# Short selling, volatility and bubbles

**Freakonometrics - Tag - R-english**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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Yesterday, I wrote a post (in French) about short-selling in financial market since some journalists claimed that it was well-known that short -selling does increase volatility on financial market. Not only in French speaking journals actually, since we can read on http://www.forbes.com that « in a market with restrictions on short-selling, volatility is reduced ». But things are not that simple. For instance http://www.optionsatoz.com/ explains it from a theoretical point of view. But we can also look at the data. For instance, we can compare the stock price of *Air China*,

exchanged in Shanghai in blue (where short-selling is forbidden) and in Hong Kong in rouge (where short-selling is allowed), since @Igor gave me the tickers of those stocks

library(tseries)X<-get.hist.quote("0753.HK")Y<-get.hist.quote("601111.SS")plot(Y[,4],col="blue",ylim=c(0,30))lines(X[,4],col="red")

But as @alea_ pointed out, one asset is expressed here in Yuans renminbi, and the other one in HK dollars. So I downloaded the exchange rate from http://www.oanda.com/

Z=read.table("http://freakonometrics.blog.free.fr/public/data/change-cny-hkd.csv",header=TRUE,sep=";",dec=",")D=as.Date(as.character(Z$date),"%d/%m/%y")z=as.numeric(Z$CNY.HKD)plot(D,z,type="l")X2=X[,4]for(t in 1:length(X2)){ X2[t]=X2[t]*z[D==time(X2[t])]} X2=X[,4]plot(Y[,4],col="blue",ylim=c(0,30))lines(X2,col="red")

Now both stocks are expressed in the same currency. To compare returns volatility, a first idea can be to use GARCH models,

RX=diff(log(X2))RY=diff(log(Y[,4]))Xgarch = garch(as.numeric(RX))SIGMAX=predict(Xgarch)Ygarch = garch(as.numeric(RY))SIGMAY=predict(Ygarch)plot(time(Y)[-1],SIGMAY[,1],col="blue",type="l")lines(time(X2)[-1],SIGMAX[,1],col="red")

But volatility is here too eratic. So an alternative can be to use exponentially-weighted moving averages, where simple recursive relationships are considered

or equivalently

The code is not great, but it is easy to understand,

moy.ew=function(x,r){ m=rep(NA,length(x)) for(i in 1:length(x)){ m[i]=weighted.mean(x[1:i], rev(r^(0:(i-1))))} return(m)} sd.ew=function(x,r,m){ sd=rep(NA,length(x)) for(i in 1:length(x)){ sd[i]=weighted.mean((x[1:i]-m[i])^2, rev(r^(0:(i-1))))} return(sd)} q=.97MX=moy.ew(RX,q)SX=sd.ew(RX,q,MX)MY=moy.ew(RY,q)SY=sd.ew(RY,q,MY)plot(time(Y)[-1],SY,col="blue",type="l")lines(time(X2)[-1],SX,col="red")

And now we have something less erratic, so we can focus now on the interpretation.

It is also possible to look on the difference between those two series of volatility, areas in blue means that in Shanghai (again, where short-selling is forbidden) returns are more volatile than in Hong Kong, and areas in redare periods where returns are more volatile in Hong Kong,

a=time(X2)[which(time(X2)%in%time(Y))]b=SY[which(time(Y)%in%time(X2))]- SX[which(time(X2)%in%time(Y))]n=length(a)a=a[-n];b=b[-n]plot(a,b,col="black",type="l")polygon(c(a,rev(a)),c(pmax(b,0),rep(0,length(a))), col="blue",border=NA)polygon(c(a,rev(a)),c(pmin(b,0),rep(0,length(a))), col="red",border=NA)

So clearly, there is nothing clear that can be said... Sometimes, volatility is higher in Hong Kong, and sometimes, it is higher in Shanghai. But if we look at the *price*, instead of looking at *volatility*,

a=time(X2)[which(time(X2)%in%time(Y))]b=as.numeric(Y[which(time(Y)%in%time(X2)),4])- as.numeric(X2[which(time(X2)%in%time(Y))])n=length(a)a=a[-n];b=b[-n]plot(a,b,col="black",type="l")polygon(c(a,rev(a)),c(pmax(b,0),rep(0,length(a))), col="blue",border=NA)polygon(c(a,rev(a)),c(pmin(b,0),rep(0,length(a))), col="red",border=NA)

Here, it looks like bans on short-selling creates bubbles. Might not not be a *good* thing.

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