Some Basic Theory of

Finance

Introduction to Pricing: Single Period Models

Let us begin with a very simple example designed to illustrate the no-arbitrage

approach to pricing derivatives. Consider a stock whose price at present is $s.

Over a given period, the stock may move either up or down, up to a value su

where u > 1 with probability p or down to the value sd where d < 1 with

probability 1 ’ p. In this model, these are the only moves possible for the stock

in a single period. Over a longer period, of course, many other values are

possible. In this market, we also assume that there is a so-called risk-free bond

available returning a guaranteed rate of r% per period. Such a bond cannot

default; there is no random mechanism governing its return which is known

upon purchase. An investment of $1 at the beginning of the period returns a

guaranteed $(1 + r) at the end. Then a portfolio purchased at the beginning

of a period consisting of y stocks and x bonds will return at the end of the

period an amount $x(1 + r) + ysZ where Z is a random variable taking

13

14 CHAPTER 2. SOME BASIC THEORY OF FINANCE

values u or d with probabilities p and 1 ’ p respectively. We permit owning

a negative amount of a stock or bond, corresponding to shorting or borrowing

the correspond asset for immediate sale.

An ambitious investor might seek a portfolio whose initial cost is zero (i.e.

x + ys = 0) such that the return is greater than or equal to zero with positive

probability. Such a strategy is called an arbitrage. This means that the investor

is able to achieve a positive probability of future pro¬ts with no down-side risk

with a net investment of $0. In mathematical terms, the investor seeks a point

(x, y) such that x + ys = 0 (net cost of the portfolio is zero) and

x(1 + r) + ysu ≥ 0,

x(1 + r) + ysd ≥ 0

with at least one of the two inequalities strict (so there is never a loss and a

non-zero chance of a positive return). Alternatively, is there a point on the line

y = ’ 1 x which lies above both of the two lines

s

1+r

y=’ x

su

1+r

y=’ x

sd

and strictly above one of them? Since all three lines pass through the origin,

we need only compare the slopes; an arbitrage will NOT be possible if

1+r 1 1+r

’ ·’ ·’ (2.1)

sd s su

and otherwise there is a point (x, y) permitting an arbitrage. The condition for

no arbitrage (2.1) reduces to

d u

(2.2)

<1<

1+r 1+r

So the condition for no arbitrage demands that (1 + r ’ u) and (1 + r ’ d)

have opposite sign or d · (1 + r) · u. Unless this occurs, the stock always

has either better or worse returns than the bond, which makes no sense in a

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 15

free market where both are traded without compulsion. Under a no arbitrage

assumption since d · (1 + r) · u, the bond payo¬ is a convex combination or

a weighted average of the two possible stock payo¬s; i.e. there are probabilities

0 · q · 1 and (1 ’ q) such that (1 + r) = qu + (1 ’ q)d. In fact it is easy to

solve this equation to determine the values of q and 1 ’ q.

(1 + r) ’ d u ’ (1 + r)

and 1 ’ q =

q= , .

u’d u’d

Denote by Q the probability distribution which puts probabilities q and 1 ’ q

on these points su, sd. Then if S1 is the value of the stock at the end of the

period, note that

1 1 1

(qsu + (1 ’ q)sd) =

EQ (S1 ) = s(1 + r) = s

1+r 1+r 1+r

where EQ denotes the expectation assuming that Q describes the probabilities

of the two outcomes.

In other words, if there is to be no arbitrage, there exists a probability mea-

sure Q such that the expected price of future value of the stock S1 discounted

to the present using the return from a risk-free bond is exactly the present value

of the stock. The measure Q is called the risk-neutral measure and the prob-

abilities that it assigns to the possible outcomes of S are not necessarily those

that determine the future behaviour of the stock. The risk neutral measure

embodies both the current consensus beliefs in the future value of the stock and

the consensus investors™ attitude to risk avoidance. It is not usually true that

1

= s with P denoting the actual probability distribution describing

1+r EP (S1 )

the future probabilities of the stock. Indeed it is highly unlikely that an investor

would wish to purchase a risky stock if he or she could achieve exactly the same

expected return with no risk at all using a bond. We generally expect that

to make a risky investment attractive, its expected return should be greater

than that of a risk-free investment. Notice in this example that the risk-neutral

measure Q did not use the probabilities p, and 1 ’ p that the stock would go

16 CHAPTER 2. SOME BASIC THEORY OF FINANCE

up or down and this seems contrary to intuition. Surely if a stock is more likely

to go up, then a call option on the stock should be valued higher!

Let us suppose for example that we have a friend willing, in a private trans-

action with me, to buy or sell a stock at a price determined from his subjectively

assigned distribution P , di¬erent from Q. The friend believes that the stock

is presently worth

psu + (1 ’ p)sd

1

6= s since p 6= q.

EP S1 =

1+r 1+r

Such a friend o¬ers their assets as a sacri¬ce to the gods of arbitrage. If the

friend™s assessed price is greater than the current market price, we can buy on

the open market and sell to the friend. Otherwise, one can do the reverse.

Either way one is enriched monetarily (and perhaps impoverished socially)!

So why should we use the Q measure to determine the price of a given asset

in a market (assuming, of course, there is a risk-neutral Q measure and we are

able to determine it)? Not because it precisely describes the future behaviour

of the stock, but because if we use any other distribution, we o¬er an intelligent

investor (there are many!) an arbitrage opportunity, or an opportunity to make

money at no risk and at our expense.

Derivatives are investments which derive their value from that of a corre-

sponding asset, such as a stock. A European call option is an option which

permits you (but does not compel you) to purchase the stock at a ¬xed future

date ( the maturity date) or for a given predetermined price, the exercise price

of the option). For example a call option with exercise price $10 on a stock

whose future value is denoted S1 , is worth on expiry S1 ’ 10 if S1 > 10 but

nothing at all if S1 < 10. The di¬erence S1 ’ 10 between the value of the stock

on expiry and the exercise price of the option is your pro¬t if you exercises the

option, purchasing the stock for $10 and sell it on the open market at $S1 .

However, if S1 < 10, there is no point in exercising your option as you are

not compelled to do so and your return is $0. In general, your payo¬ from pur-

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 17

chasing the option is a simple function of the future price of the stock, such as

V (S1 ) = max(S1 ’ 10, 0). We denote this by (S1 ’ 10)+ . The future value of

the option is a random variable but it derives its value from that of the stock,

hence it is called a derivative and the stock is the underlying.

A function of the stock price V (S1 ) which may represent the return from a

portfolio of stocks and derivatives is called a contingent claim. V (S1 ) repre-

sents the payo¬ to an investor from a certain ¬nancial instrument or derivative

when the stock price at the end of the period is S1 . In our simple binomial

example above, the random variable takes only two possible values V (su) and

V (sd). We will show that there is a portfolio, called a replicating portfolio, con-

sisting of an investment solely in the above stock and bond which reproduces

these values V (su) and V (sd) exactly. We can determine the corresponding

weights on the bond and stocks (x, y) simply by solving the two equations in

two unknowns

x(1 + r) + ysu = V (su)

x(1 + r) + ysd = V (sd)

V (su)’y— su

V (su)’V (sd)

Solving: y — = and x— = By buying y — units of

.

su’sd 1+r

stock and x— units of bond, we are able to replicate the contingent claim V (S1 )

exactly- i.e. produce a portfolio of stocks and bonds with exactly the same

return as the contingent claim. So in this case at least, there can be only one

possible present value for the contingent claim and that is the present value

of the replicating portfolio x— + y — s. If the market placed any other value

on the contingent claim, then a trader could guarantee a positive return by a

simple trade, shorting the contingent claim and buying the equivalent portfolio

or buying the contingent claim and shorting the replicating portfolio. Thus this

is the only price that precludes an arbitrage opportunity. There is a simpler

18 CHAPTER 2. SOME BASIC THEORY OF FINANCE

expression for the current price of the contingent claim in this case: Note that

1 1

(qV (su) + (1 ’ q)V (sd))

EQ V (S1 ) =

1+r 1+r

1 1+r’d u ’ (1 + r)

= ( V (su) + V (sd))

1+r u’d u’d

= x— + y — s.

In words, the discounted expected value of the contingent claim is equal to

the no-arbitrage price of the derivative where the expectation is taken using the

Q-measure. Indeed any contingent claim that is attainable must have its price

determined in this way. While we have developed this only in an extremely

simple case, it extends much more generally.

Suppose we have a total of N risky assets whose prices at times t = 0, 1,

j j

are given by (S0 , S1 ), j = 1, 2, ..., N. We denote by S0 , S1 the column vector of

initial and ¬nal prices

⎛ ⎞ ⎛ ⎞

1 1

S0 S1

⎜ ⎟ ⎜ ⎟

⎜2 ⎟ ⎜2 ⎟

⎜ S0 ⎟ ⎜ S1 ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜. ⎟ ⎜. ⎟

⎜ ⎟ ⎜ ⎟

S0 = ⎜ ⎟ , S1 = ⎜ ⎟

⎜. ⎟ ⎜. ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜. ⎟ ⎜. ⎟

⎝ ⎠ ⎝ ⎠

N N

S0 S1

where at time 0, S0 is known and S1 is random. Assume also there is a riskless

asset (a bond) paying interest rate r over one unit of time. Suppose we borrow

money (this is the same as shorting bonds) at the risk-free rate to buy wj units

P j

of stock j at time 0 for a total cost of wj S0 . The value of this portfolio at

P j j

wj (S1 ’ (1 + r)S0 ). If there are weights wj so that

time t = 1 is T (w) =

this sum is always non-negative, and P (T (w) > 0) > 0, then this is an arbitrage

opportunity. Similarly, by replacing the weights wj by their negative ’wj ,

there is an arbitrage opportunity if for some weights the sum is non-positive

and negative with positive probability. In summary, there are no arbitrage op-

INTRODUCTION TO PRICING: SINGLE PERIOD MODELS 19

portunities if for all weights wj P (T (w) > 0) > 0 and P (T (w) < 0) > 0 so

T (w) takes both positive and negative values. We assume that the moment

P j j

generating function M (w) = E[exp( wj (S1 ’ (1 + r)S0 ))] exists and is an an-

alytic function of w.Roughly the condition that the moment generating function

is analytic assures that we can expand the function in a series expansion in w.

This is the case, for example, if the values of S1 , S0 are bounded. The following

theorem provides a general proof, due to Chris Rogers, of the equivalence of the

no-arbitrage condition and the existence of an equivalent measure Q. Refer to

the appendix for the technical de¬nitions of an equivalent probability measure

and the existence and properties of a moment generating function M (w).

Theorem 2 A necessary and su¬cient condition that there be no arbitrage op-

j

portunities is that there exists a measure Q equivalent to P such that EQ (S1 ) =

j

1

for all j = 1, ..., N.

1+r S0

P j j

wj (S1 ’ (1 + r)S0 ))] and

Proof. De¬ne M (w) = E exp(T (w)) = E[exp(

consider the problem

min ln(M (w)).

w

The no-arbitrage condition implies that for each j there exists µ > 0,

j j

P [S1 ’ (1 + r)S0 > µ] > 0

and therefore as wj ’ ∞ while the other weights wk , k 6= j remain ¬xed,

X j j j j

wj (S1 ’(1+r)S0 ))] > C exp(wj µ)P [S1 ’(1+r)S0 > µ] ’ ∞ as wj ’ ∞.

M (w) = E[exp(

Similarly, M (w) ’ ∞ as wj ’ ’∞. From the properties of a moment gen-

erating function (see the appendix) M (w) is convex, continuous, analytic and

‚M

M (0) = 1. Therefore the function M (w) has a minimum w— satisfying =0

‚wj

or

‚M (w)

= 0 or (2.3)

‚wj

j j

E[S1 exp(T (w))] = (1 + r)S0 E[exp(T (w))]

20 CHAPTER 2. SOME BASIC THEORY OF FINANCE

or

j

E[exp(T (w))S1 ]

j

S0 = .

(1 + r)E[exp(T (w))]

De¬ne a distribution or probability measure Q as follows; for any event A,

EP [IA exp(w0 S1 )]

Q(A) = .

EP [exp(w0 S1 )]

The Radon-Nikodym derivative (see the appendix) is

exp(w0 S1 )]

dQ

= .

EP [exp(w0 S1 )]

dP

dQ

Since ∞ > > 0, the measure Q is equivalent to the original probability mea-

dP

sure P (in the intuitive sense that it has the same support). When we calculate

expected values under this new measure, note that for each j,

dQ j

j

EQ (S1 ) = EP [ S]

dP 1

j

EP [S1 exp(w0 S1 )]

=

EP [exp(w0 S1 )]

j

= (1 + r)S0 .

or

1

j j

S0 = EQ (S1 ).

1+r

Therefore, the current price of each stock is the discounted expected value of the

future price under this “risk-neutral” measure Q.

Conversely if

1

j j

S0 , for all j (2.4)

EQ (S1 ) =

1+r

holds for some measure Q then EQ [T (w)] = 0 for all w and this implies that the

random variable T (w) is either identically 0 or admits both positive and negative

values. Therefore the existence of the measure Q satisfying (2.4) implies that

there are no arbitrage opportunities.

The so-called risk-neutral measure Q is constructed to minimize the cross-

entropy between Q and P subject to the constraints E(S1 ’ (1 + r)S0 ) = 0

MULTIPERIOD MODELS. 21

where cross-entropy is de¬ned in Section 1.5. If there N possible values of the

random variables S1 and S0 then (2.3) consists of N equations in N unknowns

and so it is reasonable to expect a unique solution. In this case, the Q measure

is unique and we call the market complete.

The theory of pricing derivatives in a complete market is rooted in a rather

trivial observation because in a complete market, the derivative can be replicated

with a portfolio of other marketable securities. If we can reproduce exactly the

same (random) returns as the derivative provides using a linear combination of

other marketable securities (which have prices assigned by the market) then the

derivative must have the same price as the linear combination of other securities.

Any other price would provide arbitrage opportunities.

Of course in the real world, there are costs associated with trading, these

costs usually related to a bid-ask spread. There is essentially a di¬erent price for

buying a security and for selling it. The argument above assumes a frictionless

market with no trading costs, with borrowing any amount at the risk-free bond

rate possible, and a completely liquid market- any amount of any security can be

bought or sold. Moreover it is usually assumed that the market is complete and

it is questionable whether complete markets exist. For example if a derivative

security can be perfectly replicated using other marketable instruments, then

what is the purpose of the derivative security in the market? All models,

excepting those on Fashion File, have de¬ciencies and critics. The merit of the

frictionless trading assumption is that it provides an accurate approximation

to increasingly liquid real-world markets. Like all useful models, this permits

tentative conclusions that should be subject to constant study and improvement.

Multiperiod Models.

When an asset price evolves over time, the investor normally makes decisions

about the investment at various periods during its life. Such decisions are made

22 CHAPTER 2. SOME BASIC THEORY OF FINANCE

with the bene¬t of current information, and this information, whether used

or not, includes the price of the asset and any related assets at all previous

time periods, beginning at some time t = 0 when we began observation of the

process. We denote this information available for use at time t as Ht . Formally,

Ht is what is called a sigma-¬eld (see the appendix) generated by the past, and

there are two fundamental properties of this sigma-¬eld that will use. The ¬rst

is that the sigma-¬elds increase over time. In other words, our information

about this and related processes increases over time because we have observed

more of the relevant history. In the mathematical model, we do not “forget”

relevant information: this model ¬ts better the behaviour of youthful traders

than aging professors. The second property of Ht is that it includes the value

of the asset price S„ , „ · t at all times „ · t. In measure-theoretic language, St

is adapted to or measurable with respect to Ht . Now the analysis above shows

that when our investment life began at time t = 0 and we were planning for the

next period of time, absence of arbitrage implies a risk-neutral measure Q such

1

that EQ ( 1+r S1 ) = S0 . Imagine now that we are in a similar position at time

t, planning our investment for the next unit time. All expected values should

be taken in the light of our current knowledge, i.e. given the information Ht .

An identical analysis to that above shows that under the risk neutral measure

Q, if St represents the price of the stock after t periods, and rt the risk-free

one-period interest rate o¬ered that time, then

1

St+1 |Ht ) = St . (2.5)

EQ (

1 + rt

Suppose we let Bt be the value of $1 invested at time t = 0 after a total

of t periods. Then B1 = (1 + r0 ), B2 = (1 + r0 )(1 + r1 ), and in general

Bt = (1 + r0 )(1 + r1 )...(1 + rt’1 ). Since the interest rate per period is announced

at the beginning of this period, the value Bt is known at time t ’ 1. If you

owe exactly $1.00 payable at time t, then to cover this debt you should have an

MULTIPERIOD MODELS. 23

investment at time t = 0 of $E(1/Bt ), which we might call the present value

of the promise. In general, at time t, the present value of a certain amount

$VT promised at time T (i.e. the present value or the value discounted to the

present of this payment) is

Bt

|Ht ).

E(VT

BT

Now suppose we divide (2.5) above by Bt. We obtain

St+1 1 1 1 St

|Ht ) = EQ ( St+1 |Ht ) = St+1 |Ht ) =

EQ ( EQ ( .

Bt+1 Bt (1 + rt ) Bt 1 + rt Bt

(2.6)

Notice that we are able to take the divisor Bt outside the expectation since Bt

is known at time t (in the language of Appendix 1, Bt is measurable with re-

spect to Ht+1 ). This equation (2.6) describes an elegant mathematical property

shared by all marketable securities in a complete market. Under the risk-neutral

measure, the discounted price Yt = St /Bt forms a martingale. A martingale

is a process Yt for which the expectation of a future value given the present is

equal to the present i.e.

E(Yt+1 |Ht ) = Yt .for all t. (2.7)

Properties of a martingale are given in the appendix and it is easy to show that

for such a process, when T > t,

E(YT |Ht ) = E[...E[E(YT |HT ’1 )|HT ’2 ]...|Ht ] = Yt . (2.8)

A martingale is a fair game in a world with no in¬‚ation, no need to consume

and no mortality. Your future fortune if you play the game is a random vari-

able whose expectation, given everything you know at present, is your present

fortune.

Thus, under a risk-neutral measure Q in a complete market, all marketable

securities discounted to the present form martingales. For this reason, we often

refer to the risk-neutral measure as a martingale measure. The fact that prices of

24 CHAPTER 2. SOME BASIC THEORY OF FINANCE

marketable commodities must be martingales under the risk neutral measure has

many consequences for the canny investor. Suppose, for example, you believe

that you are able to model the history of the price process nearly perfectly, and

it tells you that the price of a share of XXX computer systems increases on

average 20% per year. Should you use this P ’measure in valuing a derivative,

even if you are con¬dent it is absolutely correct, in pricing a call option on

XXX computer systems with maturity one year from now? If you do so, you are

o¬ering some arbitrager another free lunch at your expense. The measure Q,

not the measure P , determines derivative prices in a no-arbitrage market. This

also means that there is no advantage, when pricing derivatives, in using some

elaborate statistical method to estimate the expected rate of return because this

is a property of P not Q.

What have we discovered? In general, prices in a market are determined as

expected values, but expected values with respect to the measure Q. This is true

in any complete market, regardless of the number of assets traded in the market.

For any future time T > t, and for any derivative de¬ned on the traded assets

Bt

in a market whose value at time t is given by Vt , EQ ( BT VT |Ht ] = Vt = the

market price of the derivative at time t. So in theory, determining a reasonable

price of a derivative should be a simple task, one that could be easily handled

by simulation. Suppose we wish to determine a suitable price for a derivative

whose value is determined by some stock price process St . Suppose that at

time T > t, the value of the derivative is a simple function of the stock price at

that time VT = V (ST ). We may simply generate many simulations of the future

value of the stock and corresponding value of the derivative ST , V (ST ) given the

current store of information Ht . These simulations must be conducted under the

measure Q. In order to determine a fair price for the derivative, we then average

the discounted values of the derivatives, discounted to the present, over all the

simulations. The catch is that the Q measure is often neither obvious from

the present market prices nor statistically estimable from its past. It is given

MULTIPERIOD MODELS. 25

implicitly by the fact that the expected value of the discounted future value of

traded assets must produce the present market price. In other words, a ¬rst

step in valuing any asset is to determine a measure Q for which this holds. Now

in some simple models involving a single stock, this is fairly simple, and there

is a unique such measure Q. This is the case, for example, for the stock model

above in which the stock moves in simple steps, either increasing or decreasing

at each step. But as the number of traded assets increases, and as the number

of possible jumps per period changes, a measure Q which completely describes

the stock dynamics and which has the necessary properties for a risk neutral

measure becomes potentially much more complicated as the following example

shows.

Solving for the Q Measure.

Let us consider the following simple example. Over each period, a stock price

provides a return greater than, less than, or the same as that of a risk free

investment like a bond. Assume for simplicity that the stock changes by the

factor u(1 + r) (greater) or (1 + r) (the same) d(1 + r)(less) where u > 1 > d =

1/u. The Q probability of increases and decreases is unknown, and may vary

from one period to the next. Over two periods, the possible paths executed by

this stock price process are displayed below assuming that the stock begins at

time t = 0 with price S0 = 1.

[FIGURE 2.1 ABOUT HERE]

In general in such a tree there are three branches from each of the nodes

at times t = 0, 1 and there are a total of 1 + 3 = 4 such nodes. Thus, even

if we assume that probabilities of up and down movements do not depend on

how the process arrived at a given node, there is a total of 3 — 4 = 12 unknown

parameters. Of course there are constraints; for example the sum of the three

probabilities on branches exiting a given node must add to one and the price

26 CHAPTER 2. SOME BASIC THEORY OF FINANCE

Figure 2.1: A Trinomial Tree for Stock Prices

process must form a martingale. For each of the four nodes, this provides two

constraints for a total of 8 constraints, leaving 4 parameters to be estimated.

We would need the market price of 4 di¬erent derivatives or other contingent

claims to be able to generate 4 equations in these 4 unknowns and solve for

them. Provided we are able to obtain prices of four such derivatives, then we

can solve these equations. If we denote the risk-neutral probability of ™up™ at

each of the four nodes by p1 , p2 , p3 , p4 then the conditional distribution of St+1

given St = s is:

Stock value su(1 + r) s(1 + r) sd(1 + r)

u’d u’1

1’ = 1 ’ kpi

Probability pi 1’d pi 1’d pi = cpi

Consider the following special case, with the risk-free interest rate per period

r, u = 1.089, S0 = $1.00. We also assume that we are given the price of four

call options expiring at time T = 2. The possible values of the price at time

T = 2 corresponding to two steps up, one step up and one constant, one up

one down, etc. are the values of S(T ) in the set

{1.1859, 1.0890, 1.0000, 0.9183, 0.8432}.

Now consider a “call option” on this stock expiring at time T = 2 with strike

MULTIPERIOD MODELS. 27

price K. Such an option has value at time t = 2 equal to (S2 ’ K) if this is

For brevity we denote this by (S2 ’ K)+ . The

positive, or zero otherwise.

present value of the option is EQ (S2 ’ K)+ discounted to the present, where

K is the exercise price of the option and S2 is the price of the stock at time 2.

Thus the price of the call option at time 0 is given by

V0 = EQ (S2 ’ K)+ /(1 + r)2

Assuming interest rate r = 1% per period, suppose we have market prices of four

call options with the same expiry and di¬erent exercise prices in the following

table;

K =Exercise Price V0 =Call Option Price

T =Maturity

0.867 2 0.154

0.969 2 .0675

1.071 2 .0155

1.173 2 .0016

If we can observe the prices of these options only, then the equations to be

solved for the probabilities associated with the measure Q equate the observed

price of the options to their theoretical price V0 = E(S2 ’ K)+ /(1 + r)2 .

1

(1.186 ’ 1.173)p1 p2

0.0016 =

(1.01)2

1

[(1.186 ’ 1.071)p1 p2 + (1.089 ’ 1.071){p1 (1 ’ kp2 ) + (1 ’ kp1 )p2 }]

0.0155 =

(1.01)2

1

[0.217p1 p2 + 0.12{p1 (1 ’ kp2 ) + (1 ’ kp1 )p2 }

0.0675 =

(1.01)2

+ 0.031{(1 ’ kp1 )(1 ’ kp2 ) + cp1 p2 + cp1 p4 )}

1

[0.319p1 p2 + 0.222{p1 (1 ’ kp2 ) + (1 ’ kp1 )p2 }

0.154 =

(1.01)2

+ 0.133{(1 ’ kp1 )(1 ’ kp2 ) + cp1 p2 + cp1 p4 )}

+ 0.051{{cp1 (1 ’ kp4 ) + (1 ’ kp1 )cp3 }].

28 CHAPTER 2. SOME BASIC THEORY OF FINANCE

While it is not too di¬cult to solve this system in this case one can see that

with more branches and more derivatives, this non-linear system of equations

becomes di¬cult very quickly. What do we do if we observe market prices for

only two derivatives de¬ned on this stock, and only two parameters can be

obtained from the market information? This is an example of what is called

an incomplete market, a market in which the risk neutral distribution is not

uniquely speci¬ed by market information. In general when we have fewer

equations than parameters in a model, there are really only two choices

(a) Simplify the model so that the number of unknown parameters and the

number of equations match.

(b) Determine additional natural criteria or constraints that the parameters

must satisfy.

In this case, for example, one might prefer a model in which the probability

of a step up or down depends on the time, but not on the current price of the

stock. This assumption would force equal all of p2 = p3 = p4 and simplify the

system of equations above. For example using only the prices of the ¬rst two

derivatives, we obtain equations, which, when solved, determine the probabilities

on the other branches as well.

1

(1.186 ’ 1.173)p1 p2

0.0016 =

(1.01)2

1

[(1.186 ’ 1.071)p1 p2 + (1.089 ’ 1.071){p1 (1 ’ kp2 ) + (1 ’ kp1 )p2 }]

0.0155 =

(1.01)2

This example re¬‚ects a basic problem which occurs often when we build a

reasonable and ¬‚exible model in ¬nance. Frequently there are more parameters

than there are marketable securities from which we can estimate these parame-

ters. It is quite common to react by simplifying the model. For example, it

is for this reason that binomial trees (with only two branches emanating from

each node) are often preferred to the trinomial tree example we use above, even

though they provide a worse approximation to the actual distribution of stock

MULTIPERIOD MODELS. 29

returns.

In general if there are n di¬erent securities (excluding derivatives whose value

is a function of one or more of these) and if each security can take any one of m

di¬erent values, then there are a total of mn possible states of nature at time

t = 1. The Q measure must assign a probability to each of them. This results in

a total of mn unknown probability values, which, of course must add to one, and

result in the right expectation for each of n marketable securities. To uniquely

determine Q we would require a total of mn ’ n ’ 1 equations or mn ’ n ’ 1

di¬erent derivatives. For example for m = 10, n = 100, approximately one with

a hundred zeros, a prohibitive number, are required to uniquely determine Q.

In a complete market, Q is uniquely determined by marketed securities, but

in e¬ect no real market can be complete. In real markets, one asset is not

perfectly replicated by a combination of other assets because there is no value

in duplication. Whether an asset is a derivative whose value is determined by

another marketed security, together with interest rates and volatilities, markets

rarely permit exact replication. The most we can probably hope for in practice

is to ¬nd a model or measure Q in a subclass of measures with desirable features

under which

Bt

V (ST )|Ht ] ≈ Vt for all marketable V. (2.9)

EQ [

BT

Even if we had equalities in (2.9), this would represent typically fewer equa-

tions than the number of unknown Q probabilities so some simpli¬cation of the

model is required before settling on a measure Q. One could, at one™s peril,

ignore the fact that certain factors in the market depend on others. Similar

stocks behave similarly, and none may be actually independent. Can we, with

any reasonable level of con¬dence, accurately predict the e¬ect that a lowering

of interest rates will have on a given bank stock? Perhaps the best model

for the future behaviour of most processes is the past, except that as we have

seen the historical distribution of stocks do not generally produce a risk-neutral

30 CHAPTER 2. SOME BASIC THEORY OF FINANCE

measure. Even if historical information provided a ¬‚awless guide to the future,

there is too little of it to accurately estimate the large number of parameters

required for a simulation of a market of reasonable size. Some simpli¬cation of

the model is clearly necessary. Are some baskets of stocks independent of other

combinations? What independence can we reasonably assume over time?

As a ¬rst step in simplifying a model, consider some of the common measures

of behaviour. Stocks can go up, or down. The drift of a stock is a tendency in

one or other of these two directions. But it can also go up and down- by a lot

or a little. The measure of this, the variance or variability in the stock returns

is called the volatility of the stock. Our model should have as ingredients these

two quantities. It should also have as much dependence over time and among

di¬erent asset prices as we have evidence to support.

Determining the Process Bt .

We have seen in the last section that given the Q or risk-neutral measure, we can,

at least in theory, determine the price of a derivative if we are given the price Bt

of a risk-free investment at time t (in ¬nance such a yardstick for measuring and

discounting prices is often called a “numeraire”). Unfortunately no completely

liquid risk-free investment is traded on the open market. There are government

treasury bills which, depending on the government, one might wish to assume

are almost risk-free, and there are government bonds, usually with longer terms,

which complicate matters by paying dividends periodically. The question dealt

with in this section is whether we can estimate or approximate an approximate

risk-free process Bt given information on the prices of these bonds. There are

typically too few genuinely risk-free bonds to get a detailed picture of the process

Bs , s > 0. We might use government bonds for this purpose, but are these

genuinely risk-free? Might not the additional use of bonds issued by other large

corporations provide a more detailed picture of the bank account process Bs ?

DETERMINING THE PROCESS BT . 31

Can we incorporate information on bond prices from lower grade debt? To

do so, we need a simple model linking the debt rating of a given bond and the

probability of default and payo¬ to the bond-holders in the event of default. To

begin with, let us assume that a given basket of companies, say those with a

common debt rating from one of the major bond rating organisations, have a

common distribution of default time. The thesis of this section is that even if

no totally risk-free investment existed, we might still be able to use bond prices

to estimate what interest rate such an investment would o¬er.

We begin with what we know. Presumably we know the current prices of

marketable securities. This may include prices of certain low-risk bonds with

face value F , the value of the bond on maturity at time T. Typically such a bond

pays certain payments of value dt at certain times t < T and then the face value

of the bond F at maturity time T, unless the bond-holder defaults. Let us assume

for simplicity that the current time is 0. The current bond prices P0 provide some

information on Bt as well as the possibility of default. Suppose we let „ denote

the random time at which default or bankruptcy would occur. Assume that the

e¬ect of possible default is to render the payments at various times random so

for example dt is paid provided that default has not yet occurred, i.e. if „ > t,

and similarly the payment on maturity is the face value of the Bond F if default

has not yet occurred and if it has, some fraction of the face value pF is paid.

When a real bond defaults, the payout to bondholders is a complicated function

of the hierarchy of the bond and may occur before maturity, but we choose this

model with payout at maturity in any case for simplicity. Then the current

price of the bond is the expected discounted value of all future payments, so

X 1 pF F

I(„ · T ) +

P0 = EQ ( ds I(„ > s) + I(„ > T ))

Bs BT BT

{s;0<s<T }

X

’1

’1

ds EQ [Bs I(„ > s)] + F EQ [BT (p + (1 ’ p)I(„ > T ))]

=

{s;0<s<T }

32 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The bank account process Bt that we considered is the compounded value at

time of an investment of $1 deposited at time 0. This value might be random

but the interest rate is declared at the beginning of each period so, for example,

Bt is completely determined at time t ’ 1. In measure-theoretical language, Bt

is Ht’1 measurable for each t. With Q is the risk-neutral distribution

X

’1

’1

P0 = EQ { ds Bs Q(„ > s|Hs’1 ) + F BT (p + (1 ’ p)Q(„ > T |HT ’1 ))}.

{s;0<s<T }

This takes a form very similar to the price of a bond which does not default but

with a di¬erent bank account process. Suppose we de¬ne a new bank account

f

process Bs , equivalent in expectation to the risk-free account, but that only

pays if default does not occur in the interval. Such a process must satisfy

f

EQ (Bs I(„ > s)|Hs’1 ) = Bs .

f

From this we see that the process Bs is de¬ned by

Bs

f on the set Q[„ > s|Hs’1 ] > 0.

Bs =

Q[„ > s|Hs’1 ]

In terms of this new bank account process, the price of the bond can be rewritten

as

X

g g

’1 ’1 ’1

P0 = EQ { ds Bs + (1 ’ p)F BT + pF BT }.

{s;0<s<T }

If we subtract from the current bond price the present value of the guaranteed

payment of pF, the result is

X

g g

’1 ’1 ’1

P0 ’ = EQ { ds Bs + (1 ’ p)F BT }.

pF EQ (BT )

{s;0<s<T }

This equation has a simple interpretation. The left side is the price of the

bond reduced by the present value of the guaranteed payment on maturity F p.

The right hand side is the current value of a risk-free bond paying the same

f

dividends, with interest rates increased by replacing Bs by Bs and with face

value F (1 ’ p) all discounted to the present using the bank account process

DETERMINING THE PROCESS BT . 33

f

Bs . In words, to value a defaultable bond, augment the interest rate using the

probability of default in intervals, change the face value to the potential loss of

face value on default and then add the present value of the guaranteed payment

on maturity.

Typically we might expect to be able to obtain prices of a variety of bonds

issued on one ¬rm, or ¬rms with similar credit ratings. If we are willing to

assume that such ¬rms share the same conditional distribution of default time

f

Q[„ > s|Hs’1 ] then they must all share the same process Bs and so each

observed bond price P0 leads to an equation of the form

X g’1

ds vs + (1 ’ p)F BT + pF vT .

e

P0 =

{s;0<s<T }

g’1 ’1

in the unknowns vs = EQ (Bs ), ...s · T. and vT = EQ (BT ). If we assume

e

that the coupon dates of the bonds match, then k bonds of a given maturity

e

T and credit rating will allow us to estimate the k unknown values of vs . Since

the term vT is included in all bonds, it can be estimated from all of the bond

prices, but most accurately from bonds with very low risk.

Unfortunately, this model still has too many unknown parameters to be

generally useful. We now consider a particular case that is considerably simpler.

While it seems unreasonable to assume that default of a bond or bankruptcy

of a ¬rm is unrelated to interest rates, one might suppose some simple model

which allows a form of dependence. For most ¬rms, one might expect that

the probability of survival another unit time is negatively associated with the

interest rate. For example we might suppose that the probability of default in

the next time interval conditional on surviving to the present is a function of

the current interest rate, for example

a + (b ’ 1)rt

ht = Q(„ = t|„ ≥ t, rt ) = .

1 + a + brt

The quantity ht is a more natural measure of the risk at time t than are

other measures of the distribution of „ and the function ht is called the hazard

34 CHAPTER 2. SOME BASIC THEORY OF FINANCE

function. If the constant b > 1+a, then the“hazard” ht increases with increasing

interest rates, otherwise it decreases. In case the default is independent of the

interest rates, we may put b = 1 + a in which case the hazard is a/(1 + a). Then

on the set [„ ≥ s]

1 + rs e

f e

Bs = Bs’1 = (1 + a + brs )Bs’1

1 ’ hs

which means that the bond is priced using a similar bank account process but

one for which the e¬ective interest rate is not rs but a + brs . The di¬erence

a + (b ’ 1)rs between the e¬ective interest rate and rs is usually referred to as

the spread and this model justi¬es using a linear function to model this spread.

Now suppose that default is assumed independent of the past history of interest

rates under the risk-neutral measure Q. In this case, b = 1 + a and the spread

is a(1 + rs ) ' a ' a/(1 + a) provided both a and rs is small. So in this case

the spread gives an approximate risk-neutral probability of default in a given

time interval, conditional on survival to that time.

We might hope that the probabilities of default are very small and follow a

relatively simple pattern. If the pattern is not perfect, then little harm results

provided that indeed the default probabilities are small. Suppose for example

that the time of default follows a geometric distribution so that the hazard is

constant ht = h = a/(1 + a). Then

f

Bs = (1 + a)s Bs for s > 0.

f

Bs grows faster than Bs and it grows even faster as the probability of default h

increases. The e¬ective interest rate on this account is approximately a units

per period higher.

Given only three bond prices with the same default characteristics, for ex-

ample, and assuming constant interest rates so that Bs = (1 + r)s , we may solve

for the values of the three unknown parameters (r, a, p) equations of the form

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.35

X

P0 ’ pF (1 + r)’T = (1 + a + r + ar)’s ds + (1 ’ p)F (1 + a + r + ar)’T .

0<s<T

Market prices for a minimum of three di¬erent bonds would allow us to solve

for the unknowns (r, a, p) and these are obtainable from three di¬erent bonds.

Minimum Variance Portfolios and the Capital As-

set Pricing Model.

Let us begin by building a model for portfolios of securities that captures many

of the features of market movements. We assume that by using the methods of

the previous section and the prices of low-risk bonds, we are able to determine

the value Bt of a risk-free investment at time t in the future. Normally these

values might be used to discount future stock prices to the present. However

for much of this section we will consider only a single period and the analysis

will be essentially the same with our without this discounting.

Suppose we have a number n of potential investments or securities, each

risky in the sense that prices at future dates are random. Suppose we denote

the price of these securities at time t by Si (t), i = 1, 2, ..., n. There is a better

measure of the value of an investment than the price of a security or even the

change in the price of a security Si (t) ’ Si (t ’ 1) over a period because this does

not re¬‚ect the cost of our initial investment. A common measure on investments

that allows to obtain prices, but is more stable over time and between securities

is the return. For a security that has prices Si (t) and Si (t + 1) at times t and

t + 1, we de¬ne the return Ri (t + 1) on the security over this time interval by

Si (t + 1) ’ Si (t)

Ri (t + 1) = .

Si (t)

For example a stock that moved in price from $10 per share to $11 per share

over a period of time corresponds to a return of 10%. Returns can be measured

36 CHAPTER 2. SOME BASIC THEORY OF FINANCE

in units that are easily understood (for example 5% or 10% per unit time) and

are independent of the amount invested. Obviously the $1 pro¬t obtained on

the above stock could has easily been obtained by purchasing 10 shares of a

stock whose value per share changed from $1.00 to $1.10 in the same period

of time, and the return in both cases is 10%. Given a sequence of returns and

the initial value of a stock Si (0), it is easy to obtain the stock price at time t

from the initial price at time 0 and the sequence of returns.

Si (t) = Si (0)(1 + Ri (1))(1 + Ri (2))...(1 + Ri (t))

= Si (0)Πt (1 + Ri (s)).

s=1

Returns are not added over time they are multiplied as above. A 10% return

followed by a 20% return is not a 30% return but a return equal to (1 + .1)(1 +

.2) ’ 1 or 32%. When we buy a portfolio of stocks, the individual stock returns

combine in a simple fashion to give the return on the whole portfolio. For

example suppose that we wish to invest a total amount $I(t) at time t. The

amounts will change from period to period because we may wish to reinvest

gains or withdraw sums from the account. Suppose the proportion of our total

investment in stock i at time t is wi (t) so that the amount invested in stock i is

Pn

wi (t)I(t). Note that since wi (t) are proportions, i=1 wi (t) = 1. What is the

return on this investment over the time interval from t to t + 1? At the end of

this period of time, the value of our investment is

n

X

I(t) wi (t)Si (t + 1).

i=1

If we now subtract the value invested at the beginning of the period and divide

by the value at the beginning, we obtain

P P n

I(t) n wi (t)Si (t + 1) ’ I(t) n wi (t)Si (t) X

i=1 i=1

Pn = wi (t)Ri (t + 1)

I(t) i=1 wi (t)Si (t) i=1

which is just a weighted average of the individual stock returns. Note that it

does not depend on the initial price of the stocks or the total amount that we

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.37

invested at time t. The advantage in using returns instead of stock prices to

assess investments is that the return of a portfolio over a period is a value-

weighted average of the returns of the individual investments.

When time is measured continuously, we might consider de¬ning returns by

using the de¬nition above for a period of length h and then reducing h. In other

words we could de¬ne the instantaneous returns process as

Si (t + h) ’ Si (t)

lim .

Si (t)

h’0

In most cases, the returns over shorter and shorter periods are smaller and

smaller, and approach the limit zero so some renormalization is required above.

It seems more sensible to consider returns per unit time and then take a limit

i.e.

Si (t + h) ’ Si (t)

Ri (t) = lim .

hSi (t)

h’0

Notice that by the de¬nition of the derivative of a logarithm and assuming that

this derivative is well-de¬ned,

d ln(Si (t)) 1d

= Si (t)

dt Si (t) dt

Si (t + h) ’ Si (t)

= lim

hSi (t)

h’0

= Ri (t)

In continuous time, if the stock price process Si (t) is di¬erentiable, the natural

de¬nition of the returns process is the derivative of the logarithm of the stock

price. This de¬nition needs some adjustment later because the most common

continuous time models for asset prices does not result in a di¬erentiable process

Si (t). The solution we will use then will be to adopt a new concept of an integral

and recast the above in terms of this integral.

38 CHAPTER 2. SOME BASIC THEORY OF FINANCE

The Capital Asset Pricing Model (CAPM)

We now consider a simpli¬ed model for building a portfolio based on quite basic

properties of the potential investments. Let us begin by assuming a single period

so that we are planning at time t = 0 investments over a period ending at time

t = 1. We also assume that investors are interested in only two characteristics of

a potential investment, the expected value and the variance of the return over

this period. We have seen that the return of a portfolio is the value-weighted

average of the returns of the individual investments so let us denote the return

on stock i by

Si (1) ’ Si (0)

Ri = ,

Si (0)

and de¬ne µi = E(Ri ) and wi the proportion of my total investment in stock i

at the beginning of the period. For brevity of notation, let R, w and µ denote

the column vectors

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

R1 w1 µ1

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ R2 ⎟ ⎜ ⎟ ⎜ ⎟

w2 µ2

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

R =⎜ ⎟,w =⎜ ⎟ ,µ =⎜ ⎟.

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ .⎟ ⎜ ⎟ ⎜ ⎟

. .

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Rn wn µn

P

wi Ri or in matrix notation w0 R. Let us

Then the return on the portfolio is i

suppose that the covariance matrix of returns is the n — n matrix Σ so that

cov(Ri , Rj ) = Σij .

We will frequently use the following properties of expected value and covariance.

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.39

Lemma 3 Suppose ⎛ ⎞

R1

⎜ ⎟

⎜ ⎟

⎜ R2 ⎟

⎜ ⎟

⎜ ⎟

⎜ .⎟

⎜ ⎟

R =⎜ ⎟

⎜ .⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ .⎟

⎝ ⎠

Rn

is a column vector of random variables Ri with E(Ri ) = µi , i = 1, ..., n and

suppose R has covariance matrix Σ. Suppose A is a non-random vector or matrix

with exactly n columns so that AR is a vector of random variables. Then AR

has mean Aµ and covariance matrix AΣA0 .

Then it is easy to see that the expected return from the portfolio with weights

P P

wi is i wi E(Ri ) = i wi µi = w0 µ and the variance is

var(w0 R) = w0 Σw.

We will need to assume that the covariance matrix Σ is non-singular, that

is it has a matrix inverse Σ’1 . This means, at least for the present, that our

model covers only risky stocks for which the variance of returns is positive. If

a risk-free investment is available (for example a secure bond whose return is

known exactly in advance), this will be handled later.

In the Capital Asset Pricing model it is assumed at the outset that investors

concentrate on two measures of return from a portfolio, the expected value and

standard deviation. These expected values and variances are computed under

the real-world probability distribution P not under some risk-neutral Q measure.

Clearly investors prefer high expected return, wherever possible, associated with

small standard deviation of return. As a ¬rst step in this direction suppose we

plot the standard deviation and expected return for the n stocks, i.e. the n

p √

points {(σi , µi ), i = 1, 2, ..., n} where µi = E(Ri ) and σi = var(Ri ) = Σii .

These n points do not consist of the set of all achievable values of mean and

40 CHAPTER 2. SOME BASIC THEORY OF FINANCE

standard of return, since we are able to construct a portfolio with a certain

proportion of our wealth wi invested in stock i.In fact the set of possible points

consists of

X

√

{( w0 Σw, w0 µ) as the vector w ranges over all possible weights such that wi = 1}.

The resulting set has a boundary as in Figure 2.2.

0.2

0.18

0.16

0.14

·=mean return

0.12

Efficient Frontier

0.1

0.08

0.06

(σ ,· )

g g

0.04

0.02

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σ =standard deviation of return

Figure 2.2: The E¬cient Frontier

[FIGURE 2.2 ABOUT HERE]

Exactly what form this ¬gure takes depends in part on the assumptions ap-

plied to the weights. Since they represent the proportion of our total investment

in each of n stocks they must add to one. Negative weights correspond to selling

short one stock so as to be able to invest more in another, and we may assume

no limit on our ability to do so. In this case the only constraint on w is the

P

constraint wi = 1. With this constraint alone, we can determine the bound-

ary of the admissible set by ¬xing the vertical component (the mean return) of

a portfolio at some value say · and then ¬nding the minimum possible standard

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.41

deviation corresponding to that mean. This allows us to determine the leading

edge or left boundary of the region. The optimisation problem is as follows

√

min w0 Σw subject to

subject to the two constraints on the weights

w0 1 = 1

w0 µ = ·.

where 1 is the column vector of n ones. Since we will often make use of the

method of Lagrange multipliers for constrained problems such as this one, we

interject a lemma justifying the method. For details, consult Apostol (1973),

Section 13.7 or any advanced calculus text.

Lemma 4 Consider the optimisation problem

min{f (w); w ∈ Rn } subject to p constraints (2.10)

of the form g1 (w) = 0, g2 (w) = 0, ..., gp (w) = 0.

Then provided the functions f, g1 , ..., gp are continuously di¬erentiable, a nec-

essary solution for a solution to (2.10) is that there is a solution in the n + p

variables (w1 , ...wn , »1 , ..., »p ) of the equations

‚

{f (w) + »1 g1 (w) + ... + »p gp (w)} = 0, i = 1, 2, ..., n

‚wi

‚

{f (w) + »1 g1 (w) + ... + »p gp (w)} = 0, j = 1, 2, ..., p.

‚»j

This constants »i are called the Lagrange multipliers and the function that

is di¬erentiated, {f (w) + »1 g1 (w) + ... + »p gp (w)} is the Lagrangian.

Let us return to our original minimization problem with one small simpli¬-

√

cation. Since minimizing w0 Σw results in the same weight vector w as does

0

minimizing w Σw we choose the latter as our objective function.

42 CHAPTER 2. SOME BASIC THEORY OF FINANCE

We introduce Lagrange multipliers »1 , »2 and we wish to solve

‚ 0

{w Σw + »1 (w0 1 ’ 1) + »2 (w0 µ ’ ·)} = 0, i = 1, 2, ..., n

‚wi

‚ 0

{w Σw + »1 (w0 1 ’ 1) + »2 (w0 µ ’ ·)} = 0, j = 1, 2.

‚»j

The solution is obtained from the simple di¬erentiation rule

‚0 ‚0

w Σw = 2Σw and µw=w

‚w ‚w

and is of the form

w = »1 Σ’1 1+»2 Σ’1 µ

with the Lagrange multipliers »1 , »2 chosen to satisfy the two constraints, i.e.

»1 10 Σ’1 µ + »2 10 Σ’1 1 = 1

»1 µ0 Σ’1 µ + »2 µ0 Σ’1 1 = ·.

Suppose we de¬ne an n — 2 matrix M with columns 1 and µ,

M =[1 µ]

and the 2 — 2 matrix A = (M 0 Σ’1 M )’1 , then the Lagrange multipliers are

given by the vector

⎛ ⎞ ⎡ ¤

»1 1

»=⎝ ⎠ = A⎣ ¦

»2 ·

and the weights by the vector

⎡ ¤

1

w = Σ’1 M A ⎣ ¦. (2.11)

·

We are now in a position to identify the boundary or the curve in Figure 2.2.

√

As the mean of the portfolio · changes, the point takes the form ( w0 Σw, ·)

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.43

with w given by (2.11). Notice that

⎡ ¤

1

0

w Σw = [ 1 · ]A0 M 0 Σ’1 ΣΣ’1 M A ⎣ ¦

·

⎡ ¤

1

= [ 1 · ]A0 M 0 Σ’1 M A ⎣ ¦

·

⎡ ¤

1

= [ 1 · ]A ⎣ ¦

·

= A11 + 2A12 · + A22 · 2 .

√

Therefore a point on the boundary (σ, ·) = ( w0 Σw, ·) satis¬es

σ 2 ’ A22 · 2 ’ 2A12 · ’ A11 = 0

or

σ 2 = A22 · 2 + 2A12 · + A11

= σg + A22 (· ’ ·g )2

2

where

10 Σ’1 µ

A12

·g = ’ (2.12)

= 0 ’1

A22 1Σ 1

2

|A|

A

σg = A11 ’ 12 =

2

A22 A22

1

(2.13)

= 0 ’1 .

1Σ 1

and the point (σg , µg ) represents the point in the region corresponding to the

minimum possible standard deviation over all portfolios. This is the most

conservative investment portfolio available with this class of securities. What

weights to do we need to put on the individual stocks to achieve this conservative

portfolio? It is easy to see that the weight vector is given by

10 Σ’1

0

(2.14)

wg =

10 Σ’1 1

44 CHAPTER 2. SOME BASIC THEORY OF FINANCE

and since the quantity 10 Σ’1 1 in the denominator is just a scale factor to insure

that the weights add to one, the amount invested in stock i is proportional to

the sum of the elements of the i™th row of the inverse covariance matrix Σ’1 .

An equation of the form

σ2 ’ A22 (· ’ ·g )2 = σg

2

represents a hyperbola since A22 > 0. Of course investors are presumed to prefer

higher returns for a given value of the standard deviation of portfolio so it is

only the upper boundary of this curve in Figure 2.2 that is e¬cient in the sense

that there is no portfolio that is strictly better (better in the sense of higher

return combined with standard deviation that is not larger).

Now let us return to a portfolio whose standard deviation and mean return

lie on the e¬cient frontier. Let us call these e¬cient portfolios. It turns out

that any portfolio on this e¬cient frontier has the same covariance with the

0

minimum variance portfolio wg R derived above.

1

Proposition 5 Every e¬cient portfolio has the same covariance with

10 Σ’1 1

0

the conservative portfolio wg R.

Proof. We noted before that such a portfolio has mean return · and stan-

dard deviation σ which satisfy the relation

σ 2 ’ A22 · 2 ’ 2A12 · ’ A11 = 0.

Moreover the weights for this portfolio are described by

⎡ ¤

1

w = Σ’1 M A ⎣ ¦. (2.15)

·

so the returns vector from this portfolio can be written as

w0 R = [ 1 · ]AM 0 Σ’1 R.

MINIMUM VARIANCE PORTFOLIOS AND THE CAPITAL ASSET PRICING MODEL.45

It is interesting to observe that the covariance of returns between this e¬cient

0

portfolio and the conservative portfolio wg R is given by