R/ButterflyTradingFunctions.R
In R-Finance/Meucci: Collection of functionality ported from the MATLAB code of Attilio Meucci.
# In order of appearance in the demo script ButterflyTrading.R
MapVol = function( sig , y , K , T )
{
# in real life a and b below should be calibrated to security-specific time series
a = -0.00000000001
b = 0.00000000001
s = sig + a/sqrt(T) * ( log(K) - log(y) ) + b/T*( log(K) - log(y) )^2
return( s )
}
#' Compute the pricing in the horizon as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106.
#'
#' @param Butterflies List of securities with some analytics computed.
#' @param X Panel of joint factors realizations
#'
#' @return PnL Matrix of profit and loss scenarios
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/HorizonPricing.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
HorizonPricing = function( Butterflies , X )
{
r = 0.04 # 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 )
############# Ram's Code: Substituted with package's own functions #################################
#
## 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 )
#
####################################################################################################
BS = BlackScholesCallPutPrice( Y, K, r, sig, t )
Butterflies[[s]]$P_T = BS$call + BS$put
PnL = cbind( PnL , Butterflies[[s]]$P_T )
}
PnL = PnL[ , -1 ]
return( PnL )
}
ViewCurveSlopeTest = function( X , p )
{
J = nrow( X ) ; K = ncol( X )
# constrain probabilities to sum to one...
Aeq = matrix( 1, 1 , J )
beq = matrix( 1 , nrow = 1 , ncol = 1 )
browser()
# ...constrain the expectation...
V = matrix( , nrow = nrow( X ) , ncol = 0 )
# Add 3 equality views
V = cbind( V , X[ , 14 ] - X[ , 13 ] ) # View 1: spread on treasuries
V = cbind( V , X[ , 14 ] - X[ , 13 ] ) # View 2: identical view (spread on treasuries)
V = cbind( V , X[ , 6 ] - X[ , 5 ] ) # View 3: difference in YHOO Vol
v = matrix( c( .0005 , 0 ) , nrow = ncol( V ) , ncol = 1 )
Aeq = rbind( Aeq , t(V) )
beq = rbind( beq , v )
# add an inequality view
# ...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 = 0.5 # ... 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 )
return( p_ )
}
#' Process the inequality view, as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106.
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewRealizedVol.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
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 = matrix( 1, 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_ )
}
#' Process the relative inequality view on median, as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008,
#' p 100-106
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewRealizedVol.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
ViewRealizedVol = function( X , p )
{
# view 2 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 = matrix( 1, 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_ )
}
#' Process views for the expectations and binding constraints as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008,
#' p 100-106
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewCurveSlope.m"
#'
#' @author Xavier Valls \email{flamejat@@gmail.com}
#' @export
ViewCurveSlope = function( X , p )
{
# view 3
J = nrow( X );
K = ncol( X );
# constrain probabilities to sum to one...
Aeq = matrix( 1, 1 , J );
beq = 1;
# ...constrain the expectation...
V = X[ , 14 ] - X[ , 13 ];
v = 0.0005;
Aeq = rbind( Aeq , t(V) );
beq = rbind( beq , v );
A = b = matrix( nrow = 0 , ncol = 0 );
# ...compute posterior probabilities
p_ = EntropyProg( p , A , b , Aeq ,beq )$p_;
return( p_ );
}
#' Computes the conditional value at risk as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106
#'
#' @param Units panel of joint factors realizations
#' @param Scenarios vector of probabilities
#' @param Conf Confidence
#'
#' @return CVaR Conditional Value at Risk
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ComputeCVaR.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com}
#' @export
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 )
}
#' Computes the long-short conditional value at risk frontier as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008, p 100-106
#'
#' @param PnL Profit and Loss scenarios
#' @param Probs vector of probabilities
#' @param Butterflies list of securities with some analytics computed.
#' @param Options list of options
#'
#' @return Exp vector of expected returns for each asset
#' @return SDev vector of security volatilities along the efficient frontier
#' @return CVaR Conditional Value at Risk for each portfolio
#' @return Composition matrix of compositions (security weights) for each portfolio along the efficient frontier
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/LongShortMeanCVaRFrontier.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com}, Xavier Valls \email{flamejat@@gmail.com}
#' @export
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 * matrix( 1,N,1) , Options$Limit * matrix( 1,N,1) ) # constant vector in the constraints A*x <= b. an 18x1 matrix
# determine expectation of minimum-variance portfolio
Exps = t(PnL) %*% Probs
Scnd_Mom = t(PnL) %*% (PnL * (Probs %*% matrix( 1,1,N) ) )
Scnd_Mom = ( Scnd_Mom + t(Scnd_Mom) ) / 2
Covs = Scnd_Mom - Exps %*% t(Exps)
Amat = rbind( Constr$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 ) )
}
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# In order of appearance in the demo script ButterflyTrading.R
MapVol = function( sig , y , K , T )
{
# in real life a and b below should be calibrated to security-specific time series
a = -0.00000000001
b = 0.00000000001
s = sig + a/sqrt(T) * ( log(K) - log(y) ) + b/T*( log(K) - log(y) )^2
return( s )
}
#' Compute the pricing in the horizon as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106.
#'
#' @param Butterflies List of securities with some analytics computed.
#' @param X Panel of joint factors realizations
#'
#' @return PnL Matrix of profit and loss scenarios
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/HorizonPricing.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
HorizonPricing = function( Butterflies , X )
{
r = 0.04 # 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 )
############# Ram's Code: Substituted with package's own functions #################################
#
## 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 )
#
####################################################################################################
BS = BlackScholesCallPutPrice( Y, K, r, sig, t )
Butterflies[[s]]$P_T = BS$call + BS$put
PnL = cbind( PnL , Butterflies[[s]]$P_T )
}
PnL = PnL[ , -1 ]
return( PnL )
}
ViewCurveSlopeTest = function( X , p )
{
J = nrow( X ) ; K = ncol( X )
# constrain probabilities to sum to one...
Aeq = matrix( 1, 1 , J )
beq = matrix( 1 , nrow = 1 , ncol = 1 )
browser()
# ...constrain the expectation...
V = matrix( , nrow = nrow( X ) , ncol = 0 )
# Add 3 equality views
V = cbind( V , X[ , 14 ] - X[ , 13 ] ) # View 1: spread on treasuries
V = cbind( V , X[ , 14 ] - X[ , 13 ] ) # View 2: identical view (spread on treasuries)
V = cbind( V , X[ , 6 ] - X[ , 5 ] ) # View 3: difference in YHOO Vol
v = matrix( c( .0005 , 0 ) , nrow = ncol( V ) , ncol = 1 )
Aeq = rbind( Aeq , t(V) )
beq = rbind( beq , v )
# add an inequality view
# ...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 = 0.5 # ... 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 )
return( p_ )
}
#' Process the inequality view, as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106.
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewRealizedVol.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
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 = matrix( 1, 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_ )
}
#' Process the relative inequality view on median, as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008,
#' p 100-106
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewRealizedVol.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com} and Xavier Valls \email{flamejat@@gmail.com}
#' @export
ViewRealizedVol = function( X , p )
{
# view 2 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 = matrix( 1, 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_ )
}
#' Process views for the expectations and binding constraints as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008,
#' p 100-106
#'
#' @param X : panel of joint factors realizations
#' @param p : vector of probabilities
#'
#' @return p_ : vector of posterior probabilities
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ViewCurveSlope.m"
#'
#' @author Xavier Valls \email{flamejat@@gmail.com}
#' @export
ViewCurveSlope = function( X , p )
{
# view 3
J = nrow( X );
K = ncol( X );
# constrain probabilities to sum to one...
Aeq = matrix( 1, 1 , J );
beq = 1;
# ...constrain the expectation...
V = X[ , 14 ] - X[ , 13 ];
v = 0.0005;
Aeq = rbind( Aeq , t(V) );
beq = rbind( beq , v );
A = b = matrix( nrow = 0 , ncol = 0 );
# ...compute posterior probabilities
p_ = EntropyProg( p , A , b , Aeq ,beq )$p_;
return( p_ );
}
#' Computes the conditional value at risk as it appears in A. Meucci, "Fully Flexible Views: Theory and Practice",
#' The Risk Magazine, October 2008, p 100-106
#'
#' @param Units panel of joint factors realizations
#' @param Scenarios vector of probabilities
#' @param Conf Confidence
#'
#' @return CVaR Conditional Value at Risk
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/ComputeCVaR.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com}
#' @export
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 )
}
#' Computes the long-short conditional value at risk frontier as it appears in A. Meucci,
#' "Fully Flexible Views: Theory and Practice", The Risk Magazine, October 2008, p 100-106
#'
#' @param PnL Profit and Loss scenarios
#' @param Probs vector of probabilities
#' @param Butterflies list of securities with some analytics computed.
#' @param Options list of options
#'
#' @return Exp vector of expected returns for each asset
#' @return SDev vector of security volatilities along the efficient frontier
#' @return CVaR Conditional Value at Risk for each portfolio
#' @return Composition matrix of compositions (security weights) for each portfolio along the efficient frontier
#'
#' @references
#' A. Meucci, "Fully Flexible Views: Theory and Practice" \url{http://www.symmys.com/node/158}
#' See Meucci script for "ButterflyTrading/LongShortMeanCVaRFrontier.m"
#'
#' @author Ram Ahluwalia \email{ram@@wingedfootcapital.com}, Xavier Valls \email{flamejat@@gmail.com}
#' @export
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 * matrix( 1,N,1) , Options$Limit * matrix( 1,N,1) ) # constant vector in the constraints A*x <= b. an 18x1 matrix
# determine expectation of minimum-variance portfolio
Exps = t(PnL) %*% Probs
Scnd_Mom = t(PnL) %*% (PnL * (Probs %*% matrix( 1,1,N) ) )
Scnd_Mom = ( Scnd_Mom + t(Scnd_Mom) ) / 2
Covs = Scnd_Mom - Exps %*% t(Exps)
Amat = rbind( Constr$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 ) )
}
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