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0
cov(wg R, [ 1 О· ]AM 0 ОЈв€’1 R)= [ 1 О· ]AM 0 ОЈв€’1 ОЈwg
вЋЎ вЋ¤
0
1
вЋ¦ ОЈв€’1 1 1
= [ 1 О· ]A вЋЈ
10 ОЈв€’1 1
Вµ0
вЋЎ вЋ¤
0 в€’1
1ОЈ 1 1
= [ 1 О· ]A вЋЈ вЋ¦
0 ОЈв€’1 1
Вµ0 ОЈв€’1 1 1
вЋЎ вЋ¤
1 1
= [ 1 О· ]вЋЈ вЋ¦
10 ОЈв€’1 1
0
1
=
10 ОЈв€’1 1
where we use the fact that, by the deп¬Ѓnition of A,
вЋЎ вЋ¤вЋЎ вЋ¤
0 в€’1 0 в€’1
1ОЈ 1 ВµОЈ 1 10
AвЋЈ вЋ¦=вЋЈ вЋ¦.
Вµ0 ОЈв€’1 1 Вµ0 ОЈв€’1 Вµ 01

Now consider two portfolios on the boundary in Figure 2.2. For each the
weights are of the same form, say
вЋЎ вЋ¤ вЋЎ вЋ¤
1 1
wp = ОЈв€’1 M A вЋЈ вЋ¦ and wq = ОЈв€’1 M A вЋЈ вЋ¦ (2.16)
О·p О·q
where the mean returns are О·p and О·q respectively. Consider the covariance
between these two portfolios
0 0 0
cov(wp R, wq R) = wp ОЈwq
вЋЎ вЋ¤
1
](M 0 ОЈв€’1 M )в€’1 вЋЈ вЋ¦
=[ 1 О·p
О·q

= A11 + A12 (О·p + О·q ) + A22 О·p О·q
вЋЎ вЋ¤
0
= var(wp R) в€’ [ 1 О·p ]A вЋЈ вЋ¦
0

О·p в€’ О·q
46 CHAPTER 2. SOME BASIC THEORY OF FINANCE

An interesting special portfolio that is a вЂњzero-betaвЂќ portfolio, one that is
0
perfectly uncorrelated with the portfolio with weights wp R. This is obtained by
setting the above covariance equal to 0 and solving we obtain

A11 + A12 О·p
О·q = в€’
A12 + A22 О·p
Вµ0 ОЈв€’1 Вµ в€’ (Вµ0 ОЈв€’1 1)О·p
= 0 в€’1 .
Вµ ОЈ 1 в€’ (10 ОЈв€’1 1)О·p

There is a simple method for determining the point (, О·q ) graphically indicated
in Figure ??. From the equation relating points on the boundary,

Пѓ2 в€’ A22 (О· в€’ О·g )2 = Пѓg
2

we obtain
в€‚О· Пѓ
=
A22 (О· в€’ О·g )
в€‚Пѓ

and so the tangent line at the point (Пѓp , О·p ) strikes the Пѓ = 0 axis at a point
О·q which satisп¬Ѓes
О·p в€’ О·q Пѓp
=
A22 (О·p в€’ О·g )
Пѓp

or
2
Пѓp
О·q = О·p в€’
A22 (О·p в€’ О·g )
2
A22 О·p + 2A12 О·p + A11
= О·p в€’
A22 О·p + A12
A11 + A12 О·p
=в€’ (2.17)
.
A12 + A22 О·p

Note that this is exactly the same mean return obtained earlier for the portfolio
0
which has zero covariance with wp R. This shows that we can п¬Ѓnd the standard
deviation and mean of this uncorrelated portfolio by constructing the tangent
line at the point (Пѓp , О·p ) and then setting О·q to be the y-coordinate of the
point where this tangent line strikes the Пѓ = 0 axis as in Figure 2.3.
MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.47

Figure 2.3: The tangent line at the point (Пѓp , О·p )

Now suppose that there is available to all investors a risk-free investment.
Such an investment typically has smaller return than those on the eп¬ѓcient
frontier but since there is no risk associated with the investment, its standard
deviation is 0. It may be a government bond or treasury bill yielding interest
rate r so it corresponds to a point in Figure 2.4 at (0, r). Since all investors are
able to include this in their portfolio, the eп¬ѓcient frontier changes. In fact if
an investor invests an amount ОІ in this risk-free investment and amount 1 в€’ ОІ
(this may be negative) in the risky portfolio with standard deviation and mean
return (Пѓp , О·p ) then the resulting investment has mean return

0
E(ОІr + (1 в€’ ОІ)wp R) = ОІr+(1 в€’ ОІ)О· p

and standard deviation of return
q
0
V ar(ОІr + (1 в€’ ОІ)wp R) = (1 в€’ ОІ)Пѓp .

This means that every point on a line joining (0, r) to points in the risky portfolio
are now attainable and so the new set of attainable values of (Пѓ, О·) consists of a
cone with vertex at (0, r),the region shaded in Figure 2.4. The eп¬ѓcient frontier
48 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Figure 2.4: _____

is now the line L in Figure 2.4. The point m is the point at which this line is
tangent to the eп¬ѓcient frontier determined from the risky investments. Under
this theory, this point has great signiп¬Ѓcance.

Lemma 6 The value-weighted market average corresponds to the point of tan-
gency m of the line to the risky portfolio eп¬ѓcient frontier.

From (2.17) the point m has standard deviation, mean return О·m which
solves

A11 + A12 О·m
r=в€’
A12 + A22 О·m
Вµ ОЈ Вµ в€’ (Вµ0 ОЈв€’1 1)О·m
0 в€’1
= 0 в€’1
Вµ ОЈ 1 в€’ (10 ОЈв€’1 1)О·m

and this gives
Вµ0 ОЈв€’1 Вµ в€’ r(Вµ0 ОЈв€’1 1)
О·m = .
Вµ0 ОЈв€’1 1 в€’ r(10 ОЈв€’1 1)
MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.49

The corresponding weights on individual stocks are given by
вЋЎ вЋ¤
1
wm = ОЈв€’1 M A вЋЈ вЋ¦.
О·m
вЋЎ вЋ¤
A11 + A12 О·m
= ОЈв€’1 [1 Вµ] вЋЈ вЋ¦
A12 + A22 О·m
вЋЎ вЋ¤
в€’r
= cОЈв€’1 [1 Вµ] вЋЈ вЋ¦ , where c = A12 + A22 О·m
1

= cОЈв€’1 (Вµв€’r1).

These market weights depend essentially on two quantities. If R denotes the
correlation matrix
ОЈij
Rij =
Пѓi Пѓj
в€љ
where Пѓi = ОЈii is the standard deviation of the returns from stock i, and

Вµi в€’ r
О»i =
Пѓi

is the standardized excess return or the price of risk, then the weight wi on
stock i is such that
wi Пѓi в€ќ Rв€’1 О» (2.18)

with О» the column vector of values of О»i . For the purpose of comparison, recall
that the conservative portfolio, one minimizing the variance over all portfolios
of risky stocks, has weights
wg в€ќ ОЈв€’1 1

which means that the weight on stock i satisп¬Ѓes a relation exactly like (2.18)
except that the mean returns Вµi have all been replaced by the same constant.
Let us suppose that stocks, weighed by their total capitalization in the mar-
ket result in some weight vector w 6= wm . When there is a risk-free investment,
m is the only point in the risky stock portfolio that lies in the eп¬ѓcient frontier
and so evidently if we are able to trade in a market index (a stock whose value
50 CHAPTER 2. SOME BASIC THEORY OF FINANCE

depends on the total market), we can п¬Ѓnd an investment which is a combination
of the risk-free investment with that corresponding to m which has the same
standard deviation as w0 R but higher expected return. By selling short the
market index and buying this new portfolio, an arbitrage is possible. In other
words, the market will not stay in this state for long.

If the market portfolio m has standard deviation Пѓm and mean О·m , then
the line L is described by the relation

О·m в€’ r
О·=r+ Пѓ.
Пѓm

For any investment with mean return О· and standard deviation of return Пѓ
to be competitive, it must lie on this eп¬ѓcient frontier, i.e. it must satisfy the
relation

Пѓ
О· в€’ r = ОІ(О·m в€’ r), where ОІ = or equivalently (2.19)
Пѓm
О·в€’r (О·m в€’ r)
= .
Пѓ Пѓm

This is the most important result in the capital asset pricing model. The excess
return of a stock О· в€’ r divided by its standard deviation Пѓ is supposed constant,
and is called the Sharpe ratio or the market price of risk. The constant ОІ called
the beta of the stock or portfolio and represents the change in the expected
portfolio return for each unit change in the market. It is also the ratio of the
standard deviations of return of the stock and the market. Values of ОІ > 1
indicate a stock that is more variable than the market and tends to have higher
positive and negative returns, whereas values of ОІ < 1 are investments that are
more conservative and less volatile than the market as a whole.

We might attempt to use this model to simplify the assumed structure of
the joint distribution of stock returns. One simple model in which (2.19) holds
is one in which all stocks are linearly related to the market index through a
simple linear regression. In particular, suppose the return from stock i, Ri , is
MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.51

related to the return from the market portfolio Rm by

Пѓi 2
Ri в€’ r = ОІi (Rm в€’ r) + ВІi , where ОІi = , and Пѓi = ОЈii .
Пѓm

The вЂњerrorsвЂќ ВІi are assumed to be random variables, uncorrelated with the
market returns Rm . This model is called the single-index model relating the
returns from the stock Ri and from the market portfolio Rm .It has the merit
that the relationship (2.19) follows immediately.
Taking variance on both sides, we obtain

2 2 2
var(Ri ) = ОІi var(Rm ) + var(ВІi ) = Пѓi + var(ВІ) > Пѓi

2
which contradicts the assumption that var(Ri ) = Пѓi . What is the cause of this
contradiction? The relationship (2.19) assumes that the investment lies on the
eп¬ѓcient frontier. Is this not a suп¬ѓcient condition for investors to choose this
investment? All that is required for rational investors to choose a particular
stock is that it forms part of a portfolio which does lie on the eп¬ѓcient frontier.
Is every risk in an eп¬ѓcient market rewarded with additional expected return?
We cannot expect the market to compensate us with a higher rate of return for
additional risks that could be diversiп¬Ѓed away. Suppose, for example, we have
two stocks with identical values of ОІ. Suppose their returns R1 and R2 both
satisfy a linear regression relation above

Ri в€’ r = ОІ(Rm в€’ r) + ВІi , i = 1, 2,

where cov(ВІ1 , ВІ2 ) = 0. Consider an investment of equal amounts in both stocks
so that the return is

R1 + R2 ВІ1 + ВІ2
= ОІ(Rm в€’ r) + .
2 2

For simplicity assume that Пѓ1 В· Пѓ2 and notice that the variance of this new
investment is
1
ОІ 2 Пѓm + [var(ВІ1 ) + var(ВІ2 )] < var(R2 ).
2
4
52 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The diversiп¬Ѓed investment consisting of the average of the two results in the
same mean return with smaller variance. Investors should not compensated for
the additional risk in stock 2 above the level that we can achieve by sensible
diversiп¬Ѓcation. In general, by averaging or diversifying, we are able to provide
an investment with the same average return characteristics but smaller variance
than the original stock. We say that the risk (i.e. var(ВІi )) associated with
stock i which can be diversiп¬Ѓed away is the speciп¬Ѓc risk, and this risk is not
rewarded with increased expected return. Only the so-called systematic risk Пѓi
which cannot by removed by diversiп¬Ѓcation is rewarded with increased expected
return with a relation like (2.19).
The covariance matrix of stock returns is one of the most diп¬ѓcult parameters
to estimate in practice form historical data. If there are n stocks in a market
(and normally n is large), then there are n(n + 1)/2 elements of ОЈ that need
to be estimated. For example if we assume all stocks in the TSE 300 index
are correlated this results in a total of (300)(301)/2 = 45, 150 parameters
to estimate. We might use historical data to estimate these parameters but
variances and covariances among stocks change over time and it is not clear
over what period of time we can safely use to estimate these parameters. In
spite of its defects, the single index model can be used to provide a simple
approximate form for the covariance matrix ОЈ of the vector of stock returns.
Notice that under the model, assuming uncorrelated random errors ВІi with
var(ВІi ) = Оґi ,

Ri в€’ r = ОІi (Rm в€’ r) + ВІi ,

we have
2 22
cov(Ri , Rj ) = ОІi ОІj Пѓm , i 6= j, var(Ri ) = ОІi Пѓm + Оґi .

Whereas n stocks would otherwise require a total of n(n + 1)/2 parameters in
the covariance matrix ОЈ of returns, the single index model allows us to reduce
2
this to the n + 1 parameters Пѓm , and Оґi , i = 1, ..., n. There is the disadvantage
MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.53

in this formula however that every pair of stocks in the same market must be
positively correlated, a feature that contradicts some observations of real market
returns.
Suppose we use this form ОЈ = ОІОІ 0 Пѓm + в€†, to estimate weights on individual
2

stocks, where в€† is the diagonal matrix with the Оґi along the diagonal and ОІ
In this case ОЈв€’1 = в€†в€’1 +
is the column vector of individual stock betas.
cв€†в€’1 ОІОІ 0 в€†в€’1 where

в€’1 1
2 P
= в€’Пѓm
P
c= в€’2 22
2 1+ i ОІi Пѓm /Оґi
Пѓm + ОІi /Оґi
i

and consequently the conservative investor by (2.14) invests in stock i propor-
tionally to the components of ОЈв€’1 1

X
1
or to + cОІi ( ОІj /Оґj )
Оґi j
1
P
or proportional to ОІi +
cОґi ( j ОІj /Оґj )

The conditional variance of Ri given the market return Rm is Оґi . Let us call this
the excess volatility for stock i. Then the weights for the conservative portfolio
are linear in the beta for the stock and the reciprocal of the excess volatility.
The weights in the market portfolio are given by
вЋЎ вЋ¤ вЋЎ вЋ¤
1 1
wm = ОЈв€’1 M A вЋЈ вЋ¦ = (в€†в€’1 + cв€†в€’1 ОІОІ 0 в€†в€’1 )[ 1 Вµ ](M 0 ОЈв€’1 M )в€’1 вЋЈ вЋ¦
О·p О·p

Minimum Variance under Q.

Suppose we wish to п¬Ѓnd a portfolios of securities which has the smallest possible
variance under the risk neutral distribution Q. For example for a given set of
weights wi (t) representing the number of shares held in security i at time t,
P
deп¬Ѓne the portfolio О (t) = wi (t)Si (t). Recall from Section 2.1 that under
a risk neutral distribution, all stocks have exactly the same expected return
as the risk-free interest rate so the portfolio О (t) will have exactly the same
54 CHAPTER 2. SOME BASIC THEORY OF FINANCE

conditional expected rate of return under Q as all the constituent stocks,

X X B(t + 1) B(t + 1)
EQ [О (t+1)|Ht ] = wi (t)EQ [Si (t+1)|Ht ] = wi (t) Si (t) = О (t).
B(t) B(t)
i i

Since all portfolios have the same conditional expected return under Q, we
might attempt to minimize the (conditional) variance of the portfolio return of
the portfolio. The natural constraint is that the cost of the portfolio is deter-
mined by the amount c(t) that we presently have to invest. We might assume
a constant investment over time, for example c(t) = 1 for all t. Alternatively,
we might wish to study a self-п¬Ѓnancing portfolio О (t), one for which past gains
(or perish the thought, past losses) only are available to pay for the current
portfolio so we neither withdraw from nor add money to the portfolio over its
lifetime. I this case c(t) = О (t). We wish to minimise

X
varQ [О (t + 1)|Ht ] subject to the constraint wi (t)Si (t) = c(t).
i

As before, the solution is quite easy to obtain, and in fact the weights are
given by the vector
вЋ› вЋћ
w (t)
вЋњ1 вЋџ
вЋњ вЋџ
вЋњ w2 (t) вЋџ
вЋњ вЋџ
вЋњ вЋџ
вЋњ вЋџ
. c(t)
вЋњ вЋџ
ОЈв€’1 S(t).
w(t) = вЋњ вЋџ= 0
вЋџ S (t)ОЈt S(t) t
в€’1
вЋњ .
вЋњ вЋџ
вЋњ вЋџ
вЋњ вЋџ
вЋњ вЋџ
.
вЋќ вЋ
wn (t)

where ОЈt = varQ (S(t + 1)|Ht ) is the instantaneous conditional covariance
matrix of S(t) under the measure Q. If my objective were to minimize risk under
the Q measure, then this portfolio is optimal for п¬Ѓxed cost. The conditional
variance of this portfolio is given by

c2 (t)
0
varQ (О (t + 1)|Ht ) = w (t)ОЈt w(t) = 0 .
S (t)ОЈв€’1 S(t)
t
MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.55

О (t+1)в€’О (t)
In terms of the portfolio return RО  (t + 1) = , if the portfolio is
О (t)

self-п¬Ѓnancing so that c(t) = О (t), the above relation states that the conditional
variance of the return RО  (t + 1) given the past is simply

1
varQ (RО  (t + 1)|Ht ) =
S 0 (t)ОЈв€’1 S(t)
t

which is similar to the form of the variance of the conservative portfolio (2.13).
Similarly, covariances between returns for individual stocks and the return
of the portfolio О  are given by exactly the same quantity, namely

1
cov(Ri (t + 1), RО  (t + 1)|Ht ) = .
S 0 (t)ОЈв€’1 S(t)
t

Let us summarize our п¬Ѓndings so far. We assume that the conditional co-
variance matrix ОЈt of the vector of stock prices is non-singular. Under the risk
neutral measure, all stocks have exactly the same expected returns equal to the
risk-free rate. There is a unique self-п¬Ѓnancing minimum-variance portfolio О (t)
and all stocks have exactly the same conditional covariance ОІ with О . All stocks
have exactly the same regression coeп¬ѓcient ОІ when we regress on the minimum
variance portfolio.
Are other minimum variance portfolios conditionally uncorrelated with the
portfolio we obtained above. Suppose we deп¬Ѓne О 2 (t) similarly to minimize the
variance subject to the condition that CovQ (О 2 (t + 1), О (t + 1)|Ht ) = 0. It is
easy to see that this implies that the cost of such a portfolio at the beginning
of each period is 0. This means that in this new portfolio, there is a perfect
balance between long and short stocks, or that the value of the long and short
stocks are equal.
The above analysis assumes that our objective is minimizing the variance
of the portfolio under the risk-neutral distribution Q. Two objections could be
made. First we argued earlier that the performance of an investment should be
made through the returns , not through the stock prices. Since under the risk
neutral measure Q, the expected return from every stock is the risk-free rate of
56 CHAPTER 2. SOME BASIC THEORY OF FINANCE

return, we are left with the problem of minimizing the variance of the portfolio
return. By our earlier analysis, this is achieved when the proportion of our
total investment at each time period in stock i is chosen as the corresponding
ОЈв€’1 1
component of the vector where now ОЈt is the conditional covariance
t
10 ОЈв€’1 1
t

matrix of the stock returns. This may appear to be a diп¬Ђerent criterion and
hence a diп¬Ђerent solution, but because at each time step the stock price is a linear
function of the return Si (t + 1) = Si (t)(1 + Ri (t + 1)) the variance minimizing
portfolios are essentially the same. There is another objection however to an
analysis in the risk-neutral world of Q. This is a distribution which determines
the value of options in order to avoid arbitrage in the system, not the actual
distribution of stock prices. It is not clear what the relationship is between
the covariance matrix of stock prices under the actual historical distribution
and the risk neutral distribution Q, but observations seem to indicate a very
considerable diп¬Ђerence. Moreover, if this diп¬Ђerence is large, there is very little
information available for estimating the parameters of the covariance matrix
under Q, since historical data on the п¬‚uctuations of stock prices will be of
doubtful relevance.

Entropy: choosing a Q measure

Maximum Entropy

In 1948 in a fundamental paper on the transmission of information, C. E. Shan-
non proposed the following idea of entropy. The entropy of a distribution at-
tempts to measure the expected number of steps required to determine a given
outcome of a random variable with a given distribution when using a simple
binary poll. For example suppose that a random variable X has distribution
ENTROPY: CHOOSING A Q MEASURE 57

given by
x 0 1 2
P [X = x] .25 .25 .5
if we ask п¬Ѓrst whether the random variable is в‰Ґ 2 and
In this case,
then, provided the answer is no, if it is в‰Ґ 1, the expected number of queries to
ascertain the value of the random variable is 1+1(1/2) = 1.5. There is no more
eп¬ѓcient scheme for designing this binary poll in this case so we will take 1.5 to
be a measure of entropy of the distribution of X. For a discrete distribution,
such that P [X = x] = p(x), the entropy may be deп¬Ѓned to be
X
H(p) = E{в€’ ln(p(X))} = в€’ p(x) ln(p(x)).
x

More generally we deп¬Ѓne the entropy of an arbitrary distribution through the
form for a discrete distribution. If P is a probability measure (see the appen-
dix),
X
H(P ) = sup{в€’ P (Ei ) ln(P (Ei ))}

where the supremum is taken over all п¬Ѓnite partitions (Ei } of the space.
In the case of the above distribution, if we were to replace the natural log-
arithm by the log base 2, (ln and log2 diп¬Ђer only by a scale factor and are
therefore the corresponding measures of entropy are equivalent up a constant
P
multiple) notice that в€’ x p(x) log2 (p(x)) = .5(1) + .5(2) = 1.5, so this formula
correctly measures the diп¬ѓculty in ascertaining a random variable from a se-
quence of questions with yes-no or binary answers. This is true in general. The
complexity of a distribution may be measured by the expected number of ques-
tions in a binary poll to determine the value of a random variable having that
distribution, and such a measure results in the entropy H(p) of the distribution.
Many statistical distributions have an interpretation in terms of maximizing
entropy and it is often remarkable how well the maximum entropy principle re-
produces observed distributions. For example, suppose we know that a discrete
random variable takes values on a certain set of n points. What distribution p
58 CHAPTER 2. SOME BASIC THEORY OF FINANCE

on this set maximizes the entropy H(p)? First notice that if p is uniform on
P1 1
n points, p(x) = 1/n for all x and so the entropy is в€’ x n ln( n ) = ln(n).
Now consider the problem of maximizing the entropy H(p) for any distribution
on n points subject to the constraint that the probabilities add to one. As in
P P
(2.10), the Lagrangian for this problem is в€’ x p(x) ln(p(x)) в€’ О»{ x p(x) в€’ 1}
where О» is a Lagrange multiplier. Upon diп¬Ђerentiating with respect to p(x) for
each x, we obtain в€’ ln(p(x)) в€’ 1 в€’ О» = 0 or p(x) = eв€’(1+О») . The probabilities
evidently do not depend on x and the distribution is thus uniform. Applying
the constraint that the sum of the probabilities is one results in p(x) = 1/n
for all x. The discrete distribution on n points which has maximum entropy is
the uniform distribution. What if we repeat this analysis using additional con-
straints, for example on the moments of the distribution? Suppose for example
that we require that the mean of the distribution is some п¬Ѓxed constant Вµ and
the variance п¬Ѓxed at Пѓ 2 . The problem is similar to that treated above but with
two more terms in the Lagrangian for each of the additional constraints. The
Lagrangian becomes
X X X X
x2 p(x)в€’Вµ2 в€’Пѓ 2 }
в€’ p(x) ln(p(x))в€’О»1 { p(x)в€’1}в€’О»2 { xp(x)в€’Вµ}в€’О»3 {
x x x

whereupon setting the derivative with respect to p(x) equal to zero and ap-
plying the constraints we obtain

p(x) = exp{в€’О»1 в€’ О»2 x в€’ О»3 x2 },

with constants О»1 , О»2 , О»3 chosen to satisfy the three constraints. Since the ex-
ponent is a quadratic function of x, this is analogous to the normal distribution
except that we have required that it be supported on a discrete set of points x.
With more points, positioned more closely together, the distribution becomes
closer to the normal. Let us call such a distribution the discrete normal dis-
tribution. For a simple example, suppose that we wish to use the maximum
entropy principle to approximate the distribution of the sum of the values on
ENTROPY: CHOOSING A Q MEASURE 59

0.18

0.16

0.14

0.12

0.1
probability

0.08

0.06

0.04

0.02

0
2 3 4 5 6 7 8 9 10 11 12
value

Figure 2.5: A discrete analogue of the normal distribution compared with the
distribution of the sum of the values on two dice.

two dice. In this case the actual distribution is known to us as well as the mean
and variance E(X) = 7, var(X) = 35/6;
2 3 4 5 6 7 8 9 10 11 12
x
1 2 3 4 5 6 5 4 3 2 1
P (X = x) 36 36 36 36 36 36 36 36 36 36 36
The maximum entropy distribution on these same points constrained to have
the same mean and variance is very similar to this, the actual distribution. This
can been seen in Figure 2.5.

In fact if we drop the requirement that the distribution is discrete, or equiv-
alently take a limit with an increasing number of discrete points closer and
closer together, the same kind of argument shows that the maximum entropy
distribution subject to a constraint on the mean and the variance is the normal
distribution. So at least two well-known distributions arise out of maximum
60 CHAPTER 2. SOME BASIC THEORY OF FINANCE

entropy considerations. The maximum entropy distribution on a discrete set
of points is the uniform distribution. The maximum entropy subject to a con-
straint on the mean and the variance is a (discrete) normal distribution. There
are many other examples as well. In fact most common distributions in statis-
tics have an interpretation as a maximum entropy distribution subject to some
constraints.
Entropy has a number of properties that one would expect of a measure of
the information content in a random variable. It is non-negative, and can in
usual circumstances be inп¬Ѓnite. We expect that the information in a function
of X , say g(X), is less than or equal to the information in X itself, equal if
the function is one to one (which means in eп¬Ђect we can determine X from
the value of g(X)). Entropy is a property of a distribution, not of a random
variable. Nevertheless it is useful to be able to abuse the notation used earlier
by referring to H(X) as the entropy of the distribution of X. Then we have the
following properties

Proposition 7 H(X) в‰Ґ 0

Proposition 8 H(g(X)) В· H(X) for any function g(x)..

The information or uncertainty in two random variables is clearly greater
than that in one. The deп¬Ѓnition of entropy is deп¬Ѓned in the same fashion as
before, for discrete random variables (X, Y ),

H(X, Y ) = в€’E(ln p(X, Y ))

where p(x, y) is the joint probability function

p(x, y) = P [X = x, Y = y].

If the two random variables are independent, then we expect that the uncer-
tainty should add. If they are dependent, then the entropy of the pair (X, Y )
is less than the sum of the individual entropies.
ENTROPY: CHOOSING A Q MEASURE 61

Proposition 9 H(X, Y ) В· H(X) + H(Y ) with equality if and only if X and Y
are independent.

Let us now use the principle of maximum entropy to address an eminently
practical problem, one of altering a distribution to accommodate a known mean
value. Suppose we are interested in determining a risk-neutral distribution
for pricing options at maturity T. Theorem 1 tells us that if there is to be no
arbitrage, our distribution or measure Q must satisfy a relation of the form

EQ (eв€’rT ST ) = S0

where r is the continuously compounded interest rate, S0 is the initial (present)
value of the underlying stock, and ST is its value at maturity. Let us also
suppose that we constraint the variance of the future stock price under the
measure Q so that
varQ (ST ) = Пѓ 2 T.

Then from our earlier discussion, the maximum entropy distribution under
constraints on the mean and variance is the normal distribution so that the
probability density function of ST is
(s в€’ erT S0 )2
1
f (s) = в€љ }.
exp{в€’
2Пѓ2 T
Пѓ 2ПЂT
If we wished a maximum entropy distribution which is compatible with a
number of option prices, then we should impose these option prices as additional
constraints. Again suppose the current time t = 0 and we know the prices
Pi , i = 1, ..., n of n diп¬Ђerent call options available on the market, all on the same
security and with the same maturity T but with diп¬Ђerent strike prices Ki . The
distribution Q we assign to ST must satisfy the constraints

E(eв€’rT (ST в€’ Ki )+ ) = Pi , i = 1, ..., n (2.20)

as well as the martingale constraint

E(eв€’rT ST ) = S0 . (2.21)
62 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Once again introducing Lagrange multipliers, the probability density function
of ST will take the form
n
X
в€’rT
О»i (s в€’ Ki )+ + О»0 s}
f (s) = k exp{e
i=1

where the parameters О»0 , ..., О»n are chosen to satisfy the constraints (2.20) and
(2.21) and k so that the function integrates to 1. When п¬Ѓt to real option price
data, these distributions typically resemble a normal density, usually however
with some negative skewness and excess kurtosis. See for example Figure XXX.
There are alsoвЂњsawtoothвЂќ like appendages with teeth corresponding to each of
the n options. Note too this density is strictly positive at the value s = 0,
a feature that we may or may not wish to have. Because of the вЂќteethвЂќ, a
smoother version of the density is often used, one which may not perfectly
reproduce option prices but is nevertheless appears to be more natural.

Minimum Cross-Entropy

Normally market information does not completely determine the risk-neutral
measure Q . We will argue that while market data on derivative prices rather
than historical data should determine the Q measure, historical asset prices
can be used to п¬Ѓll in the information that is not dictated by no-arbitrage con-
siderations. In order to relate the real world to the risk-free world, we need
either suп¬ѓcient market data to completely describe a risk-neutral measure Q
(such a model is called a complete market) or we need to limit our candidate
class of Q measures somewhat. We may either deп¬Ѓne the joint distributions of
the stock prices or their returns, since from one we can pass to the other. For
convenience, suppose we describe the joint distribution of the returns process.
The conditions we impose on the martingale measure are the following;

1. Under Q, each normalized stock price Sj (t)/Bt and derivative price
Vt /Bt forms a martingale. Equivalently, EQ [Si (t+1)|Ht ] = Si (t)(1+r(t))
ENTROPY: CHOOSING A Q MEASURE 63

where r(t) is the risk free interest rate over the interval (t, t + 1). (Recall
that this risk-free interest rate r(t) is deп¬Ѓned by the equation B(t + 1) =
(1 + r(t))B(t).)

2. Q is a probability measure.

A slight revision of notation is necessary here. We will build our joint distri-
butions conditionally on the past and if P denotes the joint distribution stock
prices S(1), S(2), ...S(T ) over the whole period of observation 0 < t < T then
Pt+1 denotes the conditional distribution of S(t + 1) given Ht . Let us denote
the conditional moment generating function of the vector S(t + 1) under the
measure Pt+1 by

X
mt (u) = EP [exp(u0 S(t + 1)|Ht ] = EP [exp( ui Si (t + 1))|Ht ]
i

We implicitly assume, of course, that this moment generating function exists.
Suppose, for some vector of parameters О· we choose Qt+1 to be the exponential
tilt of Pt+1 , i.e.
exp(О· 0 s)
dQt+1 (s) = dPt+1 (s)
mt (О·)

The division by mt (О·) is necessary to ensure that Qt+1 is a probability measure.

Why transform a density by multiplying by an exponential in this way?
There are many reasons for such a transformation. Exponential families of dis-
tributions are built in exactly this fashion and enjoy properties of suп¬ѓciency,
completeness and ease of estimation. This exponential tilt resulted from maxi-
mizing entropy subject to certain constraints on the distribution. But we also
argue that the measure Q is the probability measure which is closest to P in
a certain sense while still satisfying the required moment constraint. We п¬Ѓrst
introduce cross-entropy which underlies considerable theory in Statistics and
elsewhere in Science.
64 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Cross Entropy

Consider two probability measures P and Q on the same space. Then the
cross entropy or Kullbach-Leibler вЂњdistanceвЂќ between the two measures is given
by
X Q(Ei )
H(Q, P ) = sup Q(Ei ) log
P (Ei )
{Ei }

where the supremum is over all п¬Ѓnite partitions {Ei } of the probability space.
Various properties are immediate.

Proposition 10 H(Q, P ) в‰Ґ 0 with equality if and only if P and Q are iden-
tical.

If Q is absolutely continuous with respect to P , that is if there is some
density function f (x) such that
Z
f (x)dP for all E
Q(E) =
E

then provided that f is smooth, we can also write
dQ
H(Q, P ) = EQ log( ).
dP
If Q is not absolutely continuous with respect to P then the cross entropy
H(Q, P ) is inп¬Ѓnite. We should also remark that the cross entropy is not really
a distance in the usual sense (although we used the term вЂњdistanceвЂќ in reference
to it) because in general H(Q, P ) 6= H(P |Q). For a п¬Ѓnite probability space,
there is an easy relationship between entropy and cross entropy given by the
following proposition. In eп¬Ђect the result tells us that maximizing entropy H(Q)
is equivalent to minimizing the cross-entropy H(Q, P ) where P is the uniform
distribution.

Proposition 11 If the probability space has a п¬Ѓnite number n points, and P
denotes the uniform distribution on these n points, then for any other probability
measure Q,
H(Q, P ) = n в€’ H(Q)
ENTROPY: CHOOSING A Q MEASURE 65

Now the following result asserts that the probability measure Q which is
closest to P in the sense of cross-entropy but satisп¬Ѓes a constraint on its mean
is generated by a so-called вЂњexponential tiltвЂќ of the distribution of P.

Theorem 12 : Minimizing cross-entropy.

Let f (X) be a vector valued function f (X) = (f1 (X), f2 (X), ..., fn (X)) and
Вµ = (Вµ1 , ..., Вµn ). Consider the problem

min H(Q, P )
Q

subject to the constraint EQ (fi (X)) = Вµi , i = 1, ..., n. Then the solution, if it
exists, is given by
Pn
exp(О· 0 f (X)) exp( i=1 О·i fi (X))
dQ = dP =
m(О·) m(О·)
Pn в€‚m
where m(О·) = EP [exp( i=1 О·i fi (X))] and О· is chosen so that = Вµm(О·).
в€‚О·i

The proof of this result, in the case of a discrete distribution P is a straight-
forward use of Lagrange multipliers (see Lemma 3). We leave it as a problem
at the end of the chapter.
Now let us return to the constraints on the vector of stock prices. In order
that the discounted stock price forms a martingale under the Q measure, we
require that EQ [S(t + 1)|Ht ] = (1 + r(t))S(t). This is achieved if we deп¬Ѓne Q
such that for any event A в€€ Ht ,
Z
Zt dP where
Q(A) =
A
s
X
0
О·t (St+1 в€’ St )) (2.22)
Zs = kt exp(
t=1
where kt are Ht measurable random variables chosen so that Zt forms a mar-
tingale
E(Zt+1 |Ht ) = Zt .
66 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Theorem 9 shows that this exponentially tilted distribution has the property
of being the closest to the original measure P while satisfying the condition
that the normalized sequence of stock prices forms a martingale.

There is a considerable literature exploring the links between entropy and
risk-neutral valuation of derivatives. See for example Gerber and Shiu (1994),
Avellaneda et. al (1997), Gulko(1998), Samperi (1998). In a complete or
incomplete market, risk-neutral valuation may be carried out using a martingale
measure which maximizes entropy or minimizes cross-entropy subject to some
natural constraints including the martingale constraint. For example it is easy
to show that when interest rates r are constant, Q is the risk-neutral measure
for pricing derivatives on a stock with stock price process St , t = 0, 1, ... if
and only if it is the probability measure minimizing H(Q, P ) subject to the
martingale constraint

1
(2.23)
St = EQ [ ВЇ St+1 ].
1+r

There is a continuous time analogue of (2.22) as well which we can anticipate
by inspecting the form of the solution. Suppose that St denotes the stock price
at time t where we now allow t to vary continuously in time. which we will
discuss later but (2.22) can be used to anticipate it. Then an analogue of (2.22)
could be written formally as

Z t
0
О·t dSt в€’ gt )
Zs = exp(
0

where both processes О·t and gt are вЂњpredictableвЂќ which loosely means that
they are determined in advance of observing the increment St , St+в€†t . Then the
dQ
process Zs is the analogue of the Radon-Nikodym derivative of the processes
dP

restricted to the time interval 0 В· t В· s. For a more formal deп¬Ѓnition, as well as
an explanation of how we should interpret the integral, see the appendix. This
process Zs is, both in discrete and continuous time, a martingale.
MODELS IN CONTINUOUS TIME 67

Wiener Process
3

2.5

2

1.5
W(t)

1

0.5

0

-0.5

-1
0 1 2 3 4 5 6 7 8 9 10
t

Figure 2.6: A sample path of the Wiener process

Models in Continuous Time

We begin with some oversimpliп¬Ѓed rules of stochastic calculus which can be
omitted by those with a background in Brownian motion and diп¬Ђusion. First,
we deп¬Ѓne a stochastic process Wt called the standard Brownian motion or
Wiener process having the following properties;

1. For each h > 0, the increment W (t+h)в€’W (t) has a N (0, h) distribution
and is independent of all preceding increments W (u) в€’ W (v), t > u > v >
0.

2. W (0 ) = 0 .

The fact that such a process exists is by no means easy to see. It has been an
important part of the literature in Physics, Probability and Finance at least since
the papers of Bachelier and Einstein, about 100 years ago. A Brownian motion
process also has some interesting and remarkable theoretical properties; it is
continuous with probability one but the probability that the process has п¬Ѓnite
68 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Random Walk
4

3

2

1
Sn

0

-1

-2

-3
0 2 4 6 8 10 12 14 16 18 20
n

Figure 2.7: A sample path of a Random Walk

variation in any interval is 0. With probability one it is nowhere diп¬Ђerentiable.
Of course one might ask how a process with such apparently bizarre properties
can be used to approximate real-world phenomena, where we expect functions
to be built either from continuous and diп¬Ђerentiable segments or jumps in the
process. The answer is that a very wide class of functions constructed from those
that are quite well-behaved (e.g. step functions) and that have independent
increments converge as the scale on which they move is reп¬Ѓned either to a
Brownian motion process or to a process deп¬Ѓned as an integral with respect to a
Brownian motion process and so this is a useful approximation to a broad range
of continuous time processes. For example, consider a random walk process
Pn
i=1 Xi where the random variables Xi are independent identically
Sn =
distributed with expected value E(Xi ) = 0 and var(Xi ) = 1. Suppose we plot
the graph of this random walk (n, Sn ) as below. Notice that we have linearly
interpolated the graph so that the function is deп¬Ѓned for all n, whether integer
or not.

MODELS IN CONTINUOUS TIME 69

Now if we increase the sample size and decrease the scale appropriately on
both axes, the result is, in the limit, a Brownian motion process. The vertical
в€љ
scale is to be decreased by a factor 1/ n and the horizontal scale by a factor
nв€’1 . The theorem concludes that the sequence of processes

1
Yn (t) = в€љ Snt
n

converges weakly to a standard Brownian motion process as n в†’ в€ћ. In practice
this means that a process with independent stationary increments tends to look
like a Brownian motion process. As we shall see, there is also a wide variety
of non-stationary processes that can be constructed from the Brownian motion
process by integration. Let us use the above limiting result to render some
of the properties of the Brownian motion more plausible, since a serious proof
is beyond our scope. Consider the question of continuity, for example. Since
Pn(t+h)
1
|Yn (t + h) в€’ Yn (t)| в‰€ | в€љn i=nt Xi | and this is the absolute value of an
asymptotically normally(0, h) random variable by the central limit theorem, it
is plausible that the limit as h в†’ 0 is zero so the function is continuous at t.
On the other hand note that
n(t+h)
11 X
Yn (t + h) в€’ Yn (t)
в‰€в€љ Xi
h h n i=nt

should by analogy behave like hв€’1 times a N (0, h) random variable which blows
up as h в†’ 0 so it would appear that the derivative at t does not exist. To
obtain the total variation of the process in the interval [t, t + h] , consider the
lengths of the segments in this interval, i.e.
n(t+h)
1X
в€љ |Xi |
n i=nt
Pn(t+h)
1
|Xi |
and notice that since the law of large numbers implies that i=nt
nh
в€љ
converges to a positive constant, namely E|Xi |, if we multiply by the
nh
limit must be inп¬Ѓnite, so the total variation of the Brownian motion process is
inп¬Ѓnite.
70 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Continuous time process are usually built one small increment at a time
and deп¬Ѓned to be the limit as the size of the time increment is reduced to zero.
Let us consider for example how we might deп¬Ѓne a stochastic (Ito) integral of
RT
the form 0 h(t)dWt . An approximating sum takes the form
Z nв€’1
X
T
h(t)dWt в‰€ h(ti )(W (ti+1 ) в€’ W (ti )), 0 = t0 < t1 < ... < tn = T.
0 i=0

Note that the function h(t) is evaluated at the left hand end-point of the in-
tervals [ti , ti+1 ], and this is characteristic of the Ito calculus, and an important
feature distinguishing it from the usual Riemann calculus studied in undergrad-
uate mathematics courses. There are some simple reasons why evaluating the
function at the left hand end-point is necessary for stochastic models in п¬Ѓnance.
For example let us suppose that the function h(t) measures how many shares
of a stock we possess and W (t) is the price of one share of stock at time t.
It is clear that we cannot predict precisely future stock prices and our decision
about investment over a possibly short time interval [ti , ti+1 ] must be made
at the beginning of this interval, not at the end or in the middle. Second, in
the case of a Brownian motion process W (t), it makes a diп¬Ђerence where in
the interval [ti , ti+1 ] we evaluate the function h to approximate the integral,
whereas it makes no diп¬Ђerence for Riemann integrals. As we reп¬Ѓne the parti-
Pnв€’1
tion of the interval, the approximating sums i=0 h(ti+1 )(W (ti+1 ) в€’ W (ti )),
for example, approach a completely diп¬Ђerent limit. This diп¬Ђerence is essentially
due to the fact that W (t), unlike those functions studied before in calculus, is
of inп¬Ѓnite variation. As a consequence, there are other important diп¬Ђerences in
the Ito calculus. Let us suppose that the increment dW is used to denote
small increments W (ti+1 ) в€’ W (ti ) involved in the construction of the integral.
If we denote the interval of time ti+1 в€’ ti by dt, we can loosely assert that dW
has the normal distribution with mean 0 and variance dt. If we add up a large
number of independent such increments, since the variances add, the sum has
variance the sum of the values dt and standard deviation the square root. Very
MODELS IN CONTINUOUS TIME 71

roughly, we can assess the size of dW since its standard deviation is (dt)1/2 .
Now consider deп¬Ѓning a process as a function both of the Brownian motion and
of time, say Vt = g(Wt , t). If Wt represented the price of a stock or a bond,
Vt might be the price of a derivative on this stock or bond. Expanding the
increment dV using a Taylor series expansion gives

в€‚2 dW 2
в€‚ в€‚
(2.24)
dVt = g(Wt , t)dW + g(Wt , t) + g(Wt , t)dt
в€‚W 2
в€‚W 2 в€‚t
+ (stuп¬Ђ) Г— (dW )3 + (more stuп¬Ђ) Г— (dt)(dW )2 + ....

is normal with mean 0 and standard deviation (dt)1/2 and
Loosely, dW
so dW is non-negligible compared with dt as dt в†’ 0. We can deп¬Ѓne each of the
diп¬Ђerentials dW and dt essentially by reference to the result when we integrate
both sides of the equation. If I were to write an equation in diп¬Ђerential form

dXt = h(t)dWt

then this only has real meaning through its integrated version
Z t
Xt = X0 + h(t)dWt .
0

What about the terms involving (dW )2 ? What meaning should we assign to a
R P
term like h(t)(dW )2 ? Consider the approximating function h(ti )(W (ti+1 )в€’
W (ti ))2 . Notice that, at least in the case that the function h is non-random we
are adding up independent random variables h(ti )(W (ti+1 ) в€’ W (ti ))2 each with
expected value h(ti )(ti+1 в€’ ti ) and when we add up these quantities the limit
R
is h(t)dt by the law of large numbers. Roughly speaking, as diп¬Ђerentials, we
should interpret (dW )2 as dt because that is the way it acts in an integral.
Subsequent terms such as (dW )3 or (dt)(dW )2 are all o(dt), i.e. they all
approach 0 faster than does dt as dt в†’ 0. So п¬Ѓnally substituting for (dW )2 in
2.24 and ignoring all terms that are o(dt), we obtain a simple version of ItoвЂ™s
lemma
72 CHAPTER 2. SOME BASIC THEORY OF FINANCE

1 в€‚2
в€‚ в€‚
g(Wt , t)dW + {
dg(Wt , t) = g(Wt , t) + g(Wt , t)}dt.
2
в€‚W 2 в€‚W в€‚t

This rule results, for example, when we put g(Wt , t) = Wt2 in

d(Wt2 ) = 2Wt dWt + dt

or on integrating both sides and rearranging,
Zb Z
1b
1 2 2
Wt dWt = (Wb в€’ Wa ) в€’ dt.
2 2a
a
Rb
The term a dt above is what distinguishes the Ito calculus from the Riemann
calculus, and is a consequence of the nature of the Brownian motion process, a
continuous function of inп¬Ѓnite variation.
There is one more property of the stochastic integral that makes it a valuable
tool in the construction of models in п¬Ѓnance, and that is that a stochastic integral
with respect to a Brownian motion process is always a martingale. To see this,
note that in an approximating sum
Z nв€’1
X
T
h(t)dWt в‰€ h(ti )(W (ti+1 ) в€’ W (ti ))
0 i=0

each of the summands has conditional expectation 0 given the past, i.e.

E[h(ti )(W (ti+1 ) в€’ W (ti ))|Hti ] = h(ti )E[(W (ti+1 ) в€’ W (ti ))|Hti ] = 0

since the Brownian increments have mean 0 given the past and since h(t) is
measurable with respect to Ht .
We begin with an attempt to construct the model for an Ito process or dif-
fusion process in continuous time. We construct the price process one increment
at a time and it seems reasonable to expect that both the mean and the vari-
ance of the increment in price may depend on the current price but does not
depend on the process before it arrived at that price. This is a loose description
of a Markov property. The conditional distribution of the future of the process
MODELS IN CONTINUOUS TIME 73

depends only on the current time t and the current price of the process. Let us
suppose in addition that the increments in the process are, conditional on the
past, normally distributed. Thus we assume that for small values of h, con-
ditional on the current time t and the current value of the process Xt , the
increment Xt+h в€’ Xt can be generated from a normal distribution with mean
a(Xt , t)h and with variance Пѓ 2 (Xt , t)h for some functions a and Пѓ2 called the
drift and diп¬Ђusion coeп¬ѓcients respectively. Such a normal random variable can
be formally written as a(Xt , t )dt+ Пѓ 2 (Xt , t)dWt . Since we could express XT as
P
an initial price X0 plus the sum of such increments, XT = X0 + i (Xti+1 в€’Xti ).
The single most important model of this type is called the Geometric Brown-
ian motion or Black-Scholes model. Since the actual value of stock, like the
value of a currency or virtually any other asset is largely artiп¬Ѓcial, depending on
such things as the number of shares issued, it is reasonable to suppose that the
changes in a stock price should be modeled relative to the current price. For
example rather than model the increments, it is perhaps more reasonable to
model the relative change in the process. The simplest such model of this type
is one in which both the mean and the standard deviation of the increment in
the price are linear multiples of price itself; viz. dXt is approximately nor-
mally distributed with mean aXt dt and variance Пѓ 2 Xt dt. In terms of stochastic
2

diп¬Ђerentials, we assume that

(2.25)
dXt = aXt dt + ПѓXt dWt .

Now consider the relative return from such a process over the increment dYt =
dXt /Xt . Putting Yt = g(Xt ) = ln(Xt ) note that analogous to our derivation of
ItoвЂ™s lemma
1
dg(Xt ) = g 0 (Xt )dXt + g 00 (Xt )(dX)2 + ...
2
1 1 22
{aXt dt + ПѓXt dWt .} в€’
= 2 Пѓ Xt dt
Xt 2Xt
Пѓ2
= (a в€’ )dt + ПѓdWt .
2
74 CHAPTER 2. SOME BASIC THEORY OF FINANCE

which is a description of a general Brownian motion process, a process with
Пѓ2
increments dYt that are normally distributed with mean (a в€’ and with
2 )dt

variance Пѓ 2 dt. This process satisfying dXt = aXt dt + ПѓXt dWt is called the
Geometric Brownian motion process (because it can be written in the form
Xt = eYt for a Brownian motion process Yt ) or a Black-Scholes model.
Many of the continuous time models used in п¬Ѓnance are described as Markov
diп¬Ђusions or Ito processes which permits the mean and the variance of the
increments to depend more generally on the present value of the process and
the time. The integral version of this relation is of the form
Z Z
T T
XT = X0 + a(Xt , t)dt + Пѓ(Xt , t)dWt .
0 0

We often write such an equation with diп¬Ђerential notation,

(2.26)
dXt = a(Xt , t)dt + Пѓ(Xt , t)dWt .

but its meaning should always be sought in the above integral form. The co-
eп¬ѓcients a(Xt , t) and Пѓ(Xt , t) vary with the choice of model. As usual, we
interpret 2.26 as meaning that a small increment in the process, say dXt =
Xt+h в€’ Xt (h very small) is approximately distributed according to a normal
distribution with conditional mean a(Xt , t)dt and conditional variance given by
Пѓ 2 (Xt , t)var(dWt ) = Пѓ 2 (Xt , t)dt. Here the mean and variance are conditional
on Ht , the history of the process Xt up to time t.
Various choices for the functions a(Xt , t), Пѓ(Xt , t) are possible. For the
Black-Scholes model or geometric Brownian motion, a(Xt , t) = aXt and Пѓ(Xt , t) =
ПѓXt for constant drift and volatility parameters a, Пѓ. The Cox-Ingersoll-Ross
model, used to model spot interest rates, corresponds to a(Xt , t) = A(b в€’ Xt )
в€љ
and Пѓ(Xt , t) = c Xt for constants A, b, c. The Vasicek model, also a model for
interest rates, has a(Xt , t) = A(b в€’ Xt ) and Пѓ(Xt , t) = c. There is a large num-
ber of models for most continuous time processes observed in п¬Ѓnance which can
be written in the form 2.26. So called multi-factor models are of similar form
MODELS IN CONTINUOUS TIME 75

where Xt is a vector of п¬Ѓnancial time series and the coeп¬ѓcient functions a(Xt , t)
is vector valued, Пѓ(Xt , t) is replaced by a matrix-valued function and dWt is
interpreted as a vector of independent Brownian motion processes. For techni-
cal conditions on the coeп¬ѓcients under which a solution to 2.26 is guaranteed
to exist and be unique, see Karatzas and Shreve, sections 5.2, 5.3.
As with any diп¬Ђerential equation there may be initial or boundary condi-
tions applied to 2.26 that restrict the choice of possible solutions. Solutions
to the above equation are diп¬ѓcult to arrive at, and it is often even more diп¬ѓ-
cult to obtain distributional properties of them. Among the key tools are the
Kolmogorov diп¬Ђerential equations (see Cox and Miller, p. 215). Consider the
transition probability kernel

p(s, z, t, x) = P [Xt = x|Xs = z]

in the case of a discrete Markov Chain. If the Markov chain is continuous (as it
is in the case of diп¬Ђusions), that is if the conditional distribution of Xt given Xs
is absolutely continuous with respect to Lebesgue measure, then we can deп¬Ѓne
p(s, z, t, x) to be the conditional probability density function of Xt given Xs = z.
The two equations, for a diп¬Ђusion of the above form, are:
KolmogorovвЂ™s backward equation

в€‚2
в€‚ в€‚ 12
p = в€’a(z, s) p в€’ Пѓ (z, s) 2 p (2.27)
в€‚s в€‚z 2 в€‚z

and the forward equation

1 в€‚2 2
в€‚ в€‚
p = в€’ (a(x, t)p) + (2.28)
(Пѓ (x, t)p)
2 в€‚x2
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