Prediction of imputed share returns

10. M. Neumann, G. Skiadopoulos. “Predictable dynamics in Higher-Order Risk-Neutral Moments: Evidence from the S&P 500 Options”. Journal of Financial and Quantitative Analysis, No 3, June 2013.

11. H. Hong, J. Stein. “Differences of Opinion, Short-Sales Constraints, and Market Crashes”. Review of Financial Studies, Vol 16, Is. 2, 2003.

12. H. Pesaran, R. Smith. “Estimating Long-Run Relationships from Dynamic Heterogeneous Panels”. Journal of Econometrics, Vol 68, Is. 1, July 1995.

13. H. Pesaran. “Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure”. Econometrica, Vol 74, Is. 4, June 2006.

A. Chudik, H. Pesaran, E. Tosetti. “Weak and Strong Cross Section Dependence and Estimation of Large Panels”. Econometrics Journal, No 14, June 2009.

A. Chudik, H. Pesaran. “Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors”. Journal of Econometrics, Vol 188, Is 2, 2015.

14. H. Pesaran. “Testing Weak Cross-Sectional Dependence in Large Panels”. Econometric Reviews, Vol 34, Is. 6, 2015.



Appendix 1

R-script used to calculate the RNM's

rm(list=ls())

# Loading the necessary packages.

library(data.table)

library(stringr)

library(caTools)

library(compiler)

library(tryCatchLog)

# Functions to calculate values of the contracts:

Vcontract_value<-function(SecurityID, Date)

{

tryCatch(expr={

return(trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'],

data_id$Vcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'])+

trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P'],

data_id$Vcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P']))},

error=function(e){return(NA)})

}

Vcontract_value<-cmpfun(Vcontract_value)

Wcontract_value<-function(SecurityID, Date)

{

tryCatch(expr={

return(trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'],

data_id$Wcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'])+

trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P'],

data_id$Wcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P']))},

error=function(e){return(NA)})

}

Wcontract_value<-cmpfun(Wcontract_value)

Xcontract_value<-function(SecurityID, Date)

{

tryCatch(expr={

return(trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'],

data_id$Xcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='C'])+

trapz(data_id$ImpliedStrike[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P'],

data_id$Xcontract[data_id$SecurityID==SecurityID & data_id$Date==Date & data_id$CallPut=='P']))},

error=function(e){return(NA)})

}

Xcontract_value<-cmpfun(Xcontract_value)

# Downloading the file with interest rates, parsing:

setwd('/Users/mac/Desktop/Thesis/2005')

monthrate<-fread('monthrate.csv')

names(monthrate)<-c('Date', 'InterestRate')

monthrate$Date<-as.Date(monthrate$Date, format='%d.%m.%Y')

monthrate$InterestRate<-as.numeric(monthrate$InterestRate)/100

# Loop through the necessary years to download the files with the prices of the underlying and the volatility surfaces:

for (year in 1996:2016){

setwd(paste('/Volumes/KINGSTON/Optionmetrics_data_feed/', year, sep = ''))

# Import of the necessary files, binding them into one file "data":

files<-list.files()

# Securities prices file:

sec_prices<-fread(paste('Security_price_', year, '.csv', sep=''), dec=',')

names(sec_prices)<-c('SecurityID', 'Date', 'BidLow', 'AskHigh',

'ClosePrice', 'Volume', 'TotalReturn', 'AdjustmentFactor1',

'OpenPrice', 'SharesOutstanding', 'AdjustmentFactor2')

prices<-sec_prices[,c('SecurityID', 'Date', 'ClosePrice')]

prices$SecurityID<-as.numeric(prices$SecurityID)

prices$ClosePrice<-abs(as.numeric(prices$ClosePrice))

prices$Date<-as.Date(prices$Date, format="%Y-%m-%d")

rm(sec_prices)

# Volatility Surface files:

volatility_files_names<-files[2:5]

rm(files)

data<-do.call(rbind, lapply(volatility_files_names,

function(x) fread(x, dec=',',

col.names = c('SecurityID', 'Date', 'Days','Delta',

'CallPut', 'ImpliedVolatility',

'ImpliedStrike', 'ImpliedPremium', 'Dispersion'))))

rm(volatility_files_names)

data$Date<-as.Date(data$Date, format="%Y-%m-%d")

data<-data[,c('SecurityID', 'Date', 'Days','Delta', 'CallPut', 'ImpliedVolatility',

'ImpliedStrike', 'ImpliedPremium')]

data[,c('SecurityID', 'Days', 'Delta', 'ImpliedVolatility',

'ImpliedStrike', 'ImpliedPremium')]<-lapply(

data[,c('SecurityID', 'Days', 'Delta',

'ImpliedVolatility', 'ImpliedStrike', 'ImpliedPremium')], as.numeric)

data<-data[ImpliedVolatility!=-99.99]

data<-data[ImpliedPremium!=-99.99]

data<-data[Days==30]

data<-data[abs(Delta)<50]

# Merging the files:

data<-merge(data, monthrate, by=c('Date'))

data<-merge(data, prices, by=c('SecurityID', 'Date'))

rm(prices)

data<-data[order(SecurityID, Date, CallPut, ImpliedStrike)]

# Calculation of the weights of contracts in the BKM integrals:

data[,Vcontract:=2*((1-log(ImpliedStrike/ClosePrice))/(ImpliedStrike^2))*ImpliedPremium]

data[,Wcontract:=((6*log(ImpliedStrike/ClosePrice)-3*(log(ImpliedStrike/ClosePrice))^2)/(ImpliedStrike^2))*ImpliedPremium]

data[,Xcontract:=((12*(log(ImpliedStrike/ClosePrice))^2-4*(log(ImpliedStrike/ClosePrice))^3)/(ImpliedStrike^2))*ImpliedPremium]

# Approximation of the BKM integrals using trapezoid rule to get values of contracts:

moments<-unique(data[, c('SecurityID', 'Date', 'InterestRate')])

moments$Vvalue<-0

moments$Wvalue<-0

moments$Xvalue<-0

results<-moments[1]

for (i in unique(moments$SecurityID)){

data_id<-data[SecurityID==i]

moments_id<-moments[SecurityID==i]

moments_id[,Vvalue:=mapply(Vcontract_value, moments_id$SecurityID, moments_id$Date)]

moments_id[,Wvalue:=mapply(Wcontract_value, moments_id$SecurityID, moments_id$Date)]

moments_id[,Xvalue:=mapply(Xcontract_value, moments_id$SecurityID, moments_id$Date)]

results<-rbind(results, moments_id)

}

# Calculation of the moments:

results<-results[2:nrow(results)]

results[, Mu:=exp(InterestRate*30/360)-1-exp(InterestRate*30/360)*Vvalue/2-exp(InterestRate*30/360)*Wvalue/6-exp(InterestRate*30/360)*Xvalue/24]

results[, Volatility:=((exp(InterestRate*30/360)*Vvalue-(Mu)^2)*360/30)^(1/2)]

results[, Skewness:=(exp(InterestRate*30/360)*Wvalue-3*Mu*exp(InterestRate*30/360)*Vvalue+2*(Mu)^3)/(exp(InterestRate*30/360)*Vvalue-(Mu)^2)^(3/2)]

results[, Kurtosis:=(exp(InterestRate*30/360)*Xvalue-4*Mu*exp(InterestRate*30/360)*Wvalue+6*exp(InterestRate*30/360)*(Mu)^2*Vvalue-3*(Mu)^4)/(exp(InterestRate*30/360)*Vvalue-(Mu)^2)^2]

# Saving the results:

results<-results[,c("SecurityID", "Date", "Mu", "Volatility", "Skewness","Kurtosis")]

setwd('/Users/mac/Desktop/Thesis/New Results')

write.csv2(results, file=paste(year, '_moments.csv', sep=''), row.names = FALSE, na='')

rm(data)

rm(moments)

rm(results)

rm(data_id)

rm(moments_id)

}



Appendix 2

The CD tests for the Volatility, Skewness and Kurtosis Series.

The CD tests for the Volatility, Skewness and Kurtosis Series

Volatility	Skewness	Kurtosis
Pesaran (2015) test for weak cross-sectional dependence.	Pesaran (2015) test for weak cross-sectional dependence.	Pesaran (2015) test for weak cross-sectional dependence.
H0: errors are weakly cross-sectional dependent.	H0: errors are weakly cross-sectional dependent.	H0: errors are weakly cross-sectional dependent.
CD = 303.559	CD = 164.877	CD = 80.044
p-value = 0.000	p-value = 0.000	p-value = 0.000



Appendix 3

The output of the CCE models for Volatility, Skewness and Kurtosis.

Common Correlated Effects models for Volatility, Skewness and Kurtosis. Return (-1), BidAskSpread (-1) and HistoricBeta (-1) refer to the return, bid-ask spread and beta variables lagged one period. Leverage, DefaultSpread, TermSpread, RiskFreeRate and VIX refer to the leverage ratio, default spread, term spread risk-free rate and Vix variables. The resulting coefficients, standard errors and p-values are given for these variables in the table. Coefficients of the significant coefficients at 5% level are highlight in bold.

	Volatility	Skewness	Kurtosis
	Coef. (St. Er.).	P-Value	Coef. (St. Er.).	P-Value	Coef. (St. Er.).	P-Value
Return (-1)	-0,3813	0,000	-0,6361	0,000	-0,1780	0,049
	(0,0243)		(0,0651)		(0,0904)
BidAskSpread (-1)	2,0365	0,000	0,8409	0,000	-0,2219	0,165
	(0,1229)		(0,1176)		(0,1598)
HistoricBeta (-1)	0,0851	0,002	0,0285	0,229	-0,0467	0,267
	(0,0277)		(0,0237)		(0,0421)
Leverage	-0,8454	0,264	-0,1296	0,587	-0,1889	0,673
	(0,7573)		(0,2384)		(0,4479)
DefaultSpread	3,3689	0,079	-0,3123	0,835	-0,3761	0,900
	(1,9196)		(1,4988)		(2,9962)
TermSpread	0,5271	0,539	-0,4681	0,546	0,9217	0,529
	(0,8574)		(0,7762)		(1,4644)
RiskFreeRate	1,4877	0,299	-0,4326	0,855	-0,1423	0,976
	(1,4320)		(2,3654)		(4,7638)
VIX	-0,0005	0,705	0,0005	0,532	0,0015	0,299
	(0,0012)		(0,0008)		(0,0015)
	RMSE:	0,270	RMSE:	0,400	RMSE:	0,910



Appendix 4

Summary statistics and the corralation table for the subsample of 150 stocks used for regressions.

Summary statistics for the computed RNM's for the subsample

	Volatility	Skewness	Kurtosis
Number of observations	548125	548125	548125
Mean	1,1790	-0,1139	2,4076
Median	1,0517	-0,1439	2,2058
Standart Deviation	0,5727	0,4797	1,0274
Autocorrelation	0,9401	0,6394	0,6644

Correlations between the dependent and the explanatory variables in the subsample. Vol, Skew, Kurt and Lev refer to the volatility, skewness, kurtosis and leverage variables, respectively

	Vol	Skew	Kurt	Return	Spread	Beta	Lev	DefautSpread	Term Spread	Risk-Free Rate	VIX
Vol	1,0000
Skew	0,3367	1,0000
Kurt	-0,0277	0,4503	1,0000
Return	-0,0122	-0,0951	-0,0372	1,0000
Spread	0,5650	0,1960	-0,0956	-0,0141	1,0000
Beta	0,3168	0,1295	-0,0644	0,0033	0,2318	1,0000
Lev	-0,0881	-0,0631	0,0232	-0,0021	-0,0454	-0,0612	1,0000
DefautSpread	0,2735	-0,0413	-0,0804	0,0026	0,2447	0,0441	-0,0145	1,0000
Term Spread	0,0263	-0,0722	0,0023	0,0056	-0,0094	0,0119	0,0006	0,3022	1,0000
Risk-Free Rate	0,0512	0,0944	-0,0917	-0,0071	0,1237	-0,0516	0,0089	-0,3240	-0,7924	1,0000
VIX	0,4070	0,0504	-0,1242	-0,0604	0,4137	-0,0341	0,0128	0,6440	0,1651	0,0177	1,0000

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