4 Similarities between equities, currencies, commodities and indices
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the random behavior of assets Chapter 3
8
Perez Companc
7
6
5
4
3
2
1
0
20-Feb-95 31-May-95 08-Sep-95 17-Dec-95 26-Mar-96
04-Jul-96
12-Oct-96
Figure 3.2 Perez Companc from February 1995 to November 1996.
Remembering that the returns are more important to us than the absolute level of the asset
price, I show in Figure 3.3 how to calculate returns on a spreadsheet. Denoting the asset value
on the ith day by Si , then the return from day i to day i + 1 is given by
Si+1 − Si
= Ri .
Si
(I’ve ignored dividends here, they are easily allowed for, especially since they only get paid
two or four times a year typically.) Of course, I didn’t need to use data spaced at intervals of
a day, I will comment on this later.
In Figure 3.4 I show the daily returns for Perez Companc. This looks very much like ‘noise,’
and that is exactly how we are going to model it.
The mean of the returns distribution is
R=
1
M
M
Ri
(3.2)
(Ri − R)2 ,
(3.3)
i=1
and the sample standard deviation is
1
M −1
M
i=1
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Part One mathematical and ﬁnancial foundations
Date
01-Mar-95
Perez
2.11
Return
02-Mar-95
03-Mar-95
1.90
2.18
−0.1
0.149906
−0.01081
−0.11258
06-Mar-95
2.16
07-Mar-95
08-Mar-95
09-Mar-95
1.91
1.86
1.97
10-Mar-95
2.27
0.15
13-Mar-95
14-Mar-95
2.49
2.76
0.099874
0.108565
15-Mar-95
16-Mar-95
2.61
2.67
0.021858
17-Mar-95
2.64
−0.0107
20-Mar-95
21-Mar-95
2.60
2.59
−0.01622
−0.00275
22-Mar-95
2.59
−0.00275
23-Mar-95
24-Mar-95
2.55
2.73
−0.01232
0.069307
27-Mar-95
2.91
0.064815
28-Mar-95
29-Mar-95
2.92
2.92
0.002899
0
30-Mar-95
31-Mar-95
3.12
3.14
0.069364
0.005405
03-Apr-95
04-Apr-95
3.13
3.24
0.037736
05-Apr-95
3.25
0.002597
06-Apr-95
3.28
0.007772
07-Apr-95
10-Apr-95
11-Apr-95
12-Apr-95
3.21
3.02
3.08
3.19
−0.02057
−0.06037
17-Apr-95
3.21
18-Apr-95
19-Apr-95
3.17
3.24
Average return
Standard deviation
0.002916
0.024521
=AVERAGE(C3:C463)
−0.02985
0.061538
=STDEVP(C3:C463)
−0.05426
=(B13-B12)/B12
−0.00269
0.019553
0.035616
0.007936
−0.01312
0.021277
Figure 3.3 Spreadsheet for calculating asset returns.
where M is the number of returns in the sample (one fewer than the number of asset prices).
From the data in this example we ﬁnd that the mean is 0.002916 and the standard deviation is
0.024521.
Notice how the mean daily return is much smaller than the standard deviation. This is very
typical of ﬁnancial quantities over short timescales. On a day-by-day basis you will tend to
see the noise in the stock price, and will have to wait months perhaps before you can spot the
trend.
The frequency distribution of this time series of daily returns is easily calculated, and very
instructive to plot. In Excel use Tools | Data Analysis | Histogram. In Figure 3.5 is shown the
frequency distribution of daily returns for Perez Companc. This distribution has been scaled
and translated to give it a mean of zero, a standard deviation of one and an area under the
curve of one. On the same plot is drawn the probability density function for the standardized
Normal distribution function
1 2
1
√ e− 2 φ ,
2π
where φ is a standardized Normal variable. The two curves are not identical but are fairly close.
the random behavior of assets Chapter 3
0.2
Perez Companc returns
0.15
0.1
0.05
20-Jan-97
12-Oct-96
04-Jul-96
26-Mar-96
17-Dec-95
08-Sep-95
31-May-95
−0.05
20-Feb-95
0
−0.1
−0.15
Figure 3.4 Daily returns of Perez Companc.
Probability
0.7
0.6
Perez returns
Normal
0.5
0.4
0.3
0.2
0.1
−4.5
−3.5
−2.5
−1.5
0
−0.5
0.5
Return (scaled)
1.5
2.5
3.5
4.5
Figure 3.5 Normalized frequency distribution of Perez Companc and the standardized Normal
distribution.
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Part One mathematical and ﬁnancial foundations
Figure 3.6 Glaxo–Wellcome returns histogram. Source: Bloomberg L.P.
Supposing that we believe that the empirical returns are close enough to Normal for this to
be a good approximation, then we have come a long way towards a model. I am going to write
the returns as a random variable, drawn from a Normal distribution with a known, constant,
non-zero mean and a known, constant, non-zero standard deviation:
Ri =
Si+1 − Si
= mean + standard deviation × φ.
Si
Figure 3.6 shows the returns distribution of Glaxo–Wellcome as calculated by Bloomberg.
This has not been normalized.
3.6 TIMESCALES
How do the mean and standard deviation of the returns’ time series, as estimated by (3.2) and
(3.3), scale with the time step between asset price measurements? In the example the time step
is one day, but suppose I sampled at hourly intervals or weekly, how would this affect the
distribution?
the random behavior of assets Chapter 3
Call the time step δt. The mean of the return scales with the size of the time step. That is,
the larger the time between sampling the more the asset will have moved in the meantime, on
average. I can write
mean = µ δt,
for some µ which we will assume to be constant. In the Perez Companc example we had a
mean of 0.002916 over a timescale of one day, δt = 1/252 so that
à = 252 ì 002916 = 0.735 = 73.5%.
Ignoring randomness for the moment, our model is simply
Si+1 − Si
= µ δt.
Si
Rearranging, we get
Si+1 = Si (1 + µ δt).
If the asset begins at S0 at time t = 0 then after one time step t = δt and
S1 = S0 (1 + µ δt).
After two time steps t = 2 δt and
S2 = S1 (1 + µ δt) = S0 (1 + µ δt)2 ,
and after M time steps t = M δt = T and
SM = S0 (1 + µ δt)M .
This is just
SM = S0 (1 + µ δt)M = S0 eM log(1+µ δt) ≈ S0 eµM δt = S0 eµT .
In the limit as the time step tends to zero with the total time T ﬁxed, this approximation
becomes exact. This result is important for two reasons.
First, in the absence of any randomness the asset exhibits exponential growth, just like cash
in the bank.
Second, the model is meaningful in the limit as the time step tends to zero. If I had chosen
to scale the mean of the returns distribution with any other power of δt it would have resulted
in either a trivial model (ST = S0 ) or inﬁnite values for the asset (ST = ±∞).
The second point can guide us in the choice of scaling for the random component of the
return. How does the standard deviation of the return scale with the time step δt? (Recall that
you add variances not standard deviations.) Again, consider what happens after T /δt time steps
each of size δt (i.e. after a total time of T ). Inside the square root in expression (3.3) there
are a large number of terms, T /δt of them. In order for the standard deviation to remain ﬁnite
as we let δt tend to zero, the individual terms in the expression must each be of O(δt). Since
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Part One mathematical and ﬁnancial foundations
each term is a square of a return, the standard deviation of the asset return over a time step δt
must be O(δt 1/2 ):
standard deviation = σ δt 1/2 ,
where σ is some parameter measuring the amount of randomness; the larger this parameter
the more uncertain is the return. For the moment let’s assume that it is constant. For Perez
Companc we have a standard deviation of 0.024521 over one day so that
σ =
√
252 × 0.024521 = 0.389 = 38.9%.
Putting these scalings explicitly into our asset return model
Ri =
Si+1 − Si
= µ δt + σ φ δt 1/2 .
Si
(3.4)
I can rewrite Equation (3.4) as
Si+1 − Si = µSi δt + σ Si φ δt 1/2 .
(3.5)
The left-hand side of this equation is the change in the asset price from time step i to time
step i + 1. The right-hand side is the ‘model.’ We can think of this equation as a model for a
random walk of the asset price. This is shown schematically in Figure 3.7. We know exactly
where the asset price is today but tomorrow’s value is unknown. It is distributed about today’s
value according to (3.5).
S
Asset tomorrow
Asset today
Distribution of asset
price change
t
Figure 3.7 A representation of the random walk.
the random behavior of assets Chapter 3
3.6.1
The Drift
The parameter µ is called the drift rate, the expected return or the growth rate of the asset.
Statistically it is very hard to measure since the mean scales with the usually small parameter
δt. It can be estimated by
µ=
1
M δt
M
Ri .
i=1
The unit of time that is usually used is the year, in which case µ is quoted as an annualized
growth rate.
In the classical option pricing theory the drift plays almost no role. So even though it is hard
to measure, this doesn’t matter too much.1
3.6.2
The Volatility
The parameter σ is called the volatility of the asset. It can be estimated by
1
(M − 1) δt
M
(Ri − R)2 .
i=1
Again, this is almost always quoted in annualized terms.
The volatility is the most important and elusive quantity in the theory of derivatives. I will
come back again and again to its estimation and modeling.
Because of their scaling with time, the drift and volatility have different effects on the asset
path. The drift is not apparent over short timescales for which the volatility dominates. Over
long timescales, for instance decades, the drift becomes important. Figure 3.8 is a realized path
of the logarithm of an asset, together with its expected path and a ‘conﬁdence interval.’ In
this example the conﬁdence interval represents one standard deviation. With the assumption of
Normality this means that 68% of the time the asset should be within this range. The mean
path is growing linearly in time and the conﬁdence interval grows like the square root of time.
Thus over short timescales the volatility dominates.2
3.7
ESTIMATING VOLATILITY
The most common estimate of volatility is simply
1
(M − 1) δt
M
(Ri − R)2 .
i=1
If δt is sufﬁciently small the mean return R term can be ignored.
For small δt
1
(M − 1) δt
M
(log S(ti ) − log S(ti−1 ))2
i=1
can also be used, where S(ti ) is the closing price on day ti .
1
2
In non-classical theories and in portfolio management, it does often matter, very much.
Why did I take the logarithm? Because changes in the logarithm are related to the return on the asset.
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Part One mathematical and ﬁnancial foundations
One standard
deviation above and
below the mean
Stock price
Mean
Time
Figure 3.8 Path of the logarithm of an asset, its expected path and one standard deviation above
and below.
100%
90%
80%
70%
Volatility estimate
66
60%
50%
40%
30%
20%
10%
0%
Time
Figure 3.9 The plateauing effect when using a moving window volatility estimate.
the random behavior of assets Chapter 3
It is highly unlikely that volatility is constant in time. Changing economic circumstances,
seasonality etc. will inevitably result in volatility changing with time. If you want to know the
volatility today you must use some past data in the calculation. Unfortunately, this means that
there is no guarantee that you are actually calculating today’s volatility.
Typically you would use daily closing prices to work out daily returns and then use the
past 10, 30, 100, . . . daily returns in the formula above. Or you could use returns over longer
or shorter periods. Since all returns are equally weighted, while they are in the estimate of
volatility, any large return will stay in the estimate of volatility until the 10 (or 30 or 100) days
have passed. This gives rise to a plateauing of volatility, and is totally spurious.
In Figure 3.9 is shown the spurious plateauing effect associated with a sudden large drop in
a stock price.
Since volatility is not directly observable, and because of the plateauing effect in the simple
measure of volatility, you might want to use other volatility estimates. We’ll see some more in
Chapter 49.
3.8
THE RANDOM WALK ON A SPREADSHEET
The random walk (3.5) can be written as a ‘recipe’ for generating Si+1 from Si :
Si+1 = Si 1 + µ δt + σ φ δt 1/2 .
(3.6)
We can easily simulate the model using a spreadsheet. In this simulation we must
input several parameters, a starting value for the asset, a time step δt, the drift rate µ,
the volatility σ and the total number of time steps. Then, at each time step, we must
choose a random number φ from a Normal distribution. I will talk about simulations
in depth in Chapter 80dir , for the moment let me just say that an approximation
to a Normal variable that is fast in a spreadsheet, and quite accurate, is simply to add up
twelve random variables drawn from a uniform distribution over zero to one, and subtract six:
12
RAND() − 6.
i=1
The Excel spreadsheet function RAND() gives a uniformlydistributed random variable.
In Figure 3.10 I show the details of a spreadsheet used for
simulating the asset price random walk.
3.9
THE WIENER PROCESS
So far we have a model that allows the asset to take any value after a time step. This is some
progress but we have still not reached our goal of continuous time, we still have a discrete time
step. This section is a brief introduction to the continuous-time limit of equations like (3.4).
I will start to introduce ideas from the world of stochastic modeling and Wiener processes,
delving more deeply in Chapter 4.
I am now going to use the notation d· to mean ‘the change in’ some quantity. Thus dS is
the ‘change in the asset price.’ But this change will be in continuous time. Thus we will go to
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Part One mathematical and ﬁnancial foundations
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A
Asset
Drift
Volatility
Timestep
B
C
E
F
G
Asset
0
100
0.01 98.38844
0.02 94.28005
0.03 95.40441
0.04 92.79735
=D4+$B$4
0.05 93.45168
0.06 93.99664
0.07 97.66597
0.08 96.52319
=E7*(1+$B$2*$B$4+$B$3*SQRT($B$4)*(RAND()+RAND()+RAND()+RAND()
0.09 99.58417
+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()-6))
0.1 95.77222
0.11 99.60075
0.12 99.01974
0.13 100.8729
0.14 101.2378
0.15 102.4736
0.16 102.7694
0.17 100.7347
0.18 102.7021
0.19 107.3493
0.2 109.887
0.21 108.688
0.22 110.7826
0.23 112.8932
0.24 111.0625
0.25 111.6157
0.26 112.5443
0.27 111.9805
0.28 115.6002
0.29 117.9831
0.3 115.2694
0.31 117.4374
100
0.15
0.25
0.01
D
Time
H
Figure 3.10 Simulating the random walk on a spreadsheet.
the limit δt = 0. The ﬁrst δt on the right-hand side of (3.5) becomes dt but the second term is
more complicated.
I cannot straightforwardly write dt 1/2 instead of δt 1/2 . If I do go to the zero-time step limit
then any random dt 1/2 term will dominate any deterministic dt term. Yet in our problem the
factor in front of dt 1/2 has a mean of zero, so maybe it does
not outweigh the drift after all. Clearly something subtle is
happening in the limit.
It turns out, and we will see this in Chapter 4, that because the
variance of the random term is O(δt) we can make a sensible
continuous-time limit of our discrete-time model. This brings
us into the world of Wiener processes.
I am going to write the term φ δt 1/2 as
dX.
the random behavior of assets Chapter 3
You can think of dX as being a random variable, drawn from a Normal distribution with
mean zero and variance dt:
E[dX] = 0 and E[dX2 ] = dt.
This is not exactly what it is, but it is close enough to give the right idea. This is called a
Wiener process. The important point is that we can build up a continuous-time theory using
Wiener processes instead of Normal distributions and discrete time.
3.10 THE WIDELY ACCEPTED
MODEL FOR EQUITIES,
CURRENCIES, COMMODITIES
AND INDICES
Our asset price model in the continuous-time limit, using the
Wiener process notation, can be written as
dS = µS dt + σ S dX.
(3.7)
This is our ﬁrst stochastic differential equation. It is a continuous-time model of an asset
price. It is the most widely accepted model for equities, currencies, commodities and indices,
and the foundation of so much ﬁnance theory.
We’ve now built up a simple model for equities that we are going to be using quite a lot.
You could ask, if the stock market is so random how can fund managers justify their fee? Do
they manage to outsmart the market? Are they clairvoyant or aren’t the markets random? Well,
I won’t swear that markets are random but I can say with conﬁdence that fund managers don’t
outperform the market. In Figure 3.11 is shown the percentage of funds that outperform an
index of all UK stocks. Whether we look at a one-, three-, ﬁve- or 10-year horizon we can see
that the vast majority of funds can’t even keep up with the market. And statistically speaking,
there are bound to be a few that beat the market, but only by chance. Maybe one should invest
in a fund that does the opposite of all other funds. Great idea except that the management fee
and transaction costs probably mean that that would be a poor investment too. This doesn’t
prove that markets are random, but it’s sufﬁciently suggestive that most of my personal share
exposure is via an index-tracker fund.
3.11 SUMMARY
In this chapter I introduced a simple model for the random walk of asset. Initially I built the
model up in discrete time, showing what the various terms mean, how they scale with the time
step and showing how to implement the model on a spreadsheet.
Most of this book is about continuous-time models for assets. The continuous-time version
of the random walk involves concepts such as stochastic calculus and Wiener processes. I
introduced these brieﬂy in this chapter and will now go on to explain the underlying theory of
stochastic calculus to give the necessary background for the rest of the book.
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