correction
- ## to compute the joint probability distribition of the loss and recovery
- ## inputs:
- ## dp: vector of default probabilities
- ## pp: vector of prepay probabilities
- ## w: vector of issuer weights
- ## S: vector of severities
- ## N: number of tick sizes on the grid
- ## defaultflag: if true computes the default
- ## outputs
- ## q: matrix of joint loss and recovery probability
- ## colSums(q) is the recovery distribution marginal
- ## rowSums(q) is the loss distribution marginal
- n <- length(dp)
- lu <- 1/(N-1)
- q <- matrix(0, N, N)
- q[1,1] <- 1
- for(k in 1:n){
- y1 <- (1-S[k]) * w[k]/lu
- y2 <- w[k]/lu
- j1 <- floor(y1)
- j2 <- floor(y2)
- if(defaultflag){
- x <- y2
- i <- j2
- }else{
- x <- y2-y1
- i <- floor(x)
- }
-
- ## weights <- c((i+1-x)*(j1+1-y1), (i+1-x)*(y1-j1), (x-i)*(y1-j1), (j1+1-y1)*(x-i))
- weights1 <- c((i+1-x)*(j1+1-y1), (i+1-x)*(y1-j1), (x-i)*(y1-j1), (j1+1-y1)*(x-i))
- dpsplit <- dp[k] * weights1
-
- if(defaultflag){
- weights2 <- c((i+1-x)*(j2+1-y2), (i+1-x)*(y2-j2), (x-i)*(y2-j2), (j2+1-y2)*(x-i))
- ppsplit <- pp[k] * weights2
- }else{
- ppsplit <- pp[k] * c(j2+1-y2, y2-j2)
- }
- qtemp <- matrix(0, N, N)
- qtemp[(i+1):N,(j1+1):N] <- qtemp[(i+1):N,(j1+1):N] + dpsplit[1] * q[1:(N-i),1:(N-j1)]
- qtemp[(i+1):N,(j1+2):N] <- qtemp[(i+1):N,(j1+2):N] + dpsplit[2] * q[1:(N-i), 1:(N-j1-1)]
- qtemp[(i+2):N,(j1+2):N] <- qtemp[(i+2):N,(j1+2):N] + dpsplit[3] * q[1:(N-i-1), 1:(N-j1-1)]
- qtemp[(i+2):N,(j1+1):N] <- qtemp[(i+2):N, (j1+1):N] + dpsplit[4] * q[1:(N-i-1), 1:(N-j1)]
- if(defaultflag){
- qtemp[(i+1):N,(j2+1):N] <- qtemp[(i+1):N,(j2+1):N] + ppsplit[1] * q[1:(N-i),1:(N-j2)]
- qtemp[(i+1):N,(j2+2):N] <- qtemp[(i+1):N,(j2+2):N] + ppsplit[2] * q[1:(N-i), 1:(N-j2-1)]
- qtemp[(i+2):N,(j2+2):N] <- qtemp[(i+2):N,(j2+2):N] + ppsplit[3] * q[1:(N-i-1), 1:(N-j2-1)]
- qtemp[(i+2):N,(j2+1):N] <- qtemp[(i+2):N, (j2+1):N] + ppsplit[4] * q[1:(N-i-1), 1:(N-j2)]
- }else{
- qtemp[, (j2+1):N] <- qtemp[,(j2+1):N]+ppsplit[1]*q[,1:(N-j2)]
- qtemp[, (j2+2):N] <- qtemp[,(j2+2):N]+ppsplit[2]*q[,1:(N-j2-1)]
- }
- q <- qtemp + (1-pp[k]-dp[k]) * q
- }
- q[length(q)] <- q[length(q)] + 1 - sum(q)
- return(q)
-}
-
-lossdistC <- function(p, w, S, N, defaultflag=FALSE){
- ## C version of lossdistrib2, roughly 50 times faster
- checkSymbol("lossdistrib")
- .C("lossdistrib", as.double(p), as.integer(length(p)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag), q = double(N))$q
-}
-
-lossdistCblas <- function(p, w, S, N, defaultflag=FALSE){
- ## C version of lossdistrib2, roughly 50 times faster
- checkSymbol("lossdistrib_blas")
- .C("lossdistrib_blas", as.double(p), as.integer(length(p)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag), q = double(N))$q
-}
-
-lossdistCZ <- function(p, w, S, N, defaultflag=FALSE, rho, Z, wZ){
- checkSymbol("lossdistrib_Z")
- .C("lossdistrib_Z", as.double(p), as.integer(length(p)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag),
- as.double(rho), as.double(Z), as.integer(length(Z)),
- q = matrix(0, N, length(Z)))$q
-}
-
-lossdistC.truncated <- function(p, w, S, N, T=N){
- ## C version of lossdistrib2, roughly 50 times faster
- checkSymbol("lossdistrib_truncated")
- .C("lossdistrib_truncated", as.double(p), as.integer(length(p)),
- as.double(w), as.double(S), as.integer(N), as.integer(T), q = double(T))$q
-}
-
-recovdistC <- function(dp, pp, w, S, N){
- ## C version of recovdist
- checkSymbol("recovdist")
- .C("recovdist", as.double(dp), as.double(pp), as.integer(length(dp)),
- as.double(w), as.double(S), as.integer(N), q = double(N))$q
-}
-
-lossdistC.joint <- function(p, w, S, N, defaultflag=FALSE){
- ## C version of lossdistrib.joint, roughly 20 times faster
- checkSymbol("lossdistrib_joint")
- .C("lossdistrib_joint", as.double(p), as.integer(length(p)), as.double(w),
- as.double(S), as.integer(N), as.logical(defaultflag), q = matrix(0, N, N))$q
-}
-
-lossdistC.jointblas <- function(p, w, S, N, defaultflag=FALSE){
- ## C version of lossdistrib.joint, roughly 20 times faster
- checkSymbol("lossdistrib_joint_blas")
- .C("lossdistrib_joint_blas", as.double(p), as.integer(length(p)), as.double(w),
- as.double(S), as.integer(N), as.logical(defaultflag), q = matrix(0, N, N))$q
-}
-
-
-lossdistC.jointZ <- function(dp, w, S, N, defaultflag = FALSE, rho, Z, wZ){
- ## N is the size of the grid
- ## dp is of size n.credits
- ## w is of size n.credits
- ## S is of size n.credits by nZ
- ## rho is a double
- ## Z is a vector of length nZ
- ## w is a vector if length wZ
- checkSymbol("lossdistrib_joint_Z")
- r <- .C("lossdistrib_joint_Z", as.double(dp), as.integer(length(dp)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag), as.double(rho),
- as.double(Z), as.double(wZ), as.integer(length(Z)), q = matrix(0, N, N))$q
-}
-
-lossdistC.prepay.joint <- function(dp, pp, w, S, N, defaultflag=FALSE){
- ## C version of lossdist.prepay.joint
- checkSymbol("lossdistrib_prepay_joint")
- r <- .C("lossdistrib_prepay_joint", as.double(dp), as.double(pp), as.integer(length(dp)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag), q=matrix(0, N, N))$q
- return(r)
-}
-
-lossdistC.prepay.jointZ <- function(dp, pp, w, S, N, defaultflag = FALSE, rho, Z, wZ){
- ## N is the size of the grid
- ## dp is of size n.credits
- ## pp is of size n.credits
- ## w is of size n.credits
- ## S is of size n.credits by nZ
- ## rho is a double
- ## Z is a vector of length nZ
- ## w is a vector if length wZ
- checkSymbol("lossdistrib_prepay_joint_Z")
- r <- .C("lossdistrib_prepay_joint_Z", as.double(dp), as.double(pp), as.integer(length(dp)),
- as.double(w), as.double(S), as.integer(N), as.logical(defaultflag), as.double(rho),
- as.double(Z), as.double(wZ), as.integer(length(Z)), q = matrix(0, N, N))$q
-}
-
-lossrecovdist <- function(defaultprob, prepayprob, w, S, N, defaultflag=FALSE, useC=TRUE){
- if(missing(prepayprob)){
- if(useC){
- L <- lossdistC(defaultprob, w, S, N, defaultflag)
- R <- lossdistC(defaultprob, w, 1-S, N)
- }else{
- L <- lossdistrib2(defaultprob, w, S, N, defaultflag)
- R <- lossdistrib2(defaultprob, w, 1-S, N)
- }
- }else{
- if(useC){
- L <- lossdistC(defaultprob+defaultflag*prepayprob, w, S, N, defaultflag)
- R <- recovdistC(defaultprob, prepayprob, w, S, N)
- }else{
- L <- lossdistrib2(defaultprob+defaultflag*prepayprob, w, S, N, defaultflag)
- R <- recovdist(defaultprob, prepayprob, w, S, N)
- }
- }
- return(list(L=L, R=R))
-}
-
-lossrecovdist.term <- function(defaultprob, prepayprob, w, S, N, defaultflag=FALSE, useC=TRUE){
- ## computes the loss and recovery distribution over time
- L <- array(0, dim=c(N, ncol(defaultprob)))
- R <- array(0, dim=c(N, ncol(defaultprob)))
- if(missing(prepayprob)){
- for(t in 1:ncol(defaultprob)){
- temp <- lossrecovdist(defaultprob[,t], , w, S[,t], N, defaultflag, useC)
- L[,t] <- temp$L
- R[,t] <- temp$R
- }
- }else{
- for(t in 1:ncol(defaultprob)){
- temp <- lossrecovdist(defaultprob[,t], prepayprob[,t], w, S[,t], N, defaultflag, useC)
- L[,t] <- temp$L
- R[,t] <- temp$R
- }
- }
- return(list(L=L, R=R))
-}
-
-lossrecovdist.joint.term <- function(defaultprob, prepayprob, w, S, N, defaultflag=FALSE, useC=TRUE){
- ## computes the joint loss and recovery distribution over time
- Q <- array(0, dim=c(ncol(defaultprob), N, N))
- if(useC){
- if(missing(prepayprob)){
- for(t in 1:ncol(defaultprob)){
- Q[t,,] <- lossdistC.joint(defaultprob[,t], w, S[,t], N, defaultflag)
- }
- }else{
- for(t in 1:ncol(defaultprob)){
- Q[t,,] <- lossdistC.prepay.joint(defaultprob[,t], prepayprob[,t], w, S[,t], N, defaultflag)
- }
- }
- }else{
- if(missing(prepayprob)){
- for(t in 1:ncol(defaultprob)){
- Q[t,,] <- lossdist.joint(defaultprob[,t], w, S[,t], N, defaultflag)
- }
- }else{
- for(t in 1:ncol(defaultprob)){
- Q[t,,] <- lossdist.prepay.joint(defaultprob[,t], prepayprob[,t], w, S[,t], N, defaultflag)
- }
- }
- }
- gc()
- return(Q)
-}
-
-dist.transform <- function(dist.joint){
- ## compute the joint (D, R) distribution
- ## from the (L, R) distribution using D = L+R
- distDR <- array(0, dim=dim(dist.joint))
- Ngrid <- dim(dist.joint)[2]
- for(t in 1:dim(dist.joint)[1]){
- for(i in 1:Ngrid){
- for(j in 1:Ngrid){
- index <- i+j
- if(index <= Ngrid){
- distDR[t,index,j] <- distDR[t,index,j] + dist.joint[t,i,j]
- }else{
- distDR[t,Ngrid,j] <- distDR[t,Ngrid,j] +
- dist.joint[t,i,j]
- }
- }
- }
- distDR[t,,] <- distDR[t,,]/sum(distDR[t,,])
- }
- return( distDR )
-}
-
-shockprob <- function(p, rho, Z, log.p=F){
- ## computes the shocked default probability as a function of the copula factor
- ## function is vectorized provided the below caveats:
- ## p and rho are vectors of same length n, Z is a scalar, returns vector of length n
- ## p and rho are scalars, Z is a vector of length n, returns vector of length n
- if(length(p)==1){
- if(rho==1){
- return(Z<=qnorm(p))
- }else{
- return(pnorm((qnorm(p)-sqrt(rho)*Z)/sqrt(1-rho), log.p=log.p))
- }
- }else{
- result <- double(length(p))
- result[rho==1] <- Z<=qnorm(p[rho==1])
- result[rho<1] <- pnorm((qnorm(p[rho<1])-sqrt(rho[rho<1])*Z)/sqrt(1-rho[rho<1]), log.p=log.p)
- return( result )
- }
-}
-
-shockseverity <- function(S, Stilde=1, Z, rho, p){
- ## computes the severity as a function of the copula factor Z
- result <- double(length(S))
- result[p==0] <- 0
- result[p!=0] <- Stilde * exp( shockprob(S[p!=0]/Stilde*p[p!=0], rho[p!=0], Z, TRUE) -
- shockprob(p[p!=0], rho[p!=0], Z, TRUE))
- return(result)
-}
-
-dshockprob <- function(p,rho,Z){
- dnorm((qnorm(p)-sqrt(rho)*Z)/sqrt(1-rho))*dqnorm(p)/sqrt(1-rho)
-}
-
-dqnorm <- function(x){
- 1/dnorm(qnorm(x))
-}
-
-fit.prob <- function(Z, w, rho, p0){
- ## if the weights are not perfectly gaussian, find the probability p such
- ## E_w(shockprob(p, rho, Z)) = p0
- require(distr)
- if(p0==0){
- return(0)
- }
- if(rho == 1){
- distw <- DiscreteDistribution(Z, w)
- return(pnorm(q(distw)(p0)))
- }
- eps <- 1e-12
- dp <- (crossprod(shockprob(p0,rho,Z),w)-p0)/crossprod(dshockprob(p0,rho,Z),w)
- p <- p0
- while(abs(dp) > eps){
- dp <- (crossprod(shockprob(p,rho,Z),w)-p0)/crossprod(dshockprob(p,rho,Z),w)
- phi <- 1
- while ((p-phi*dp)<0 || (p-phi*dp)>1){
- phi <- 0.8*phi
- }
- p <- p - phi*dp
- }
- return(p)
-}
-
-fit.probC <- function(Z, w, rho, p0){
- checkSymbol("fitprob")
- r <- .C("fitprob", as.double(Z), as.double(w), as.integer(length(Z)),
- as.double(rho), as.double(p0), q = double(1))
- return(r$q)
-}
-
-stochasticrecov <- function(R, Rtilde, Z, w, rho, porig, pmod){
- ## if porig == 0 (probably matured asset) then return orginal recovery
- ## it shouldn't matter anyway since we never default that asset
- if(porig == 0){
- return(rep(R, length(Z)))
- }else{
- ptilde <- fit.prob(Z, w, rho, (1-R)/(1-Rtilde) * porig)
- return(abs(1-(1-Rtilde) * exp(shockprob(ptilde, rho, Z, TRUE) - shockprob(pmod, rho, Z, TRUE))))
- }
-}
-
-stochasticrecovC <- function(R, Rtilde, Z, w, rho, porig, pmod){
- checkSymbol("stochasticrecov")
- r <- .C("stochasticrecov", as.double(R), as.double(Rtilde), as.double(Z),
- as.double(w), as.integer(length(Z)), as.double(rho), as.double(porig),
- as.double(pmod), q = double(length(Z)))
- return(r$q)
-}
-
-pos <- function(x){
- pmax(x, 0)
-}
-
-trancheloss <- function(L, K1, K2){
- pos(L - K1) - pos(L - K2)
-}
-
-trancherecov <- function(R, K1, K2){
- pos(R - 1 + K2) - pos(R - 1 +K1)
-}
-
-tranche.cl <- function(L, R, cs, K1, K2, Ngrid=nrow(L), scaled=FALSE){
- ## computes the couponleg of a tranche
- ## if scaled is TRUE, scale it by the size of the tranche (K2-K1)
- ## can make use of the fact that the loss and recov distribution are
- ## truncated (in that case nrow(L) != Ngrid
- if(K1==K2){
- return( 0 )
- }else{
- support <- seq(0, 1, length=Ngrid)[1:nrow(L)]
- size <- K2 - K1 - crossprod(trancheloss(support, K1, K2), L) -
- crossprod(trancherecov(support, K1, K2), R)
- sizeadj <- as.numeric(0.5 * (size + c(K2-K1, size[-length(size)])))
- if(scaled){
- return( 1/(K2-K1) * crossprod(sizeadj * cs$coupons, cs$df) )
- }else{
- return( crossprod(sizeadj * cs$coupons, cs$df) )
- }
- }
-}
-
-tranche.cl.scenarios <- function(l, r, cs, K1, K2, scaled=FALSE){
- ## computes the couponleg of a tranche for one scenario
- ## if scaled is TRUE, scale it by the size of the tranche (K2-K1)
- ## can make use of the fact that the loss and recov distribution are
- ## truncated (in that case nrow(L) != Ngrid
- if(K1==K2){
- return( 0 )
- }else{
- size <- K2 - K1 - trancheloss(l, K1, K2) - trancherecov(r, K1, K2)
- sizeadj <- as.numeric(0.5 * (size + c(K2-K1, size[-length(size)])))
- if(scaled){
- return( 1/(K2-K1) * crossprod(sizeadj * cs$coupons, cs$df) )
- }else{
- return( crossprod(sizeadj * cs$coupons, cs$df) )
- }
- }
-}
-
-funded.tranche.pv <- function(L, R, cs, K1, K2, scaled = FALSE){
- if(K1==K2){
- return(0)
- }else{
- size <- K2 - K1 -trancheloss(L, K1, K2) - trancherecov(R, K1, K2)
- sizeadj <- as.numeric(0.5 * (size + c(K2-K1, size[-length(size)])))
- interest <- crossprod(sizeadj * cs$coupons, cs$df)
- principal <- diff(c(0, trancherecov(R, K1, K2)))
- principal[length(principal)] <- principal[length(principal)] + size[length(size)]
- principal <- crossprod(cs$df, principal)
- if(scaled){
- pv <- (interest + principal)/(K2-K1)
- }else{
- pv <- (interest + principal)
- }
- return(pv)
- }
-}
-
-tranche.pl <- function(L, cs, K1, K2, Ngrid=nrow(L), scaled=FALSE){
- ## computes the protection leg of a tranche
- ## if scaled
- if(K1==K2){
- return(0)
- }else{
- support <- seq(0, 1, length=Ngrid)[1:nrow(L)]
- cf <- K2 - K1 - crossprod(trancheloss(support, K1, K2), L)
- cf <- c(K2 - K1, cf)
- if(scaled){
- return( 1/(K2-K1) * crossprod(diff(cf), cs$df))
- }else{
- return( crossprod(diff(cf), cs$df))
- }
- }
-}
-
-tranche.pl.scenarios <- function(l, cs, K1, K2, scaled=FALSE){
- ## computes the protection leg of a tranche
- ## if scaled
- if(K1==K2){
- return(0)
- }else{
- cf <- K2 - K1 - trancheloss(l, K1, K2)
- cf <- c(K2 - K1, cf)
- if(scaled){
- return( 1/(K2-K1) * as.numeric(crossprod(diff(cf), cs$df)))
- }else{
- return( as.numeric(crossprod(diff(cf), cs$df)))
- }
- }
-}
-
-tranche.pv <- function(L, R, cs, K1, K2, Ngrid=nrow(L)){
- return( tranche.pl(L, cs, K1, K2, Ngrid) + tranche.cl(L, R, cs, K1, K2, Ngrid))
-}
-
-tranche.pv.scenarios <- function(l, r, cs, K1, K2){
- return( tranche.pl.scenarios(l, cs, K1, K2, TRUE) +
- tranche.cl.scenarios(l, r, cs, K1, K2, TRUE))
-}
-
-
-adjust.attachments <- function(K, losstodate, factor){
- ## computes the attachments adjusted for losses
- ## on current notional
- return( pmin(pmax((K-losstodate)/factor, 0),1) )
-}
-
-tranche.pvvec <- function(K, L, R, cs){
- r <- rep(0, length(K)-1)
- for(i in 1:(length(K)-1)){
- r[i] <- 1/(K[i+1]-K[i]) * tranche.pv(L, R, cs, K[i], K[i+1])
- }
- return( r )
-}
-
-BClossdist <- function(defaultprob, issuerweights, recov, rho, Z, w,
- N=length(recov)+1, defaultflag=FALSE, n.int=500){
- ## do not use if weights are not gaussian, results would be incorrect
- ## since shockseverity is invalid in that case (need to use stochasticrecov)
- LZ <- matrix(0, N, length(Z))
- RZ <- matrix(0, N, length(Z))
- L <- matrix(0, N, ncol(defaultprob))
- R <- matrix(0, N, ncol(defaultprob))
- for(t in 1:ncol(defaultprob)){
- for(i in 1:length(Z)){
- g.shocked <- shockprob(defaultprob[,t], rho, Z[i])
- S.shocked <- shockseverity(1-recov, 1, Z[i], rho, defaultprob[,t])
- temp <- lossrecovdist(g.shocked, , issuerweights, S.shocked, N)
- LZ[,i] <- temp$L
- RZ[,i] <- temp$R
- }
- L[,t] <- LZ%*%w
- R[,t] <- RZ%*%w
- }
- list(L=L, R=R)
-}
-
-BClossdistC <- function(defaultprob, issuerweights, recov, rho, Z, w,
- N=length(issuerweights)+1, defaultflag=FALSE){
- checkSymbol("BClossdist")
- L <- matrix(0, N, dim(defaultprob)[2])
- R <- matrix(0, N, dim(defaultprob)[2])
- rho <- rep(rho, length(issuerweights))
- r <- .C("BClossdist", defaultprob, dim(defaultprob)[1], dim(defaultprob)[2],
- as.double(issuerweights), as.double(recov), as.double(Z), as.double(w),
- as.integer(length(Z)), as.double(rho), as.integer(N), as.logical(defaultflag), L=L, R=R)
- return(list(L=r$L,R=r$R))
-}
-
-BCtranche.pv <- function(defaultprob, issuerweights, recov, cs, K1, K2, rho1, rho2,
- Z, w, N=length(issuerweights)+1){
- ## computes the protection leg, couponleg, and bond price of a tranche
- ## in the base correlation setting
- if(K1==0){
- if(rho1!=0){
- stop("equity tranche must have 0 lower correlation")
- }
- }
- dK <- K2 - K1
- dist2 <- BClossdistC(defaultprob, issuerweights, recov, rho2, Z, w, N)
- if(rho1!=0){
- dist1 <- BClossdistC(defaultprob, issuerweights, recov, rho1, Z, w, N)
- }
- cl2 <- tranche.cl(dist2$L, dist2$R, cs, 0, K2)
- cl1 <- tranche.cl(dist1$L, dist1$R, cs, 0, K1)
- pl2 <- tranche.pl(dist2$L, cs, 0, K2)
- pl1 <- tranche.pl(dist1$L, cs, 0, K1)
- return(list(pl=(pl2-pl1)/dK, cl=(cl2-cl1)/dK,
- bp=100*(1+(pl2-pl1+cl2-cl1)/dK)))
-}
-
-BCtranche.delta <- function(portfolio, index, coupon, K1, K2, rho1, rho2, Z, w,
- N=length(portolio)+1, tradedate = Sys.Date()){
- ## computes the tranche delta (on current notional) by doing a proportional
- ## blip of all the curves
- ## if K2==1, then computes the delta using the lower attachment only
- ## this makes sense for bottom-up skews
- eps <- 1e-4
- portfolioplus <- portfolio
- portfoliominus <- portfolio
- cs <- couponSchedule(IMMDate(tradedate), index$maturity,"Q", "FIXED", coupon, 0, tradedate,
- IMMDate(tradedate, "prev"))
- for(i in 1:length(portfolio)){
- portfolioplus[[i]]@curve@hazardrates <- portfolioplus[[i]]@curve@hazardrates * (1 + eps)
- portfoliominus[[i]]@curve@hazardrates <- portfoliominus[[i]]@curve@hazardrates * (1- eps)
- }
- dPVindex <- indexpv(portfolioplus, index, tradedate = tradedate, clean = FALSE)$bp -
- indexpv(portfoliominus, index, tradedate = tradedate, clean = FALSE)$bp
- defaultprobplus <- 1 - SPmatrix(portfolioplus, length(cs$dates))
- defaultprobminus <- 1 - SPmatrix(portfoliominus, length(cs$dates))
- if(K2==1){
- K1adj <- adjust.attachments(K1, index$loss, index$factor)
- dPVtranche <- BCtranche.pv(defaultprobplus, issuerweights, recov, cs, 0, K1adj, 0, rho1,
- Z, w, N)$bp -
- BCtranche.pv(defaultprobminus, issuerweights, recov, cs, 0, K1adj, 0, rho1,
- Z, w, N)$bp
- delta <- (1 - dPVtranche/(100*dPVindex) * K1adj)/(1-K1adj)
- }else{
- Kmod <- adjust.attachments(c(K1, K2), index$loss, index$factor)
- dPVtranche <- BCtranche.pv(defaultprobplus, issuerweights, recov, cs, Kmod[1], Kmod[2], rho1, rho2,
- Z, w, N)$bp -
- BCtranche.pv(defaultprobminus, issuerweights, recov, cs, Kmod[1], Kmod[2], rho1, rho2,
- Z, w, N)$bp
- delta <- dPVtranche/(100*dPVindex)
- }
- ## dPVindex <- BCtranche.pv(portfolioplus, index, coupon, 0, 1, 0, 0.5, lu)$bp-
- ## BCtranche.pv(portfoliominus, index, coupon, 0, 1, 0, 0.5, lu)$bp
- return( delta )
-}
-
-BCstrikes <- function(portfolio, index, coupon, K, rho, N=101) {
- ## computes the strikes as a percentage of expected loss
- EL <- c()
- for(i in 2:length(K)){
- EL <- c(EL, -BCtranche.pv(portfolio, index, coupon, K[i-1], K[i], rho[i-1], rho[i], N)$pl)
- }
- Kmodified <- adjust.attachments(K, index$loss, index$factor)
- return(cumsum(EL*diff(Kmodified))/sum(EL*diff(Kmodified)))
-}
-
-delta.factor <- function(K1, K2, index){
- ## compute the factor to convert from delta on current notional to delta on original notional
- ## K1 and K2 original strikes
- factor <- (adjust.attachments(K2, index$loss, index$factor)
- -adjust.attachments(K1, index$loss, index$factor))/(K2-K1)
- return( factor )
-}
-
-MFupdate.prob <- function(Z, w, rho, defaultprob){
- ## update the probabilites based on a non gaussian factor
- ## distribution so that the pv of the cds stays the same.
- p <- matrix(0, nrow(defaultprob), ncol(defaultprob))
- for(i in 1:nrow(defaultprob)){
- for(j in 1:ncol(defaultprob)){
- p[i,j] <- fit.prob(Z, w, rho[i], defaultprob[i,j])
- }
- }
- return( p )
-}
-
-MFupdate.probC <- function(Z, w, rho, defaultprob){
- ## update the probabilities based on a non gaussian factor
- ## distribution so that the pv of the cds stays the same.
- p <- matrix(0, nrow(defaultprob), ncol(defaultprob))
- checkSymbol("fitprob")
- for(i in 1:nrow(defaultprob)){
- for(j in 1:ncol(defaultprob)){
- p[i,j] <- .C("fitprob", as.double(Z), as.double(w), as.integer(length(Z)),
- as.double(rho[i]), as.double(defaultprob[i,j]), q = double(1))$q
- }
- }
- return( p )
-}
-
-MFlossrecovdist.prepay <- function(w, Z, rho, defaultprob, defaultprobmod, prepayprob, prepayprobmod,
- issuerweights, recov, Ngrid=2*length(issuerweights)+1, defaultflag=FALSE){
- ## computes the loss and recovery distribution using the modified factor distribution
- n.credit <- length(issuerweights)
- n.int <- length(w)
- Rstoch <- array(0, dim=c(n.int, n.credit, ncol(defaultprob)))
- for(t in 1:ncol(defaultprob)){
- for(i in 1:n.credit){
- Rstoch[,i,t] <- stochasticrecov(recov[i], 0, Z, w, rho, defaultprob[i,t], defaultprobmod[i,t])
- }
- }
- parf <- function(i){
- dpshocked <- apply(defaultprobmod, 2, shockprob, rho=rho, Z=Z[i])
- ppshocked <- apply(prepayprobmod, 2, shockprob, rho=rho, Z=-Z[i])
- S <- 1 - Rstoch[i,,]
- dist <- lossrecovdist.term(dpshocked, ppshocked, issuerweights, S, Ngrid, defaultflag)
- }
- L <- matrix(0, Ngrid, ncol(defaultprob))
- R <- matrix(0, Ngrid, ncol(defaultprob))
- for(i in 1:length(w)){
- dist <- parf(i)
- L <- L + dist$L * w[i]
- R <- R + dist$R * w[i]
- }
- return( list(L=L, R=R) )
-}
-
-MFlossdist.joint <- function(cl, w, Z, rho, defaultprob, defaultprobmod, issuerweights, recov,
- Ngrid=2*length(issuerweights)+1, defaultflag=FALSE){
- ## rowSums(Q) is the default/loss distribution depending if
- ## defaultflag is TRUE or FALSE (default setting is FALSE)
- ## colSums(Q) is the recovery distribution
- ## so that recovery is the y axis and L/D is the x axis
- ## if we use the persp function, losses is the axes facing us,
- ## and R is the axis going away from us.
- n.credit <- length(issuerweights)
- n.int <- lenth(w)
- Rstoch <- array(0, dim=c(n.int, n.credit, ncol(defaultprob)))
- for(t in 1:ncol(defaultprob)){
- for(i in 1:n.credit){
- Rstoch[,i,t] <- stochasticrecov(recov[i], 0, Z, w, rho, defaultprob[i,t], defaultprobmod[i,t])
- }
- }
- parf <- function(i){
- pshocked <- apply(defaultprobmod, 2, shockprob, rho=rho, Z=Z[i])
- S <- 1 - Rstoch[i,,]
- dist <- lossrecovdist.joint.term(pshocked, 0, issuerweights, S, Ngrid, defaultflag)
- gc()
- return(dist)
- }
- temp <- parSapply(cl, 1:length(w), parf)
- clusterCall(cl, gc)
- Q <- array(0, dim=c(ncol(defaultprob), Ngrid, Ngrid))
- for(i in 1:length(w)){
- Q <- Q + w[i]*array(temp[,i], dim=c(ncol(defaultprob), Ngrid, Ngrid))
- }
- return( Q )
-}
-
-MFlossdist.prepay.joint <- function(w, Z, rho, defaultprob, defaultprobmod,
- prepayprob, prepayprobmod, issuerweights, recov,
- Ngrid=2*length(issuerweights)+1, defaultflag=FALSE){
- ## rowSums is the loss distribution
- ## colSums is the recovery distribution
- ## so that recovery is the y axis and L is the x axis
- ## if we use the persp function, losses is the axes facing us,
- ## and R is the axis going away from us.
- n.credit <- length(issuerweights)
- n.int <- length(w)
- Rstoch <- array(0, dim=c(n.credit, n.int, ncol(defaultprob)))
-
- for(t in 1:ncol(defaultprob)){
- for(i in 1:n.credit){
- Rstoch[i,,t] <- stochasticrecovC(recov[i], 0, Z, w, rho[i],
- defaultprob[i,t], defaultprobmod[i,t])
- }
- }
-
- Q <- array(0, dim=c(ncol(defaultprob), Ngrid, Ngrid))
- for(t in 1:ncol(defaultprob)){
- S <- 1 - Rstoch[,,t]
- Q[t,,] <- lossdistC.prepay.jointZ(defaultprobmod[,t], prepayprobmod[,t], issuerweights,
- S, Ngrid, defaultflag, rho, Z, w)
- }
- return( Q )
-}
-
-MFtranche.pv <- function(cl, cs, w, rho, defaultprob, defaultprobmod, issuerweights, recov,
- Kmodified, Ngrid=length(issuerweights)+1){
- ## computes the tranches pv using the modified factor distribution
- ## p is the modified probability so that
- n.credit <- length(issuerweights)
- Rstoch <- array(0, dim=c(n.int, n.credit, ncol(defaultprob)))
- for(t in 1:ncol(defaultprob)){
- for(i in 1:n.credit){
- Rstoch[,i,t] <- stochasticrecov(recov[i], 0, Z, w, rho, defaultprob[i,t], defaultprobmod[i,t])
- }
- }
- parf <- function(i){
- pshocked <- apply(defaultprobmod, 2, shockprob, rho=rho, Z=Z[i])
- S <- 1 - Rstoch[i,,]
- dist <- lossrecovdist.term(pshocked, 0, issuerweights, S, Ngrid)
- return( tranche.pvvec(Kmodified, dist$L, dist$R, cs))
- }
- clusterExport(cl, list("Rstoch", "p"), envir=environment())
- result <- parSapply(cl, 1:length(w), parf)
- return( list(pv=100*(1+result%*%w), pv.w=result))
-}
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