[Returnanalytics-commits] r2151 - pkg/PerformanceAnalytics/sandbox/Meucci/demo
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Fri Jul 13 11:28:02 CEST 2012
Author: mkshah
Date: 2012-07-13 11:28:02 +0200 (Fri, 13 Jul 2012)
New Revision: 2151
Modified:
pkg/PerformanceAnalytics/sandbox/Meucci/demo/ButterflyTrading.R
Log:
Including ButterflyTrading functions in the demo file since it is a case study and doesn't include any generalized method/process
Modified: pkg/PerformanceAnalytics/sandbox/Meucci/demo/ButterflyTrading.R
===================================================================
--- pkg/PerformanceAnalytics/sandbox/Meucci/demo/ButterflyTrading.R 2012-07-13 07:34:20 UTC (rev 2150)
+++ pkg/PerformanceAnalytics/sandbox/Meucci/demo/ButterflyTrading.R 2012-07-13 09:28:02 UTC (rev 2151)
@@ -1,6 +1,3 @@
-# TODO: plot efficient frontier in R
-
-
#' This script performs the butterfly-trading case study for the
#' Entropy-Pooling approach by Attilio Meucci, as it appears in
#' "A. Meucci - Fully Flexible Views: Theory and Practice -
@@ -9,7 +6,295 @@
#' Adapted from Code by A. Meucci, September 2008
#' Last version available at www.symmys.com > Teaching > MATLAB
+PlotFrontier = function( e , s , w )
+{
+ # subplot(2,1,1)
+ plot( s , e )
+ # grid on
+ # set(gca,'xlim',[min(s) max(s)])
+ #
+ # subplot(2,1,2)
+
+ xx = nrow( w ) ; N = ncol( w )
+ Data = apply( w , 1 , cumsum ) #TODO: Check. Take cumulative sum of *rows*. Try sapply?
+
+ for ( n in 1:N )
+ {
+ x = cbind( min(s) , s , max(s) )
+ y = cbind( 0 , Data[ , N-n+1 ] , 0 )
+ # hold on
+ #h = fill( x , y , cbind( .9 , .9 , .9) - mod( n , 3 ) %*% cbind( .2 , .2 , .2) )
+ }
+
+ #set(gca,'xlim',[min(s) max(s)],'ylim',[0 max(max(Data))])
+ #xlabel('portfolio # (risk propensity)')
+ #ylabel('portfolio composition')
+}
+ViewCurveSlope = function( X , p )
+{
+ # view 3 (expectations and binding constraints): slope of the yield curve will increase by 5 bp
+
+ J = nrow( X ) ; K = ncol( X )
+
+ # constrain probabilities to sum to one...
+ Aeq = ones( 1 , J )
+ beq = 1
+
+ # ...constrain the expectation...
+ V = X[ , 14 ] - X[ , 13 ]
+ v = .0005
+
+ Aeq = rbind( Aeq , t(V) )
+
+ beq = rbind( beq , v )
+
+ A = b = emptyMatrix
+
+ # ...compute posterior probabilities
+ p_ = EntropyProg( p , A , b , Aeq ,beq )$p_
+ return( p_ )
+}
+
+ViewRealizedVol = function( X , p )
+{
+ # view 2 (relative inequality view on median): bullish on realized volatility of MSFT (i.e. absolute log-change in the underlying).
+ # This is the variable such that, if larger than a threshold, a long position in the butterfly turns into a profit (e.g. Rachev 2003)
+ # we issue a relative statement on the media comparing it with the third quintile implied by the reference market model
+
+ library( matlab )
+ J = nrow( X ) ; K = ncol( X )
+
+ # constrain probabilities to sum to one...
+ Aeq = ones( 1 , J )
+ beq = 1
+
+ # ...constrain the median...
+ V = abs( X[ , 1 ] ) # absolute value of the log of changes in MSFT close prices (definition of realized volatility)
+
+ V_Sort = sort( V , decreasing = FALSE ) # sorting of the abs value of log changes in prices from smallest to largest
+ I_Sort = order( V )
+
+ F = cumsum( p[ I_Sort ] ) # represents the cumulative sum of probabilities from ~0 to 1
+
+ I_Reference = max( matlab:::find( F <= 3/5 ) ) # finds the (max) index corresponding to element with value <= 3/5 along the empirical cumulative density function for the abs log-changes in price
+ V_Reference = V_Sort[ I_Reference ] # returns the corresponding abs log of change in price at the 3/5 of the cumulative density function
+
+ I_Select = find( V <= V_Reference ) # finds all indices with value of abs log-change in price less than the reference value
+
+ a = zeros( 1 , J )
+ a[ I_Select ] = 1 # select those cases where the abs log-change in price is less than the 3/5 of the empirical cumulative density...
+
+ A = a
+ b = .5 # ... and assign the probability of these cases occuring as 50%. This moves the media of the distribution
+
+ # ...compute posterior probabilities
+ p_ = EntropyProg( p , A , b , Aeq , beq )$p_
+
+ return( p_ )
+}
+
+ViewImpliedVol = function( X , p )
+{
+ # View 1 (inequality view): bearish on on 2m-6m implied volaility spread for Google
+
+ J = nrow( X ) ; K = ncol( X )
+
+ # constrain probabilities to sum to one...
+ Aeq = ones( 1 , J )
+ beq = 1
+
+ # ...constrain the expectation...
+ V = X[ , 12 ] - X[ , 11 ] # GOOG_vol_182 (6m implied vol) - GOOG_vol_91 (2m implied vol)
+ m = mean( V )
+ s = std( V )
+
+ A = t( V )
+ b = m - s
+
+ # ...compute posterior probabilities
+ p_ = EntropyProg( p , A , b , Aeq , beq )$p_
+
+ return( p_ )
+}
+
+ComputeCVaR = function( Units , Scenarios , Conf )
+{
+ PnL = Scenarios %*% Units
+ Sort_PnL = PnL[ order( PnL , decreasing = FALSE ) ]
+
+ J = length( PnL )
+ Cut = round( J %*% ( 1 - Conf ) , 0 )
+
+ CVaR = -mean( Sort_PnL[ 1:Cut ] )
+
+ return( CVaR )
+}
+
+LongShortMeanCVaRFrontier = function( PnL , Probs , Butterflies , Options )
+{
+ library( matlab )
+ library( quadprog )
+ library( limSolve )
+
+ # setup constraints
+ J = nrow(PnL); N = ncol(PnL)
+ P_0s = matrix( , nrow = 1 , ncol = 0 )
+ D_s = matrix( , nrow = 1 , ncol = 0 )
+ emptyMatrix = matrix( nrow = 0 , ncol = 0 )
+
+ for ( n in 1:N )
+ {
+ P_0s = cbind( P_0s , Butterflies[[n]]$P_0 ) # 1x9 matrix
+ D_s = cbind( D_s , Butterflies[[n]]$Delta ) # 1x9 matrix
+ }
+
+ Constr = list()
+ Constr$Aeq = P_0s # linear coefficients in the constraints Aeq*X = beq (equality constraints)
+ Constr$beq = Options$Budget # the constant vector in the constraints Aeq*x = beq
+
+ if ( Options$DeltaNeutral == TRUE )
+ {
+ Constr$Aeq = rbind( Constr$Aeq , D_s ) # 2x9 matrix
+ Constr$beq = rbind( Constr$beq , 0 ) # 2x9 matrix
+ }
+
+ Constr$Aleq = rbind( diag( as.vector( P_0s ) ) , -diag( as.vector( P_0s ) ) ) # linear coefficients in the constraints A*x <= b. an 18x9 matrix
+ Constr$bleq = rbind( Options$Limit * ones(N,1) , Options$Limit * ones(N,1) ) # constant vector in the constraints A*x <= b. an 18x1 matrix
+
+ # determine expectation of minimum-variance portfolio
+ Exps = t(PnL) %*% Probs
+ Scnd_Mom = t(PnL) %*% (PnL * (Probs %*% ones(1,N) ) )
+ Scnd_Mom = ( Scnd_Mom + t(Scnd_Mom) ) / 2
+ Covs = Scnd_Mom - Exps %*% t(Exps)
+
+ Amat = rbind( Constr$Aeq , Constr$Aleq ) # stack the equality constraints on top of the inequality constraints
+ bvec = rbind( Constr$beq , Constr$bleq ) # stack the equality constraints on top of the inequality constraints
+
+ #if ( nrow(Covs) != length( zeros(N,1) ) ) stop("Dmat and dvec are incompatible!")
+ #if ( nrow(Covs) != nrow(Amat)) stop("Amat and dvec are incompatible!")
+
+ MinSDev_Units = solve.QP( Dmat = Covs , dvec = -1 * zeros(N,1) , Amat = -1*t(Amat) , bvec = -1*bvec , meq = length( Constr$beq) ) # TODO: Check this
+ MinSDev_Exp = t( MinSDev_Units$solution ) %*% Exps
+
+ # determine expectation of maximum-expectation portfolio
+
+ MaxExp_Units = linp( E = Constr$Aeq , F = Constr$beq , G = -1*Constr$Aleq , H = -1*Constr$bleq , Cost = -Exps , ispos = FALSE )$X
+
+ MaxExp_Exp = t( MaxExp_Units ) %*% Exps
+
+ # slice efficient frontier in NumPortf equally thick horizontal sections
+ Grid = t( seq( from = Options$FrontierSpan[1] , to = Options$FrontierSpan[2] , length.out = Options$NumPortf ) )
+ TargetExp = as.numeric( MinSDev_Exp ) + Grid * as.numeric( ( MaxExp_Exp - MinSDev_Exp ) )
+
+ # compute composition, expectation, s.dev. and CVaR of the efficient frontier
+ Composition = matrix( , ncol = N , nrow = 0 )
+ Exp = matrix( , ncol = 1 , nrow = 0 )
+ SDev = matrix( , ncol = 1 , nrow = 0 )
+ CVaR = matrix( , ncol = 1 , nrow = 0 )
+
+ for (i in 1:Options$NumPortf )
+ {
+ # determine least risky portfolio for given expectation
+ AEq = rbind( Constr$Aeq , t(Exps) ) # equality constraint: set expected return for each asset...
+ bEq = rbind( Constr$beq , TargetExp[i] )
+
+ Amat = rbind( AEq , Constr$Aleq ) # stack the equality constraints on top of the inequality constraints
+ bvec = rbind( bEq , Constr$bleq ) # ...and target portfolio return for i'th efficient portfolio
+
+ # Why is FirstDegree "expected returns" set to 0?
+ # Becasuse we capture the equality view in the equality constraints matrix
+ # In other words, we have a constraint that the Expected Returns by Asset %*% Weights = Target Return
+ Units = solve.QP( Dmat = Covs , dvec = -1*zeros(N,1) , Amat = -1*t(Amat) , bvec = -1*bvec , meq = length( bEq ) )
+
+ # store results
+ Composition = rbind( Composition , t( Units$solution ) )
+
+ Exp = rbind( Exp , t( Units$solution ) %*% Exps )
+ SDev = rbind( SDev , sqrt( t( Units$solution ) %*% Covs %*% Units$solution ) )
+ CVaR = rbind( CVaR , ComputeCVaR( Units$solution , PnL , Options$Quant ) )
+ }
+
+ colnames( Composition ) = c( "MSFT_vol_30" , "MSFT_vol_91" , "MSFT_vol_182" ,
+ "YHOO_vol_30" , "YHOO_vol_91" , "YHOO_vol_182" ,
+ "GOOG_vol_30" , "GOOG_vol_91" , "GOOG_vol_182" )
+
+ return( list( Exp = Exp , SDev = SDev , CVaR = CVaR , Composition = Composition ) )
+}
+
+
+MapVol = function( sig , y , K , T )
+{
+ # in real life a and b below should be calibrated to security-specific time series
+
+ a=-.00000000001
+ b= .00000000001
+
+ s = sig + a/sqrt(T) * ( log(K) - log(y) ) + b/T*( log(K) - log(y) )^2
+
+ return( s )
+}
+
+HorizonPricing = function( Butterflies , X )
+{
+ r = .04 # risk-free rate
+ tau = 1/252 # investment horizon
+
+ # factors: 1. 'MSFT_close' 2. 'MSFT_vol_30' 3. 'MSFT_vol_91' 4. 'MSFT_vol_182'
+ # securities: 1. 'MSFT_vol_30' 2. 'MSFT_vol_91' 3. 'MSFT_vol_182'
+
+ # create a new row called DlnY and Dsig
+ # create a new row called 'DlnY'. Assign the first row (vector) of X to this DlnY for the 1:3 securities
+ for ( s in 1:3 ) { Butterflies[[s]]$DlnY = X[ , 1 ] }
+
+ # assign the 2nd row of X to a new element called Dsig
+ Butterflies[[1]]$Dsig=X[ , 2 ]
+ Butterflies[[2]]$Dsig=X[ , 3 ]
+ Butterflies[[3]]$Dsig=X[ , 4 ]
+
+ # factors: 5. 'YHOO_close' 6. 'YHOO_vol_30' 7. 'YHOO_vol_91' 8. 'YHOO_vol_182'
+ # securities: 4. 'YHOO_vol_30' 5. 'YHOO_vol_91' 6. 'YHOO_vol_182'
+ for ( s in 4:6 ) { Butterflies[[s]]$DlnY=X[ , 5 ] }
+
+ Butterflies[[4]]$Dsig=X[ , 6 ]
+ Butterflies[[5]]$Dsig=X[ , 7 ]
+ Butterflies[[6]]$Dsig=X[ , 8 ]
+
+ # factors: # 9. 'GOOG_close' 10. 'GOOG_vol_30' 11. 'GOOG_vol_91' 12. 'GOOG_vol_182'
+ # securities: 7. 'GOOG_vol_30' 8. 'GOOG_vol_91' 9. 'GOOG_vol_182'
+ for ( s in 7:9 ) { Butterflies[[s]]$DlnY=X[ , 9 ] }
+
+ Butterflies[[7]]$Dsig=X[ , 10 ]
+ Butterflies[[8]]$Dsig=X[ , 11 ]
+ Butterflies[[9]]$Dsig=X[ , 12 ]
+
+ PnL = matrix( NA , nrow = nrow(X) )
+
+ for ( s in 1:length(Butterflies) )
+ {
+ Y = Butterflies[[s]]$Y_0 * exp(Butterflies[[s]]$DlnY)
+ ATMsig = apply( cbind( Butterflies[[s]]$sig_0 + Butterflies[[s]]$Dsig , 10^-6 ) , 1 , max )
+ t = Butterflies[[s]]$T - tau
+ K = Butterflies[[s]]$K
+ sig = MapVol(ATMsig , Y , K , t )
+
+ # library(RQuantLib) # this function can only operate on one option at a time, so we use fOptions
+ # C = EuropeanOption( type = "call" , underlying = Y , strike = K , dividendYield = 0 , riskFreeRate = r , maturity = t , volatility = sig )$value
+ # P = EuropeanOption( type = "put" , underlying = Y , strike = K , dividendYield = 0 , riskFreeRate = r , maturity = t , volatility = sig )$value
+
+ # use fOptions to value options
+ library( fOptions )
+ C = GBSOption( TypeFlag = "c" , S = Y , X = K , r = r , b = 0 , Time = t , sigma = sig )
+ P = GBSOption( TypeFlag = "p" , S = Y , X = K , r = r , b = 0 , Time = t , sigma = sig )
+
+ Butterflies[[s]]$P_T = C at price + P at price
+ PnL = cbind( PnL , Butterflies[[s]]$P_T )
+ }
+ PnL = PnL[ , -1 ]
+
+ return( PnL )
+}
+
###################################################################
#' Load panel X of joint factors realizations and vector p of respective probabilities
#' In real life, these are provided by the estimation process
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