Becausethe observations in the white noise process are IID, the forecast chart in Figure 11.2 has a mean of µ = 200 and standard deviation of σ = 10, as do all of the observations FIGURE
Trang 1CHAPTER 11 Simulating Financial Time Series
In financial modeling, we encounter two main types of time-series data:
1 Observations that appear to be independent and identically distributed (IID).
2 Observations that do not appear to be IID because they follow a trend or some
other pattern over time
Financial theory provides a compelling argument—the efficient markets
hypothe-sis—that returns on investments must be independent over time because no one
has access to information not already available to someone else If returns areindependent, however, prices will be dependent over time and we will require a way
to model that dependence This chapter presents some models that can be used forprojecting future returns, asset prices, and other financial times series in simulationmodels for risk analysis
WHITE NOISE
A white noise process is defined to be one that generates data appearing to be IID.
It takes its name from the fact that no specific frequency or pattern dominates in
a spectral analysis of the observations, similar to white light, or the noise of staticemitted from an AM radio that is not tuned in to a station
The model for a white noise process is
where µ is a constant, and t is a sequence of uncorrelated random variables
identically distributed with mean zero and finite variance for t = 1, , T The probability distribution of tis not necessarily normal, but if it is the process is said
to be Gaussian white noise named after the eighteenth-century mathematician, Carl
F Gauss, who studied the properties of the normal distribution
For example, we can simulate observations from a Gaussian white noise processwith Crystal Ball by placing several uncorrelated Normal(0,10) assumptions in a
column, adding a constant, say µ= 200, and plotting the results as was done in thefile RandomWalk.xls Figure 11.1 shows the model In cells B6:B35 are Crystal Ball
147
Trang 2FIGURE 11.1 Model to compare a white noise process to arandom walk Note that rows 8 through 33 are hidden.
normal(0,10) assumptions that we denote as t for t = 1, , 30 A Gaussian white
noise process was generated in cells C6:C35 using Expression 11.1, and a time seriesplot of one realization of the process appears in the center of Figure 11.1 Noticehow the independence of the observations in the white noise process is manifested
in the choppiness of its plot For the white noise process, no matter where each
Trang 3Simulating Financial Time Series 149
observation falls, the next observation is equally likely to be above or below themean of 200 This characteristic causes the choppy look
using the values in B6:B35 for t , t = 1, , 30.
A time series plot of one realization of the random walk process appears in thelower time series plot in Figure 11.1 Notice how the random walk process exhibits
a meandering pattern The first few points are below the mean, then once the plotgoes above the mean, it tends to stay above for a while, then heads down and goesbelow the mean again before eventually heading back up Even though the changes
in the level of the random walk are independent, the levels themselves are dependentover time This dependence causes more variability in the levels of the random walkprocess than is evident in the levels of the white noise process
The aggregate effect of the dependence of the levels of the random walkcompared to the white noise process can be seen in Figures 11.2 and 11.3 Becausethe observations in the white noise process are IID, the forecast chart in Figure 11.2
has a mean of µ = 200 and standard deviation of σ = 10, as do all of the observations
FIGURE 11.2 Forecast chart for the observation at time t= 30 forthe random process
Trang 4FIGURE 11.3 Forecast chart for the observation at time t= 30 forthe random walk.
W t in the white noise process for t≥ 1 The mean of the additive random walkprocess is also 200 for every observation, but the standard deviation grows largerevery time period because we are adding on another random change It can be
shown that each value Y t of the random walk process has a mean of µ= 200 and
a standard deviation of σ√
t In Figure 11.3 you can see that the standard deviation
(10√
30= 54.77) is much greater than the standard deviation (10) of the forecast
in Figure 11.2 The scales of the horizontal axes of these two plots were specified
to be equal so that the difference in variability between the white noise process andrandom walk was apparent However, the scales of the vertical axes in Figures 11.2and 11.3 are different Figure 11.4 is another illustration of the differences in theseforecasts with an overlay chart for cells C35 and D35
For a dynamic illustration of the difference between white noise and a randomwalk, see the file RandomWalk.xls In Run Preferences, set Run Mode to Demoand watch the time series plots to see the difference in behavior when the simulation
is running The white noise process will bounce almost entirely within the 3σ bounds
of 170 to 230 at every point in time, while the random walk will exhibit increasing
variability as t gets larger.
AUTOCORRELATION
Chapter 4 showed how to calculate both Pearson and Spearman correlations betweentwo variables with Excel When checking for independence of a series of values over
time, we calculate the autocorrelation, which is the correlation coefficient of the
values in the series that are separated by a specific length of time In this context,
Trang 5Simulating Financial Time Series 151
FIGURE 11.4 Overlay chart to compare the time t= 30 observationsfrom a white noise process and a random walk
the prefix auto–means same, so the autocorrelation is the correlation of the values
in a time series with other values within the same series Sometimes authors refer to
autocorrelation by the term serial correlation to emphasize the correlation within a
time series
While the correlation coefficients for values separated by two or more timeperiods are also of interest in time series analysis, for our purposes it is sufficient
to think only about first-order autocorrelation, which is the correlation between
values in a time series that are separated by one unit of time Thus, first-order
autocorrelation is also called Lag-1 autocorrelation Unless specified otherwise, the term autocorrelation in this chapter is meant to refer to first-order autocorrelation.
It is usually true with financial time series that if the first-order autocorrelation isnear zero, then the rest of the autocorrelation coefficients will also be near zero.However, for time series that exhibit seasonality, higher-order autocorrelation could
be significant while lag-1 autocorrelation is low
To calculate the first-order autocorrelation coefficient for the white noise processvalues in cells C5:C35, we entered into cell C3 the Excel formula =CORREL(C5:C34,C6:C35) As shown in Chapter 4, this calculates the Pearson correlation for thetwo arrays C5:C34 and C6:C35 Likewise, cell D3 holds the Excel formula
=CORREL(D5:D34,D6:D35) to find the first-order autocorrelation coefficient forthe random walk time series in cells D6:D35 Note that there are other methods
to calculate the autocorrelation coefficient having more appeal to purists, but Exceldoes not yet include these other methods in its arsenal of statistical functions Formore discussion of this point and other methods for calculating autocorrelationcoefficients, see pages 330–340 of Priestley (1981), or section 2.2 of Tsay (2002)
Trang 6Of course, with more work, you can always use Excel to calculate the lation coefficient by one of the other methods For example, another way to calculatethe first-order autocorrelation coefficient, ˆρ1, for observed values y t , t = 1, 2, , T is
t=1y t /T This version of the autocorrelation was calculated in cell Q6
for the white noise process in cells C6:C35 of RandomWalk.xls
To see how these autocorrelation coefficients vary during simulation trials, theyhave been defined as Crystal Ball forecasts Figure 11.5 shows the forecast chart forcell C3, the autocorrelation coefficient for the white noise process values By the waythese values were generated, we know that they are independent over time, so theirtrue autocorrelation is zero However, in any given simulation trial the calculated(sample) autocorrelation coefficient can differ from zero simply because of samplingerror It can be shown that the sampling error for the first-order autocorrelation
coefficient calculated for an IID time series of length T has a standard deviation
of approximately 1/√
T, so we would expect the standard deviation of the 10,000
values plotted in Figure 11.5 to be 1/√
30= 0.183 and roughly 95 percent of the
values to fall within the two standard error interval (−0.366, 0.366) Figure 11.5
shows that 95.86 percent of the observations actually fell within that interval duringthe 10,000 simulation trials, which agrees with what we expect Furthermore, the
FIGURE 11.5 Forecast chart for the autocorrelation coefficient for therandom process
Trang 7Simulating Financial Time Series 153
FIGURE 11.6 Forecast chart for the first-order autocorrelationcoefficient for the random walk process
sample standard deviation of the distribution in Figure 11.5 is 0.178, which is alsoclose to its expected value of 0.183
Figure 11.6 shows the autocorrelation for cell D3, the autocorrelation coefficientfor the random walk time series All values of the random walk autocorrelationcoefficient were significantly larger than zero, which is what we expect because thelevels of the random walk process are not independent over time
To check for a white noise process in practice, you can use the following teststatistic First, calculate the first-order autocorrelation coefficient, ˆρ1, from the T time-series observations Then find Z = ˆρ1
√
T If the absolute value of Z is greater
than two (|Z| > 2), conclude that the observations do not come from a white noiseprocess If you reach this conclusion, then you must decide how best to model thetime series if you want to use Crystal Ball to generate potential future values of thetime series The rest of this chapter describes some models for you to consider Thereare many models that might be applied, but we show a few of the more popularmodels for generating future values of financial times series with Crystal Ball.Selected models for simulating financial time series are popular because of some
‘‘stylized facts’’ recognized by finance practitioners, and listed in McNeil, Frey, andEmbrechts (2005) For series of daily returns, exchange rates, and commodity prices:
■ Return series are not IID although they show little serial correlation
■ Conditional expected returns are close to zero
■ Volatility appears to vary over time
■ Return series are leptokurtic or heavy-tailed
■ Extreme returns appear in clusters
Trang 8Varying volatility and clustered, leptokurtic returns can be modeled with some type
of mixture model The remainder of this chapter describes some models that can beincorporated in risk analysis spreadsheet models
ADDITIVE RANDOM WALK WITH DRIFT
The model for an additive random walk with drift is
a white noise process
Generating Values from a Scalar Random Walk with Drift
Process
To simulate potential future values of a time series that you think follows an additiverandom walk with drift process, take the first differences of the time series and fit aCrystal Ball assumption to them This is illustrated in Figure 11.7
The values of the time series Y t for t = 1, 2, , 20 in cells B4:B23 of
RandomWalkWithDrift.xls are quarterly sales of an industrial product The firstdifferences are found by entering =B5-B4 in Cell C5 and copying this formula downthrough cell C23 The autocorrelation coefficient of the first differences is calculated
in cell D4 as 0.184, which is smaller than the two-standard-error value of 0.447calculated in cell D7 This, combined with the apparent statistical stationarity wesee in the time series plot of the differences in Figure 11.7 lets us conclude that thedifferences can be modeled with Crystal Ball as though they are IID
To generate potential future values of the sales time series, we used CrystalBall’s distribution-fitting procedure to fit a Triangular(-69.54,17.99,100.93) distri-bution to the values in cells C5:C23 and used that distribution to specify CrystalBall assumptions in cells C25:C29 The values in B25:B29 are calculated usingExpression 11.3 Cell B25 has the formula =B23+C25 Cell B26 has the formula
=B25+C26, and this was copied and pasted to cells B27 and B28
You can forecast as many steps ahead as desired using the random walkmodel, but realize that in doing so you are assuming implicitly that the distributiongenerating the differences remains stationary over the future period for which yougenerate values The adequacy of this assumption depends on the context It maywell be adequate for a few steps ahead, but the variance of the random walk modelincreases linearly with time, so for prolonged use of the model you will want to updatethe model by fitting distributions to the new data value changes as you observe them
Trang 9Simulating Financial Time Series 155
FIGURE 11.7 Crystal Ball model on the ‘‘Scalar Random Walk’’
worksheet of RandomWalkWithDrift.xls for forecasting a timeseries with a random walk with drift process Cells C25:C29 areCrystal Ball assumptions, and B25:B29 are Crystal Ballforecasts Note that rows 8 through 22 are hidden
Forecasting with Vector Random Walk Model
You can also use the random walk model to simulate observations from time seriesthat have both autocorrelation and correlation between series This is illustrated inFigure 11.8 for the sales of three industrial products labeled X, Y, and Z in columns
B, C, and D The procedure for forecasting more than one (that is, a vector) timeseries is similar to forecasting a single (scalar) time series However, with a vectorrandom walk model, we take into account the correlation between changes in timeseries at the same time period as well as using the random walk model to induceautocorrelation among the levels of the time series
Trang 10FIGURE 11.8 Crystal Ball model on the ‘‘VectorRandom Walk’’ worksheet of
RandomWalkWithDrift.xlsfor forecasting a vector timeseries with a random walk with drift process
Cells E29:G33 are Crystal Ball assumptions, andB29:D33are Crystal Ball forecasts Note that rows 19through 27 are hidden
In cells E5:G28 of Figure 11.8, we found the first differences of the X, Y, and
Ztime series in cells B4:B28 The autocorrelations in cells E35:G35 indicate thatthe differences follow a random process Using Crystal Ball’s Batch Fit feature, wemodeled the changes in X, Y, and Z as Normal distributions with parameters thatyou will find in the file Figure 11.9 shows the correlation matrix for the changes incells M11:O13
Again, you can forecast as many steps ahead as desired using the vector randomwalk model, but realize that you are assuming implicitly that the random processesgenerating the differences remain stationary in regard to their distributions and theircross correlations
Trang 11Simulating Financial Time Series 157
FIGURE 11.9 Information generated byCrystal Ball’s Batch Fit tool on the firstdifferences of the X, Y, and Z times series infile RandomWalkWithDrift.xls
MULTIPLICATIVE RANDOM WALK MODEL
If the time series of returns on a financial asset are IID, then we can use amultiplicative model to generate potential future prices of the asset This is illustrated
in Figure 11.10, which has data obtained from finance.yahoo.com Cells B8:B170hold the monthly adjusted closing prices of the exchange traded fund (ETF) based
on the Standard & Poor’s 500 Composite Stock Price Index with sticker symbol
SPY Denote these prices as S t for t = 1, , 163 Note that the historical prices are listed in reverse chronological order, with S163 in cell B8, down to S1 in
cell B170 In cells C8:C169, we have calculated the gross returns R t = S t /S t−1 for
12× 4.09 = 14.17 percent.
Cell G13 is defined as a Crystal Ball forecast, and its chart is shown inFigure 11.11 Using our methodology, a 95 percent certainty interval for the price