Fitting Distributions to Daily S&P 500 Price Changes

Compare logistic and lognormal distributions fitted to S&P 500 daily price changes to determine which better represents financial return data

A logistic distribution provides a good fit for daily fractional price changes in financial markets. Estimate and compare the logistic and lognormal distributions for the S&P 500 index’s daily fractional price changes from January 1, 2000, to January 1, 2009, and evaluate which model better represents the data.

Retrieve daily fractional price changes of the S&P 500 index between 2000 and 2009 as a time series:

In[1]:=

sp500=TimeSeriesMap[Normal[#,Quantity]&,FinancialData["SP500","FractionalChange",{{2000,1,1},{2009,1,1},"Day"}]]

Out[1]=

TimeSeries

Time: 04 Jan 2000 to 31 Dec 2008
Data points: 2262



Fit a logistic distribution to the observed daily price changes to estimate its parameters μ and β:

In[2]:=

edist=EstimatedDistribution[sp500,LogisticDistribution[μ,β]]

Out[2]=

LogisticDistribution[-0.0000220739,0.00665232]

Display a histogram of daily changes and overlay the fitted logistic probability density function for visual comparison:

In[3]:=

h=Histogram[sp500,Automatic,"ProbabilityDensity"];Show[h,Plot[PDF[edist,x],{x,-.1,.1},PlotStyleThick,PlotRangeAll]]

Out[3]=

Find the probability that the daily fractional price increase exceeds 0.5 %:

In[4]:=

Probability[x>0.005,xedist]

Out[4]=

0.319747

Find the mean fractional price change:

In[5]:=

Mean[edist]

Out[5]=

-0.0000220739

Generate and plot a 30-day synthetic sequence of fractional price changes drawn from the fitted logistic model:

In[6]:=

ListPlot[RandomVariate[edist,30],JoinedTrue]

Out[6]=

Fit a LogNormalDistribution to the adjusted data and overlay both fits on the histogram to compare their shapes visually:

In[7]:=

LNestimate=EstimatedDistribution[1+sp500,LogNormalDistribution[μ,σ]];Show[h,Plot[{PDF[edist,x],PDF[LNestimate,x+1]},{x,-0.1,0.1},PlotStyle{Directive[Thick,Blue],Directive[Thick,Orange]},PlotLegendsPlaced[{"Logistic Fit","LogNormal Fit"},{.8,.8}]]]

Out[7]=

Perform hypothesis tests to evaluate how well each distribution fits the data and present p-values side-by-side for comparison. Higher p-values indicate that the observed data are consistent with the corresponding model; the logistic distribution generally yields a better fit for symmetric price-change data.

In[8]:=

logisticP=DistributionFitTest[sp500,edist,"TestDataTable"]

Out[8]=

	Statistic	P-Value
Cramér-von Mises	1.33283	0.000416538

In[9]:=

lognormalP=DistributionFitTest[1+sp500,LNestimate,"TestDataTable"]

Out[9]=

	Statistic	P-Value
Cramér-von Mises	6.76732	2.22045× -16 10

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Fitting Distributions to Daily S&P 500 Price Changes

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Fitting Distributions to Daily S&P 500 Price Changes

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