Quantitative thoughts

Usually, intraday volatility exhibits a “smile” – it is high at open and close and it is lower during the trading day.

DJI index, 5 min. intervals, CET time:

MOS stock, 5 s. intervals, CET time:

Because many readers of this blog are trading Nasdaq OMX Baltic stocks, it is worth to share my findings about volatility in these markets. It was surprising for me, that none of the stocks in OMX Baltic markets follow “smile” pattern as it was show above.

APG1L, 30 min. intervals, CET time:

CTS1L, 30 min. intervals, CET time:

TEO1L, 30 min. intervals, CET time:

SRS1L, 30 min. intervals, CET time:

MRK1T, 30 min. intervals, CET time:

OEG1T, 30 min. intervals, CET time:

TAL1T, 30 min. intervals, CET time:

UKB1L, 30 min. intervals, CET time:

TVEAT, 30 min. intervals, CET time:

TKM1T, 30 min. intervals, CET time:

I can find only one reason, why these stocks do not follow the “smile” pattern – liquidity. During the summer these markets are very dry and the quotes are 1.5 months long. Let me know, if you are interested in this report with longer period, then I will repeat it later.

How many times have you been disappointed by nice trading system, because neither trading cost or slippage or bid/ask spread were included into back-test results? Did you find difficult to back-test a portfolio in R or many portfolios with different stocks? Blotter package is supposed to solve these problems.

In really – it is complicated. I spent a couple of days to start using it. There was one bug in the demo example, but the rest was in my code. You should remember:
1. Time zone must be specified:

Sys.setenv(TZ="GMT")

2. xts object must be created with explicit index class:

getSymbols('^GSPC',from='2000-01-01',<strong>index.class=c("POSIXt","POSIXct")</strong>)

3. When the test is conducted all blotter related values are written into .blotter environment. If you want repeat the same test, then you need to get rid of all .blotter values. So, you have to run something like this:

rm(list=ls(envir=.blotter),envir=.blotter)

You can find the author’s comment about this issue here: http://n4.nabble.com/Blotter-package-problem-with-example-tp1018634p1572911.html
4. It is slow. If you want to find the fittest rule from a bunch of them – try avoid using blotter, instead, use pure cumulative return and later on, include blotter.

Let’s test this rule with blotter as an example:
short SPY if today’s low is higher than yesterday’s close and close next day.

require(xts)
require(quantmod)
Sys.setenv(TZ="GMT")
initDate='2000-01-01'
getSymbols('SPY',from=initDate,index.class=c("POSIXt","POSIXct"))
SPY<-adjustOHLC(SPY,use.Adjusted=T)
 
#yesterday's price
tmp<-lag(SPY,1)
 
# if today's low is higher than yesterday's close 1, else 0
signal<-ifelse(Lo(SPY)>Cl(tmp),1,0)
signal[1]<-0
 
#let's plot cumulitative return to make sure, that we are on the right path
tmp<-lag(Delt(Cl(SPY)),-1)
tmp[length(tmp)]<-0
plot(cumprod(signal*tmp+1))
 
SPY<-Cl(SPY)
symbols<-c('SPY')
 
rm(list=ls(envir=.blotter),envir=.blotter)
initPortf(ltportfolio,symbols, initDate=initDate)
initAcct(ltaccount,portfolios=c(ltportfolio), initDate=initDate,initEq=initEq)
 
signal<-signal['2000-02-01::']
 
for(i in 1:length(signal))
{
	currentDate= time(signal)[i]
	equity = getEndEq(ltaccount, currentDate)
	position = getPosQty(ltportfolio, Symbol=symbols[1], Date=currentDate)	
 
	if(position==0)
	{
		#open a new position if signal is >0
		if(signal[i]>0)
		{
			closePrice<-as.double(Cl(SPY[currentDate]))
			unitSize = as.numeric(trunc((equity/closePrice)))
                        # 5$ per transaction
			addTxn(ltportfolio, Symbol=symbols[1],  TxnDate=currentDate, TxnPrice=closePrice, TxnQty = -unitSize , TxnFees=5, verbose=T)
		}
 
	}
	else
	{
		#position is open. If signal is 0 - close it.
		if(as.double(signal[i])==0 &amp;&amp; position<0)
		{
			position = getPosQty(ltportfolio, Symbol=symbols[1], Date=currentDate)
			closePrice<-as.double((Cl(SPY[currentDate])))#as.double(get(symbols[1])[i+100])
 
                        # 5$ per transaction
			addTxn(ltportfolio, Symbol=symbols[1],  TxnDate=currentDate, TxnPrice=closePrice, TxnQty = -position , TxnFees=5, verbose=T)
 
		}
 
	}
	updatePortf(ltportfolio, Dates = currentDate)
	updateAcct(ltaccount, Dates = currentDate)
	updateEndEq(ltaccount, Dates = currentDate)
 
	equity = getEndEq(ltaccount, currentDate)
 
}
plot(getAccount(ltaccount)[["TOTAL"]]$End.Eq)

The result (it will take time to get it…):

Bespoke blogged about average monthly returns of the DJI and emphasized April. Before jumping on that information, let’s check some weak points.
In that post, only average returns are presented. We need at least extreme points (min;max) and confidence ranges. Second problem – the normal market have upward trend and we need to get rid of that. To do so, either we have subtract the  rolling mean (a tough way) or use logarithmic prices (that’s the easy way!).

Instead of DJI, I took S&P500 from 1950 until now.

Sure, average return in April is above 0, but based on historical data, negative return is possible. I conducted t-test, where null hypothesis was, that average return is equal to zero. I got p-value of 0.0042, so null hypothesis can be rejected (return is above 0).

The graph below shows cumulative return of investment, investing only in April. Keep in mind, that this is log scale and real return would be higher.

Conlusion: based on this data, expect positive return in April.

R code

require(xts)
require(quantmod)
require(ggplot2)
getSymbols('^GSPC',from='1950-01-01')
return<-Delt(Cl(to.monthly(log(GSPC))))
return[1]<-0
temp<-as.double(format(index(return),'%m'))
 
temp<-data.frame(as.double(return),as.numeric(temp))
qplot(factor(as.numeric(temp[,2])),as.double(return),data=temp,geom = "boxplot",ylab='Returns',xlab='Months')
 
t.test(return[which(format(index(return),'%m')=='04')])
plot(cumprod(return[which(format(index(return),'%m')=='04')]+1),main='Cumulative return, 1950-present')

Inspired by CXO group report, I did a rerun of the same strategy on my data. Easter’s dates can be find at wikipedia. Overall, my results are similar to CXO group’s results.

In the graph below, I plotted daily returns on Easter week (Monday to Thursday) from 1982 to 2009. I prefer this way of showing things, where the range, minimum, maximum and mean of returns are presented.
It is clear, that only returns on Thursdays are above zero and t-test confirms that:
t = 2.235, df = 27, p-value = 0.03389. The rest is close to random.

The graph below shows daily returns one week after Easter holidays. Although Monday looks like negative day, but it lacks significance:
t = -1.1517, df = 27, p-value = 0.2595 (should be less that 0.1).
The rest is noise.

In summary, only returns on Thursdays have positive bias.

#R-Language code to repeat this test.
require(quantmod)
require(xts)
easter<-as.matrix(read.table(file='data/easter.csv'))
getSymbols('^GSPC',from='1980-01-01')
 
GSPC.delta<-Delt(log(Cl(GSPC)))
GSPC.delta<-Delt(Cl(GSPC))
 
#returns during the week before Easter
#returns on Thursdays
nasa<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')-3)]),4)
colnames(nasa)<-c('ret','weekday')
 
#Monday to Wednesday
for(i in 1:3)
{
tmp<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')-(3+i))]),(4-i))
colnames(tmp)<-c('ret','weekday')
rbind(nasa,tmp)
nasa<-rbind(nasa,tmp)
}
t.test(nasa$ret[nasa$weekday==4])
 
require(ggplot2)
qplot(factor(as.numeric(nasa$week)),as.double(nasa$ret),data=nasa,geom = "boxplot",ylab='Returns',xlab='Weekdays')
 
#returns during the week after Easter
 
GSPC.delta<-Delt(Cl(GSPC))
nasa<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')+1)]),1)
colnames(nasa)<-c('ret','weekday')
for(i in 2:5)
{
print(i)
tmp<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')+i)]),i)
colnames(tmp)<-c('ret','weekday')
rbind(nasa,tmp)
nasa<-rbind(nasa,tmp)
}
t.test(nasa$ret[nasa$weekday==1]);t.test(nasa$ret[nasa$weekday==2])
qplot(factor(as.numeric(nasa$week)),as.double(nasa$ret),data=nasa,geom = "boxplot",ylab='Returns',xlab='Weekdays')

Recently, I was busy testing the following strategy:

If SPY and VIX daily returns are positive, then short SPY at close and keep it for one day.

The strategy is dump simple and it has very good feature – short side. There are not so many successful short side strategies. For testing purpose I used daily Yahoo data from 1995 until present. For commissions and bid/ask spread I used 5 $ fee per 10 000 $ trade. Here we go:

Annualized Return                     0.0421
Annualized Std Dev                   0.0488
Annualized Sharpe (Rf=0%)        0.8621
t = 3.3787, df = 3811, p-value = 0.0007356

Up to this point is was relatively easy to make a test (the true is, that I spent some time cracking and hacking blotter package, but I will write about it in separate post). My second objective was the improvement of this strategy.

One of the way to understand the strategy is to look how the components are related to each other or correlated. To do that, I took daily returns of SPY and VIX at day 0 and plotted against SPY next day’s (day+1) returns.

What can I tell by looking at this plot? I couldn’t figure out any linear relation between returns of SPY and VIX at day+0 and returns of SPY at day+1. Should I try something like random forest method?

I tried to add some TA flavors, like RSI, but the improvements were not very significant. Simplicity is genius!

It is know, that the first day of the month provides bullish edge. According to Quantifiable edges not all the months are equal. So, I made a test on S&P500 index, from January, 1980 until February, 2010. It is true, March isn’t the best month to run this strategy.

Only 3 months have significant results based on p-values:
“month 5, p-value 0.0399233570186162″
“month 7, p-value 0.0466800163648646″
“month 11, p-value 0.0218919220125013″

p.s. if somebody is interested in R-Language code to repeat this test, then let me know.

Last spring I read “Quantitative Trading” by Ernest P. Chan. In his book, he suggested to buy gas futures contract at the end of February and sell it later, in March. Today, I decided to test this strategy by using R-language.
The most important thing for such investigation is data. For this purpose, I used this public data: www.eia.doe.gov
I made 2 test – for the first one, I used futures contract, which will be settled after 4 months and for the second one, I used gas spot price.
In the first plot, we can see monthly price returns scaled by months (1-12). It is clear, that mean of March’s returns are above 0. What is encouraging in this plot is that the ranges of returns are above 0 as well (meaning, that majority of March’s returns was above 0).
Anova test gives p-value below 0.01 – some months in the group has different mean (that supports seasonality idea). March’s t-test gives these values: t = 2.8064, df = 15, p-value = 0.01329.

I used gas spot prices to generate second plot. It is similar to the first one, except that March’s returns has longer tail. It is worth to note, that September stands as a positive month as well. The statistics for this test are not so strong, as it was in the first example.
P-value of Anova test is only 0.11 and March’s t-test is:
t = 1.3791, df = 15, p-value = 0.1881

R-language code to run these tests:

#------gas contract 4 -----
library(xts)
library(ggplot2)
library(quantmod)
 
tmp<-as.matrix(read.table(file='gas.contract.4.csv',sep=';'))
gas<-as.xts(as.double(tmp[,2]),order.by=as.POSIXct(tmp[,1]))
plot(gas)
gas.monthly<-to.monthly(gas)
gas.monthly.delta<-Delt(Cl(gas.monthly))
gas.monthly.delta[1]<-0
 
gas.factor<-as.double(format(index(gas.monthly.delta),'%m'))
tmp<-data.frame(as.double(gas.monthly.delta),as.numeric(gas.factor))
qplot(factor(as.numeric(gas.factor)),as.double(gas.monthly.delta),data=tmp,geom = "boxplot",ylab='Returns',xlab='Months')
 
anova(lm(as.double(gas.monthly.delta)~gas.factor))
t.test(((gas.monthly.delta[which(format(index(gas.monthly.delta),'%m')=='03')]))[,1])
 
#----gas contract spot -------
tmp<-as.matrix(read.table(file='gas.csv',sep=';'))
gas<-as.xts(as.double(tmp[,2]),order.by=as.POSIXct(tmp[,1]))
 
plot(gas)
colnames(gas)<-c('Close')
 
gas.monthly<-to.monthly(gas)
gas.monthly.delta<-Delt(Cl(gas.monthly))
gas.monthly.delta[1]<-0
 
gas.factor<-as.double(format(index(gas.monthly.delta),'%m'))
tmp<-data.frame(as.double(gas.monthly.delta),as.numeric(gas.factor))
 
qplot(factor(as.numeric(gas.factor)),as.double(gas.monthly.delta),data=tmp,geom = "boxplot",ylab='Returns',xlab='Months')
 
anova(lm(as.double(gas.monthly.delta)~gas.factor))
t.test(((gas.monthly.delta[which(format(index(gas.monthly.delta),'%m')=='03')]))[,1])

Recently, Orion securities have issued a “BUY” recomendation for Cugar ETF. Because, neither I follow the recommendations nor I’m big fan of TA (I have to admit, that I was…), I decided to check sugar price seasonality.

Voila, the mean of monthly returns are presented in the graph. February, April and May tend to be negative and June and July show positive returns.
BUT! Don’t forget to ask – are these results significant? P-value for July is 14%, 34% for April. The rest is above 50%. So, keep in mind, that these results are very weak…

More descriptive plot:

I found a description of supposed to be profitable strategy on Bloomberg. The strategy is simple – buy S&P500 index on close and sell it on next day open.

So, I tested this claim and got nice P/L curve:

Yes, since 1993 this strategy has generated the profit >300%. But, neither commissions or slippage are included:)

Let’s consider 0.0007% commissions+slippage(7$ per 10000 trade). There we go:

Is it good? I will look for somethig better…

Recently, friend of my, got investment advise from his broker – long gas, because the price of gas is very low compared to oil. The broker didn’t indicate neither profit target or stop loss…
I got hooked on the idea.

First of all, I ran linear regression on monthly prices of oil and gas from 01-12-1993 to 01-09-2009. I was skeptical about ‘oil vs gas’ idea, but after seeing this graph it became interesting. R-Square is high 0.7385. Check this out: (the prices are adjusted with log())

The spread between oil and gas indicates, that it was very high back in September.

Finally, I took two funds: USO (oil’s ETF) and UNG (gas’s ETF) to see how does it look on real time (ok ok daily) data. As we can see, in late September the spread reached a new high. The problem is, that data set is very limited and we can see only ~3 years.

And the last one – green stands for oil’s log(price) and blue one for gas’s log(price).