1 Correlation, Co-Integration and Multi-Factor Models
Tải bản đầy đủ - 0trang
142
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
with orthogonal increments dz1 and dz2 . The coefﬁcient of linear correlation between x1
and x2 , ρ12 , is
ρ12 =
E[dx1 dx2 ]
E[dx12 ]E[dx22 ]
σ11 σ21
=
(σ11 σ21 )2 + (σ11 σ22 )2
For simplicity set the drift terms to zero, and consider the new variable y = x1 + λx2 :
dy = σ11 dz1 + λ (σ21 dz1 + σ22 dz2 )
= (σ11 + λσ21 ) dz1 + λσ22 dz2
(5.3)
In general the variable y will be normally distributed, with zero mean and instantaneous
variance var(y) = (σ11 + λσ21 )2 + (λσ22 )2 dt. After a ﬁnite time t, its distribution will
be
y ∼ N (0, [(σ11 + λσ21 )2 + (λσ22 )2 ]t)
(5.4)
Now choose λ such that the coefﬁcient in dz1 for the increment dy is zero:
λ=−
σ11
σ21
(5.5)
It is easy to see that this value of λ gives the lowest possible variance (dispersion) to the
variable y:
2
y ∼ N (0, λ2 σ22
t),
λ=−
σ11
σ21
(5.6)
If λ were equal to −1 the variable y would simply give the spread between x1 and x2 ,
and in this case
y ∼ N (0, [(σ11 − σ21 )2 + (σ22 )2 ]t),
λ = −1
(5.7)
Equation (5.7) tells us that, no matter how strong the correlation between the two variables
might be, as long as it is not 1, the variance of their spread, or, for that matter, of any
linear combination between them, will grow indeﬁnitely over time. Therefore, a linear
correlation coefﬁcient does not provide a mechanism capable of producing long-term
‘cohesion’ between diffusive state variables.
Sometimes this is perfectly appropriate. At other times, however, we might believe
that, as two stochastic variables move away from each other, there should be ‘physical’
(ﬁnancial) mechanisms capable of pulling them together. This might be true, for instance,
for yields or forward rates. Conditional on our knowing that, say, the 9.5-year yield in 10
years’ time is at 5.00%, we would expect the 10-year yield to be ‘not too far apart’, say,
somewhere between 4.50% and 5.50%. In order to achieve this long-term effect by means
of a correlation coefﬁcient we might be forced to impose too strong a correlation between
the two yields for the short-term dynamics between the two variables to be correct. Or,
conversely, a correlation coefﬁcient calibrated to the short-term changes in the two yields
5.1 CORRELATION, CO-INTEGRATION AND MULTI-FACTOR MODELS
143
is likely to predict a variance for the difference between the two yields considerably
higher than what we might consider reasonable.
In order to describe a long-term link between two variables (or, indeed, a collection of
variables) we require a different concept, namely co-integration. In general, co-integration
occurs when two time series are each integrated of order b, but some linear combination
of them is integrated of order a < b. Typically, for ﬁnance applications, b = 1 and
a = 0. (For a discussion of co-integration see Alexander (2001) for a simple introduction,
or Hamilton (1994) for a more thorough treatment.) In a diffusive context, this means
that the spread(s) between the co-integrated variables is of the mean-reverting type. More
generally, Granger (1986) and Engle and Granger (1987) have shown that if a set of
variables is co-integrated an error-correction model, i.e. a process capable of pulling
them together, must be present among them.
Why is this relevant in the context of our discussion? Let us lay down the ‘facts’ in
order.
1. Several studies suggest that forward rates of a given yield curve should be cointegrated. See, for example, Alexander (2001). Let us accept this as a fact.
2. In the real world, if these forward rates are co-integrated, they will not disperse ‘too
much’ relative to each other, even after very long time periods.
3. Just positing a correlation structure among the forward rates is not capable of providing the necessary long-term cohesion mechanism among forward rates in a manner
consistent with their short-term dynamics.
4. In a diffusive setting, in order to describe simultaneously the short-term correlation
and the long-term co-integration among forward rates one must introduce errorcorrection (mean-reverting) terms.
These statements are all true in the real world. They would seem to suggest that, even
if individually the forward rates followed exactly a diffusive process, in order to price a
long-dated option it would be inadequate to describe the nature of their co-dependence
simply by means of a correlation matrix. This statement however is not necessarily correct.
Even if the underlying forward rates are indeed co-integrated, and even if a correlated
diffusion disperses them far too much when compared with the real world, this does not
matter for option pricing because, in the diffusive setting, the error-correction (meanreverting) term appears in the drift of the state variables. And we have seen in Chapter 4,
the real-world drift does not matter (in a perfect Black-and-Scholes world) for option
pricing.1
This important caveat is the exact counterpart of the statement proved in Section 4.7
that quadratic variation, and not variance, is what matters for single-asset Black-andScholes option pricing. In a single-asset case, what matters are the local ‘vibrational’
properties of the underlying, not how much it will disperse after a ﬁnite period of time.
Similarly, in a multi-factor setting, it is guessing the relationship between the local vibrations of the underlying assets relative to each other that will allow the trader to set up a
successful hedging strategy, not estimating their long-term relative dispersion. In a Blackand-Scholes world, option traders do not engage in actuarial pricing (for which variance
and long-term relative dispersion do matter), but do engage in local riskless replication.
1 See
also the discussion in Sections 18.2 and 18.3.
144
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
Therefore in what follows I will focus on the correlation matrix as the only mechanism
necessary to describe the link between a set of underlying variables, even if we know
that in the real world this might not be appropriate. In particular, we will consider timedependent volatilities and an imperfect correlation as the only mechanisms relevant for
derivatives pricing to produce changes in the shape of a given yield curve. This topic is
addressed in the next section.
5.1.1 The Multi-Factor Debate
Historically, the early response to the need of creating mechanisms capable of producing changes in the shape of the yield curve was to invoke multi-factor models. Taking
inspiration from the results of principal component analysis (PCA), the early (one-factor)
models were seen as capable of moving the ﬁrst eigenvector (the level of the curve), but
ineffectual in achieving a change in slope, curvature, etc. At least two or three independent
modes of deformation were therefore seen as necessary to produce the required degree of
shape change in the yield curve.
The PCA eigenmodes are orthogonal by construction. When the ‘reference axes’ of
the eigenvectors are translated from the principal components back to the forward rates,
however, the latter become imperfectly instantaneously correlated. Therefore the need
to produce a change in shape in the yield curve made early modellers believe that the
introduction of several, imperfectly correlated, Brownian shocks would be the solution to
the problem. Furthermore, Rebonato and Cooper (1995) showed that, in order to model
a ﬁnancially convincing instantaneous correlation matrix, a surprisingly large number of
factors was required. Therefore, the reasoning went, to produce what we want (changes
in the shape of the yield curve) we require a sufﬁciently rich and realistic instantaneous
correlation matrix, and this, in turn, requires many Brownian factors. In the early-to-mid1990s high-dimension yield-curve models became, in the eyes of traders and researchers,
the answer and the panacea to the pricing problems of the day.
To some extent this is correct. Let us not lose sight, however, of what we are trying
to achieve, and let us not confuse our goals with (some of) the means to achieve them.
The logical structure of the problem is schematically as follows.
• The introduction of complex derivatives payoffs highlights the need for models that
allow the yield curves to change shape. This is a true statement, and a model that
allows changes in shape in the yield curve is our goal.
• If one works with instantaneously imperfectly correlated forward rates, the yield
curve will change shape over time. This is also a true statement.
• If we want to recover a ﬁnancially convincing instantaneous correlation structure,
many factors are needed. This, too, is a true statement.
• Therefore the imperfect correlation among rates created by a many-factor model is
what we require in order to obtain our goal. This does not necessarily follow.
The last step, in fact, implies that an imperfect degree of instantaneous correlation
is the only mechanism capable of producing signiﬁcant relative moves among rates.
But this is not true. Imposing an instantaneous correlation coefﬁcient less than one can
certainly produce some degree of independent movement in a set of variables that follows
5.1 CORRELATION, CO-INTEGRATION AND MULTI-FACTOR MODELS
145
a diffusion. But this is neither the only, nor, very often, the most ﬁnancially desirable
tool by means of which a change in the shape of the yield curve can be achieved. What
else can produce our goal (the change in shape in the yield curve)? I show below that
introducing time-dependent volatilities can constitute a powerful, and often ﬁnancially
more desirable, alternative mechanism in order to produce the same effect.
If this is the case, in the absence of independent ﬁnancial information it is not clear
which mechanism one should prefer. Indeed, the debate about the number of factors
‘really’ needed for yield-curve modelling was still raging as late as 2002: Longstaff et al.
(2000a) argue that a billion dollars are being thrown away in the swaption market by
using low-dimensionality models; Andersen and Andreasen (2001) rebut that even a onefactor model, as long as well implemented, is perfectly adequate; Joshi and Theis (2002)
provide an alternative perspective on the topic. Without entering this debate (I express
my views on the matter in Rebonato (2002)), in the context of the present discussion one
can say that the relative effectiveness and the ﬁnancial realism of the two mechanisms
that can give rise to changes in the shape of the yield curve must be carefully weighed
in each individual application.
It therefore appears that our goal (i.e. the ability to produce a change in the shape
of the yield curve) can be produced by different combinations of the two mechanisms
mentioned so far. If this is the case, perhaps different ‘amounts’ of de-correlation and of
time variation of the volatility can produce similar effects, in so far as changes in the shape
of the yield curve are concerned. Is there a quantitative measure of how successful we
have been in creating the desired effect, whatever means we have employed ? Surely, this
indicator cannot be the coefﬁcient of instantaneous correlation, because, as stated above
and proved later in this chapter, a substantial de-coupling between different portions of
the yield curve can occur even when we use a one-factor model. The answer is positive,
and the quantity that takes into account the combined effects of time-dependent volatilities
and of imperfect correlations is the terminal correlation, which I deﬁne and discuss in
Section 5.3. In the following sections I will therefore tackle the following topics:
• I will draw a distinction between instantaneous and terminal correlation.
• I will show that it is the terminal and not the instantaneous correlation that matters
for derivatives pricing.
• I will then proceed to show how a non-constant instantaneous volatility can give rise
to substantial terminal de-correlation among the underlying variables even when the
instantaneous correlation is very high (or, indeed, perfect).
• Having explained why we need a judicious combination of time-dependent volatilities and imperfect instantaneous correlation, I will discuss how a naăve Monte
Carlo simulation can be carried out in this setting.
• Finally, I will show how a conceptually more satisfactory and computationally
more efﬁcient Monte Carlo valuation can be carried out in this time-dependent
environment.
The last point has an intrinsic (i.e. computational) interest. It is even more important,
however, for its conceptual relevance. Since this book deals only in passing with numerical
techniques, the main reason for discussing it in this chapter is that it shows with great
146
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
clarity how terminal correlation enters the pricing of derivatives products, and how the
role of instantaneous correlation is, in a way, secondary. Again, instantaneous correlation
is just one of the mechanisms available to produce what we really want (namely, terminal
correlation).
5.2
The Stochastic Evolution of Imperfectly
Correlated Variables
In order to get a qualitative feel for the problem, let us begin by considering the evolution
of two log-normally distributed quantities (rates or prices), that we shall denote by x1 and
x2 , respectively:
dx1
= µ1 dt + σ1 (t) dw1
x1
dx2
= µ2 dt + σ2 (t) dw2
x2
(5.8)
(5.9)
The two Brownian increments, dw1 and dw2 , are assumed to be imperfectly correlated,
and we will therefore write
E[dw1 dw2 ] = ρ dt
(5.10)
Note that we have explicitly allowed for the possibility of time dependence in the two
volatilities. Also, we have appended an index (1 or 2) to the volatility symbol to emphasize
that by σ1 (t) and σ2 (t) we denote the volatility of different prices or rates at the same
time, t (and not the volatility of the same spot process at different times). Let us then
choose a ﬁnal time horizon, T , and let us impose that the unconditional variance of each
variable over this time horizon should be exactly the same:
T
0
T
σ1 (u)2 du =
σ2 (u)2 du = σ 2 T
(5.11)
0
For the sake of simplicity, let us ﬁnally assume that both variables start at time zero from
the same value:
x1 (0) = x2 (0) = x0
(5.12)
Let us now run two Monte Carlo simulations of the stock price processes out to time
T . For the ﬁrst simulation, we will use constant volatilities for σ1 and σ2 (and therefore,
given requirement (5.11), the volatility is the same for x1 and x2 and equal to σ for both
variables), but a coefﬁcient of instantaneous correlation less than one. When running the
second simulation, we will instead use two time-dependent instantaneous volatilities as in
Equations (5.8) and (5.9) (albeit constrained by Equation (5.11)), but perfect correlation.
To be more speciﬁc, for the ﬁrst variable, x1 , we will assign the instantaneous volatility
σ1 (t) = σ0 exp(−νt),
0≤t ≤T
(5.13)
5.2 THE STOCHASTIC EVOLUTION
147
with σ0 = 20%, ν = 0.64 and T = 4 years. The second variable, x2 , will have an
instantaneous volatility given by
σ2 (t) = σ0 exp[−ν(T − t)],
0≤t ≤T
(5.14)
The reader can check that the variance Equation (5.11) is indeed satisﬁed.
Having set up the problem in this manner, we can simulate the joint processes for
x1 and x2 in the two different universes. In the time-dependent case the two variables
were subjected to the same Brownian shocks. In the constant-volatility case two different Brownian shocks, correlated as per Equation (5.10), were allowed to shock the two
variables. After running a large number of simulations, we ignored our knowledge of the
volatility of the two processes and evaluated the correlation between the changes in the
logarithms of the two variables in the two simulations using the expression
i
ρ1,2 =
i
ln x1i − ln x 1
ln x1i − ln x 1
ln x2i − ln x 2
2
i
ln x2i − ln x 2
(5.15)
2
The results of these trials are shown in Figures 5.1 and 5.2.
Looking at these two ﬁgures, it is not surprising to discover that the same sample
correlation (in this case ρ1,2 = 34.89%) was obtained despite the fact that the two decorrelation-generating mechanisms were very different.
For clarity of exposition, in this stylized case a very strongly time-varying volatility
was assigned to the two variables. It is therefore easy to tell which ﬁgure is produced by which mechanism: note how in Figure 5.1 the changes for Series 1 are large
0.06
0.04
0.02
Change
0
−0.02
0
5
10
15
20
25
30
Series1
Series2
−0.04
−0.06
−0.08
−0.1
−0.12
−0.14
Time-step
Figure 5.1 Changes in the variables x1 and x2 . The two variables were subjected to the same
random shocks (instantaneous correlation = 1). The ﬁrst variable (Series 1) had an instantaneous
volatility given by σ1 (t) = σ0 exp(−νt), 0 ≤ t ≤ T , with σ0 = 20%, ν = 0.64 and T = 4 years.
The second variable (Series 2) had an instantaneous volatility given by σ2 (t) = σ0 exp[−ν(T − t)],
0 ≤ t ≤ T . The empirical sample correlation turned out to be 34.89%.
148
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
0.15
0.1
Change
0.05
0
0
5
10
15
20
25
30
−0.05
Series1
Series2
−0.1
−0.15
−0.2
Time-step
Figure 5.2 Changes in the variables x1 and x2 . The two variables were subjected to different
random shocks (instantaneous correlation = 35.00%). Both variables had the same constant instantaneous volatility of σ0 = 20%. The empirical sample correlation turned out to be 34.89%.
0.12
0.1
0.08
Change
0.06
0.04
0.02
Series1
Series2
0
−0.02
0
5
10
15
20
25
30
−0.04
−0.06
−0.08
−0.1
Time-step
Figure 5.3 Can the reader guess, before looking below, whether this realization was obtained
with constant volatility and a correlation of 85%, or with a correlation of 90% and a decay constant
ν of 0.2? The sample correlation turned out to be 85%.
at the beginning and small at the end (and vice versa for Series 2), while they have
roughly the same magnitude for the two series in Figure 5.2. In a more realistic case,
however, where the correlation is high but not perfect and the decay factor ν not as
pronounced, it can become very difﬁcult to distinguish the two cases ‘by inspection’. See
Figure 5.3.
5.2 THE STOCHASTIC EVOLUTION
149
Table 5.1 The data used to produce Figure 5.4. Note the greater decrease
in sample correlation produced by the non-constant volatility when the
instantaneous correlation is high.
Decay constant
Instantaneous correlation
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.973944 0.90183
0.837356
0.80604
0.705876 0.675758
0.626561 0.470971 0.475828
0.334509 0.330294 0.332757
0.25877
0.172208 0.173178
0.062323 −0.12066 −0.09665
0.8
0.720101
0.573349
0.338425
0.285563
0.129632
0.091779
In principle, it is of course possible to analyse the two time series separately beforehand in order to establish the possible existence of time dependence in the volatility
function. Armed with this information, the trader could, again in principle, analyse the
joint dynamics of the two variables, and estimate an instantaneous correlation coefﬁcient. In practice, however, these statistical studies are fraught with difﬁculties, and,
especially if the instantaneous volatility is mildly time dependent and the correlation relatively high, the task of disentangling the two effects can be extremely difﬁcult. See
Figure 5.3.
Unfortunately, the case of mildly-varying instantaneous volatilities and of relatively
high instantaneous correlations is the norm rather than the exception when one deals with
the dynamics of forward rates belonging to the same yield curve. The combined effects
of the two de-correlating mechanisms are priced in the relative implied volatilities of caps
and swaptions (see the discussion in Section 10.3), and even relatively ‘stressed’ but still
realistic assumptions for the correlation and volatility produce rather ﬁne differences in
the relative prices (of the order of one to three percentage points – vegas – in implied
volatility).
In order to study the relative importance of the two possible mechanisms to produce decorrelation, Table 5.1 and Figure 5.4 show the sample correlation between log-changes in
the two time series obtained by running many times the simulation experiment described
in the captions to Figures 5.1 and 5.2, with the volatility decay constant (ν) and the instantaneous correlation shown in the table. More precisely, the ﬁrst row displays the sample
correlation obtained for a series of simulations conducted using perfect instantaneous correlation and more and more strongly time-dependent volatilities (decay constants ν of 0.2,
0.4, 0.6 and 0.8); the second row displays the sample correlation obtained with the same
time-dependent volatilities and an instantaneous correlation of 0.8; and so on.
The important conclusion that one can draw from these data is that a non-constant
instantaneous volatility brings about a relatively more pronounced de-correlation when
the instantaneous correlation is high. In particular, when this latter quantity is zero, a nonconstant instantaneous volatility does not bring about any further reduction in the sample
correlation (apart from adding some noise). From these observations one can therefore
conclude that the volatility-based de-correlation mechanism should be of greater relevance
in the case of same-currency forward rates, than in the case of equities or FX rates.
150
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
Sample correlation
0.85
0.65
0.45
0.25
0.8
0.65-0.85
0.45-0.65
0.25-0.45
0.05-0.25
−0.15-0.05
0.6
0.05
0.4
−0.15
1
0.8
0.6
0.4
Instantaneous correlation
0.2
Decay constant
0.2
0
Figure 5.4 The sample correlation evaluated along the path for a variety of combinations of
the instantaneous correlation (from 1 to 0) and of the decay constant ν (from 0.2 to 0.8). Note
how the decrease in sample correlation introduced by the time dependence of volatility is more
pronounced the higher the instantaneous correlation. Regimes of high instantaneous correlations
are more commonly found in the same-currency interest-rate case, than for equities or FX.
In this section I have tried to give a qualitative feel for the impact on the sample
correlation of a time-dependent instantaneous volatility, and of a less-than-perfect instantaneous correlation. At this stage I have kept the discussion at a very qualitative level. In
particular, it is not obvious at this point why the terminal, rather then the instantaneous,
correlation should be of relevance for option pricing. The discussions in Chapters 2 and
4 about volatilities and variances should make us rather cautious before deciding on the
basis of ‘intuition’ which quantities matter when it comes to pricing an option. The purpose of the next section is therefore to identify in a more precise manner the quantities that
affect the joint stochastic evolution of correlated ﬁnancial variables, in so far as option
pricing is concerned.
The analysis will be carried out by considering a ‘thought Monte Carlo experiment’,
but the main focus is more on the conceptual part, rather than on the description of a
numerical technique. In order to carry out and analyse these Monte Carlo experiments we
will have to discretize Ito integrals, and to make use of some basic results in stochastic
integration. These topics are therefore brieﬂy introduced, but in a very non-technical
manner. I have reported some results without proof, and provided very sketchy and handwaving proofs for others. For the reader who would like to study the matter more deeply,
the references provided below can be of assistance. For a clear treatment intermediate
in length between an article and a slim book, I can recommend the course notes by
Shreve (1997). Standard, book-length, references are then Oksendal (1995), Lamberton
and Lapeyre (1991), Neftci (1996) and Baxter and Rennie (1996). If the reader were to
fall in love with stochastic calculus, Karatzas and Shreve (1991) is properly the bible,
but the amount of work required is substantial. Finally, good and simple treatments of
selected topics can be found in Bjork (1998) (e.g. for stochastic integrals) and Pliska
(1997) (e.g. for ﬁltrations).
5.3 THE ROLE OF TERMINAL CORRELATION
5.3
151
The Role of Terminal Correlation in the Joint
Evolution of Stochastic Variables
In what follows we will place ourselves in a perfect Black(-and-Scholes) world. In particular, in addition to the usual assumptions about the market structure, we will require
that the spot or forward underlying variables should be log-normally distributed. As a
consequence, we will ignore at this stage the possibility of any smiles in the implied
volatility. Smile effects are discussed in Part II. Our purpose is to identify what quantities
are essential in order to carry out the stochastic part of the evolution of the underlying
variables. We will obtain the fundamental result that, in addition to volatilities, a quantity that we will call ‘terminal correlation’ will play a crucial role. This quantity will be
shown to be in general distinct from the instantaneous correlation; in particular, it can
assume very small values even if the instantaneous correlation is perfect. In this sense,
the treatment to be found in this section formalizes and justiﬁes the discussion in the
previous sections of this chapter, and in Chapter 2. En route to obtaining these results we
will also indicate how efﬁcient Monte Carlo simulations can be carried out.
In order to obtain these results it is important to recall some deﬁnitions regarding
stochastic integrals. This is undertaken in the next section.
5.3.1 Deﬁning Stochastic Integrals
Let us consider a stochastic process, σ , and a standard Wiener process, Z, deﬁned over an
interval [a b].2 Since σ can be a stochastic process, and not just a deterministic function
of time, one cannot simply require that it should be square integrable. A more appropriate
condition is that the expectation of its square should be integrable over the interval [a b]:
b
a
E[σ 2 (u)] du < ∞
(5.16)
The second requirement that should be imposed on the process σ is that it should be
adapted to the ﬁltration generated by the Wiener process, Z.3 Our goal is to give a
meaning to the expression
b
a
σ (u) dZ(u)
(5.17)
for all functions σ satisfying the two conditions above. The task is accomplished in two
steps.
Step 1: Let us divide the interval [a b] into n subintervals, with t0 = a, . . . , tn−1 = b.
Given this partition of the interval [a b] we can associate to σ a new function, σ , deﬁned
2 The
treatment in this sub-section follows closely Bjork (1998).
intuitive deﬁnition of adaptness is the following. Let S and σ be stochastic processes. If the value of
σ at time t, σt , can be fully determined by the realization of the process S up to time t, then the process σ is
said to be adapted to S. For instance, let S be a stock price process and σ be a volatility function of the form
σ (St , t). Then the value of the volatility σ at time t is completely known if the realization of the stock price,
S, is known, and the volatility process is said to be adapted to the stock price process.
3 An
152
CHAPTER 5 INSTANTANEOUS AND TERMINAL CORRELATION
to be equal to σ on the initial point of each subinterval:
σ (t) = σ (t)
for t = t0 , t1 , . . . , tn−1
(5.18)
and to be piecewise constant over each interval, [tk tk+1 ], with k = 0, 1, . . . , n − 1. If this
is the case, the function σ can be more simply indexed by k, rather than by a continuous
time argument, and can be denoted by the symbol σk : σ (t) = σk (t) = σk for t ∈
[tk tk+1 ], with k = 0, 1, . . . , n − 1.
The elementary Ito integral between a and b, In (a, b), of the function σn (t) is then
deﬁned by the expression
b
In (a, b) =
a
σn (u) dz(u) =
σk [Z(tk+1 ) − Z(tk )]
(5.19)
k=0,n−1
A comment is in order. In deﬁning non-stochastic (e.g. Riemann) integrals, in the limit
it does not matter whether the point in the interval where the function is evaluated is chosen
to be at the beginning, at the end or anywhere else. When dealing with stochastic integrals,
however, the choice of the evaluation point for the function does make a difference, and
the limiting process leads to different results depending on its location. In order to ensure
that a sound ﬁnancial (‘physical’) interpretation can be given to the integral, it is important
that the evaluation point should be made to coincide with the left-most point in the interval.
Step 2: Equation (5.19) deﬁnes a series of elementary integrals, each one associated
with a different index n, i.e. with the number of subintervals [tk tk+1 ] into which we have
subdivided the interval [a b]. If there exists a limit for the sequence of functions In (a, b)
as the integer n tends to inﬁnity, then the Ito integral over the integral [a b], I (t; a, b),
is deﬁned as
I (t; a, b) ≡
b
a
b
σ (u) dz(u) = lim In (a, b) = lim
n→∞
= lim
n→∞
n→∞ a
σn (u)dz(u)
σk [Z(tk+1 ) − Z(tk )]
(5.20)
k=0,n−1
As deﬁned, the Ito integral itself can be regarded as stochastic variable, I . It is therefore
pertinent to ask questions about it such as its expectation or its variance. One can prove
the following:
• The expectation taken at time a of the Ito integral I (t; a, b) is zero:
b
Ea
a
σ (u) dz(u) = 0
(5.21)
• The variance of the Ito integral is linked to the time integral of the expectation of
the square of σ by the following relationship:
b
var
a
σ (u) dz(u) = Ea
b
a
2
σ (u) dz(u)
=
b
a
Ea [σ (u)2 ] du
(5.22)