sandbox/FFEV/HermiteGrid_CVaR_Recursion.R

In braverock/PortfolioAnalytics: Portfolio Analysis, Including Numerical Methods for Optimization of Portfolios

# This script illustrates the discrete Newton recursion  to process views on CVaR according to Entropy Pooling
# This script complements the article
#	"Fully Flexible Extreme Views"
#	by A. Meucci, D. Ardia, S. Keel
#	available at www.ssrn.com
# The most recent version of this code is available at
# MATLAB Central - File Exchange

# Prior market model (normal) on grid
emptyMatrix = matrix( nrow=0 , ncol=0 )
market.mu   = 0.0
market.sig2 = 1.0
market.pdf = function(x) dnorm( x , mean = market.mu , sd = sqrt(market.sig2) )
market.cdf = function(x) pnorm( x , mean = market.mu , sd = sqrt(market.sig2) )
market.rnd = function(x) rnorm( x , mean = market.mu , sd = sqrt(market.sig2) ) 
market.inv = function(x) qnorm( x , mean = market.mu , sd = sqrt(market.sig2) )
market.VaR95 = market.inv(0.05)
market.CVaR95 = integrate( function( x ) ( x * market.pdf( x ) ), -100, market.VaR95)$val / 0.05

tmp = ( ghqx - min( ghqx ) )/( max( ghqx ) - min( ghqx ) ) # rescale GH zeros so they belong to [0,1]
epsilon = 1e-10
Lower = market.inv( epsilon )
Upper = market.inv( 1 - epsilon )
X = Lower + tmp * ( Upper - Lower ) # rescale mesh

p = integrateSubIntervals( X, market.cdf )
p = normalizeProb( p )
J = nrow( X )

# Entropy posterior from extreme view on CVaR: brute-force approach
  
# view of the analyst
view.CVaR95 = -3.0

# Iterate over different VaR95 levels
nVaR95 = 100
VaR95  = seq(view.CVaR95, market.VaR95, len=nVaR95)
p_     = matrix(NaN, nrow = J, ncol = nVaR95 )
s_     = matrix(NaN, nrow = nVaR95, ncol = 1 )
KLdiv  = matrix(NaN, nrow = nVaR95, ncol = 1)

for ( i in 1:nVaR95 ) {
  idx = as.matrix( X <= VaR95[i] )
  s_[i] = sum(idx)
  posteriorEntropy = EntropyProg(p, t( idx ),  as.matrix( 0.05 ), rbind( rep(1, J), t( idx * X ) ), rbind( 1, 0.05 * view.CVaR95 ) )
  p_[,i] = posteriorEntropy$p_
  KLdiv[i] = posteriorEntropy$optimizationPerformance$ml
}
                                 
# Display results
plot( s_, KLdiv )
dummy = min( KLdiv )
idxMin = which.min( KLdiv ) 
plot( s_[idxMin], KLdiv[idxMin] )

tmp = p_[, idxMin]
tmp = tmp / sum( tmp )
plot( X, tmp )
x = seq(min(X), max(X), len = J);
tmp = market.pdf(x)
tmp = tmp / sum(tmp)
plot(x, tmp)
plot(market.CVaR95, 0)
plot(view.CVaR95, 0)

# Entropy posterior from extreme view on CVaR: Newton Raphson approach

s = emptyMatrix

# initial value
idx = as.matrix( cumsum(p) <= 0.05 )
s[1] = sum(idx)
posteriorEntropy = EntropyProg(p, t( idx ), as.matrix( 0.05 ), rbind( rep(1, J), t( idx * X ) ), rbind( 1, 0.05 * view.CVaR95) )                                 
KLdiv = as.matrix( posteriorEntropy$optimizationPerformance$ml )
p_ = posteriorEntropy$p_
                                                                               
# iterate
doStop = 0
i = 1
while ( !doStop ) {
  i = i + 1
  
  idx = cbind( matrix(1, 1, s[i - 1] ), matrix(0, 1, J - s[i-1] ) )
  posteriorEntropy1 = EntropyProg(p, idx, as.matrix( 0.05 ), rbind( matrix(1, 1, J), t( t(idx) * X) ), rbind( 1, 0.05 * view.CVaR95 ) )
  # [dummy, KLdiv_s]  = optimizeEntropy(p, [idx'; (idx .* X)'], [0.05; 0.05 * view.CVaR95], [ones(1, J); X'], [1; view.mu]);
  
  idx = cbind( matrix(1, 1, s[i - 1] + 1 ), matrix(0, 1, J - s[i - 1] - 1) )
  posteriorEntropy2 = EntropyProg(p, idx, as.matrix( 0.05 ), rbind( matrix(1, 1, J), t( t(idx) * X) ), rbind( 1, 0.05 * view.CVaR95 ) )
  # [dummy, KLdiv_s1] = optimizeEntropy(p, [idx'; (idx .* X)'], [0.05; 0.05 * view.CVaR95], [ones(1, J); X'], [1; view.mu]);
  
  idx = cbind( matrix(1, 1, s[i - 1] + 2 ), matrix(0, 1, J - s[i - 1] - 2) )
  posteriorEntropy3 = EntropyProg(p, idx, as.matrix( 0.05 ), rbind( matrix(1, 1, J), t( t(idx) * X) ), rbind( 1, 0.05 * view.CVaR95 ) )
  # [dummy, KLdiv_s2] = optimizeEntropy(p, [idx'; (idx .* X)'], [0.05; 0.05 * view.CVaR95], [ones(1, J); X'], [1; view.mu]);
  
  # first difference
  DE  = posteriorEntropy2$optimizationPerformance$ml - posteriorEntropy1$optimizationPerformance$ml
  
  # second difference
  D2E = posteriorEntropy3$optimizationPerformance$ml - 2 * posteriorEntropy2$optimizationPerformance$ml + posteriorEntropy1$optimizationPerformance$ml
  
  # optimal s
  s = rbind( s, round( s[i - 1] - (DE / D2E) ) ) 
  
  tmp = emptyMatrix
  idx  = cbind( matrix( 1, 1, s[i] ), matrix( 0, 1, J - s[i] ) )
  tempEntropy = EntropyProg(p, idx, as.matrix( 0.05 ), rbind( matrix(1, 1, J), t( t(idx) * X) ), rbind( 1, 0.05 * view.CVaR95 ) )
  # [tmp.p_, tmp.KLdiv] = optimizeEntropy(p, [idx'; (idx .* X)'], [0.05; 0.05 * view.CVaR95], [ones(1, J); X'], [1; view.mu]);
  p_ = cbind( p_, tempEntropy$p_ )
  KLdiv = rbind( KLdiv, tempEntropy$optimizationPerformance$ml ) #ok<*AGROW>
  
  # if change in KLdiv less than one percent, stop
  if( abs( ( KLdiv[i] - KLdiv[i - 1] ) / KLdiv[i - 1] )  < 0.01 ) { doStop = 1 }
}

# Display results

N = length(s)

plot(1:N, KLdiv)
x = seq(min(X), max(X), len = J)
tmp = market.pdf(x)
tmp = tmp / sum(tmp)
plot( t( X ), tmp )
plot( t( X ), p_[, ncol(p_)] )
plot( market.CVaR95, 0.0 )
plot( view.CVaR95, 0.0 )

# zoom here
plot( t( X ), tmp )
plot( t( X ), p_[, 1] )
plot( t( X ), p_[, 2] )
plot( t( X ), p_[, ncol(p_)] )
plot( market.CVaR95, 0 )
plot( view.CVaR95, 0 )