5. Linear pricing and risk neutral pricing

(5.1) Concepts of arbitrage

(5.2) Portfolio choice under utility maximization

(5.3) Finite state models and state prices

(5.4) Risk neutral pricing

A security is a random payoff variable d. The payoff is revealed and

obtained at the end of the period. Associated with a security is

price P .

1

5.1 Concept of arbitrage

Type A arbitrage

• If an investment produces an immediate positive reward with

no future payoff (either positive or negative), that investment is

said to be a type A arbitrage.

• If you invest in a type A arbitrage, you obtain money immediately

and never have to pay anything. You invest in a security that

pays zero with certainty but has a negative price. It seems quite

reasonable to assume that such thing does not exist.

2

Linear pricing follows from the assumption that there is no possibility

of type A arbitrage.

• Consider the security 2d we could buy this double security at the

reduced price, and then break it apart and sell the two halves at

price P for each half. We would obtain a net profit of 2P−P ′ and

then have no further obligation, since we sold what we bought.

We have an immediate profit, and hence have found a type A

arbitrage.

3

• If d1 and d2 are securities with prices P1 and P2, the price of

the security d1 + d2 must be P1 +P2. For if the price of d1 + d2were P ′ < P1 +P2, we could purchase the combined security for

P ′, then break it into d1 and d2 and sell these for P1 and P2,

respectively. As a result we would obtain a profit of P1+P2−P′ >

0.

• As before, this argument can be reversed if P ′ > P1+P2. Hence

the price of d1 + d2 must be P1 + P2.

• In general, therefore, the price of αd1 + βd2 must be equal to

αP1 + βP2.

4

Portfolios

Suppose now that there are n securities d1, d2, · · · , dn. A portfolio

of these securities is represented by an n-dimensional vector θ =

(θ1, θ2, · · · , θn). The ith component θi represents the amount of

security i in the portfolio. The payoff of the portfolio is the random

variable

d =n∑

i=1

θidi.

Under the assumption of no type A arbitrage, the price of the port-

folio θ is found by linearity. Thus the total price is

P =n∑

i=1

θiPi

which is a more general expression of linear pricing.

5

Type B Arbitrage

• If an investment has nonpositive cost but has positive probability

of yielding a positive payoff and no probability of yielding a neg-

ative payoff, that investment is said to be a type B arbitrage.

• In other words, a type B arbitrage is a situation where an indi-

vidual pays nothing (or a negative amount) and has a chance of

getting something. An example would be a free lottery ticket

— you pay nothing for a ticket, but have a chance of winning a

prize. Clearly, such tickets are rare in securities markets.

6

5.2 Portfolio choice under utility maximization

• Consider the portfolio problem of an investor who uses an ex-

pected utility criterion to rank alternative.

• If x is a random variable, we write x ≥ 0 to indicate that the

variable is never less than zero. We write x > 0 to indicate that

the variable is never less than zero and it is strictly positive with

some positive probability.

• Suppose that an investor has a strictly increasing utility function

U and an initial wealth W . There are n securities d1, d2, · · · , dn.

The investor wishes to form a portfolio to maximize the expected

utility of final wealth, say, x. We let the portfolio be defined

by θ = (θ1, θ2, · · · , θn), which gives the amount of the various

securities.

7

The investor’s problem is

maximize E[U(x)]

subject ton∑

i=1

θidi = x

x ≥ 0n∑

i=1

θiPi ≤W.

The investor must select a portfolio with total cost no greater than

the initial wealth W (the last constraint), that the final wealth x is

defined by the portfolio choice (the first constraint), that this final

wealth must be nonnegative in every possible outcome (the second

constraint), and that the investor wishes to maximize the expected

utility of this final wealth.

8

Portfolio choice theorem

Suppose that U(x) is continuous and increases toward infinity as

x → ∞. Suppose that there is a portfolio θ0 such thatn∑

i=1

θ0i di > 0.

Then the optimal portfolio problem has a solution if and only if

there is no arbitrage possibility.

Proof:

We shall only prove the only if portion of the theorem.

Suppose that there is a type A arbitrage produced by a portfolio

θ = (θ1, θ2, · · · , θn). Using this portfolio, it is possible to obtain

additional initial wealth without affecting the final payoff. Hence

arbitrary amounts of the portfolio θ0 can be purchased.

9

This implies that E[U(x)] does not have a maximum, because given

a feasible portfolio, that portfolio can be supplemented by arbitrary

amounts of θ0 to increase E[U(x)].

If there is a type B arbitrage, it is possible to obtain (at zero or

negative cost) an asset that has payoff x > 0 (with nonzero prob-

ability of being positive). We can acquire arbitrarily large amounts

of this asset to increase E[U(x)] arbitrarily.

Hence if there is a solution, there can be no type A or type B

arbitrage.

10

Existence of a solution and characterization of the solution

• We assume that there are no arbitrage opportunities and hence

there is an optimal portfolio, which we denote by θ∗. We also

assume that the corresponding payoff x∗ =n∑

i=1

θ∗i di satisfies x∗ >

0.

• We can immediately deduce that the inequalityn∑

i=1

θiPi ≤W will

be met with equality at the solution; otherwise some positive

fraction of the portfolio θ0 (or θ∗) could be added to improve

the result.

11

• To derive the equation satisfied by the solution, we substitute

x =n∑

i=1

θidi in the objective and ignore the constraint x ≥ 0

since we have assumed that it is satisfied by strict inequality.

The problem therefore becomes

maximize E

U

n∑

i=1

θidi

subject ton∑

i=1

θiPi = W.

12

By introducing a Lagrange multiplier λ for the constraints, and using

x∗ =n∑

i=1

θ∗i di for the payoff of the optimal portfolio, the necessary

conditions are found by differentiating the Lagrangian

L = E

U

n∑

i=1

θidi

− λ

n∑

i=1

θiPi −W

with respect to each θi. This gives

E[U ′(x∗)di] = λPi

for i = 1,2, · · · , n.

The original budget constraintn∑

i=1

θiPi = W is one more equation.

Altogether, there are n+1 equations for the n+1 unknowns θ1, θ2, · · · , θn

and λ.

13

These equations are very important because they serve two roles.

1. They give enough equations to actually solve the optimal port-

folio problem.

2. Since these equations are valid if there are no arbitrage oppor-

tunities, they provide a valuable characterization of prices under

the assumption of no arbitrage.

If there is a risk-free asset with total return R, then when di = R

and Pi = 1. Thus,

λ = E[U ′(x∗)]R.

Substitute this value of λ yields

E[U ′(x∗)di]

RE[U ′(x∗)]= Pi.

14

Portfolio pricing equation

If x∗ =n∑

i=1

θ∗i di is a solution to the optimal portfolio problem, then

E[U ′(x∗)di] = λPi

for i = 1,2, · · · , n, where λ > 0. If there is a risk-free asset with

return R, then

E[U ′(x∗)di]

RE[U ′(x∗)]= Pi

for i = 1,2, · · · , n.

15

Example (A film venture)

An investor is considering the possibility of investing in a venture

to produce an entertainment film. He has learned that there are

essentially three possible outcomes, as shown in Table. One of

these outcomes will occur in 2 years. He also has the opportunity

to earn 20% risk free over this period.

The Film Venture

Return Probability

High success 3.0 0.3Moderate success 1.0 0.4

Failure 0.0 0.3Risk free 1.2 1.0

There are three possible outcomes with associated total returns and

probabilities shown. There is also a risk-free opportunity with total

return 1.2.

16

• He wants to know whether he should invest money in the film

venture; and if so, how much?

• The expected return is 0.3 × 3 + 0.4 × 1 + 0.3 × 0 = 1.3, which

is somewhat better than what can be obtained risk free. How

much would you invest in such a venture?

• The investor decides to use U(x) = ln x as a utility function.

His problem is to select amounts θ1 and θ2 of the two available

securities, the film venture and the risk-free opportunity, each

of which has a unit price of 1. Hence his problem is to select

(θ1, θ2 to solve

maximize [0.3 ln(3θ1 + 1.2θ2) + 0.4 ln(θ1 + 1.2θ2) + 0.3 ln(1.2θ2)]

subject to θ1 + θ2 = W.

17

The necessary conditions are

0.9

3θ1 + 1.2θ2+

0.4

θ1 + 1.2θ2= λ

0.36

3θ1 + 1.2θ2+

0.48

θ1 + 1.2θ2+

0.36

1.2θ2= λ.

These two equations, together with constraint θ1 + θ2 = W , can be

solved for the unknown θ1, θ2, and λ.

The result is θ1 = 0.089W, θ2 = 0.911W , and λ = 1/W .

In other words, the investor should commit 8.9% of his wealth to

this venture; the rest should be placed in the risk-free security.

18

Log-optimal pricing

• We use the optimal x∗ to recover the price. We shall choose

U(x) = lnx and W = 1 as a special case to investigate. The

final wealth variable x∗ is then the one that is associated with the

portfolio that maximizes the expected logarithm of final wealth.

We denote this x∗ by R∗, since R∗ is the return that is optimal

for the logarithmic utility. We refer to R∗ as the log-optimal

return.

• Sinced

dxln x =

1

x, the price equation becomes

E

(diR∗

)= λPi

for all i.

19

Since this is valid for every security i, it is valid for the log-optimal

portfolio itself. This portfolio has price 1, and therefore we find

that

1 = E

(R∗

R∗

)

= λ.

Thus we have found the value of λ for this case.

If there is a risk-free asset, the portfolio pricing equation is valid for

it as well. The risk-free asset has a payoff identically equal to 1 and

price 1/R, where R is the total risk-free return. Hence we find

E(1/R∗) = 1/R.

Therefore we know that the expected value of 1/R∗ is equal to 1/R.

20

Using the value of λ = 1, the pricing equation becomes

Pi = E

(diR∗

).

Since this is true for any security i by linearity, it is also true for any

portfolio.

Log-optimal pricing The price P of any security (or portfolio) with

dividend d is

P = E

(d

R∗

)

where R∗ is the return on the log-optimal portfolio.

21

Finite state models

• Suppose that there are a finite number of possible states that

describe the possible outcomes of a specific investment situa-

tion. At the initial time it is know only that one of these states

will occur. At the end of the period, one specific state will be

revealed.

• States define uncertainty in a very basic manner. It is not even

necessary to introduce probabilities of the states. In an impor-

tant sense, probabilities are irrelevant for pricing relations.

• A security is defined within the context of states as a set of

payoffs — on payoff for each possible state (again without ref-

erence to probabilities). Hence a security is represented by a

vector of the form d = (d1, d2, · · · dS). Associated with a security

is a price P .

22

State prices

• A special form of security is one that has a payoff in only one

state. Indeed, we can define the S elementary state securities

es = 〈0,0, · · · ,0,1,0, · · · ,0〉, where the 1 is the component s for

s = 1,2, · · · , S. If such a security exists, we denote its price by

ψs.

• The security d = (d1, d2, · · · , dS) can be expressed as a combina-

tion of the elementary state securities as d =S∑

s=1

dses, and hence

by the linearity of pricing, the price of d must be

P =S∑

s=1

dsψs.

23

• If the elementary state securities do not exist, it may be possible

to construct them artificially by combining securities that do

exist. For example, in a two-state world, if 〈1,1〉 and 〈1,−1〉

exist, then one-half the sum of these two securities is equivalent

to the first elementary state security 〈1,0〉.

Question Whether a given set of securities can generate all ele-

mentary state securities.

24

Positive state prices

If a complete set of elementary securities exists or can be con-

structed as a combination of existing securities, it is important that

their prices be positive. Otherwise there would be an arbitrage op-

portunity.

To see this, suppose an elementary state security es had a zero or

negative price. That security would then present the possibility of

obtaining something (a payoff of 1 if and the state s occurs) for

nonpositive cost. This is type B arbitrage. So if elementary state

securities actually exist or can be constructed as combination of

other securities, their prices must be positive to avoid arbitrage.

25

Positive state prices theorem A set of positive state prices exist

if and only if there are no arbitrage opportunities.

Proof:

Suppose first that there are positive state prices. Then it is clear

that no arbitrage is possible. To see this, suppose a security d can

be constructed with d ≥ 0. We have d = 〈d1, d2, · · · , dS〉 with ds ≥ 0

for each s = 1,2, · · · , S. The price of d is P =S∑

s=1

ψsds, which since

ψs > 0 for all s, gives P ≥ 0. Indeed P > 0 if d 6= 0 and P = 0 if

d = 0. Hence there is no arbitrage possibility.

26

To prove the converse, we assume that there are no arbitrage op-

portunities, and we make use of the result on the portfolio choice

problem. This proof requires some additional assumptions. We

assume there is a portfolio θ0 such thatn∑

i=1

θ0i di > 0. We assign

positive probabilities ps, s = 1,2, · · · , S, to the state arbitrarily, withS∑

i=1

ps = 1, and we select a strictly increasing utility function U .

Since there is no arbitrage, there is, by the portfolio choice theo-

rem, a solution to the optimal portfolio choice problem. We assume

that the optimal payoff has x∗ > 0.

27

The necessary conditions show that for any security d with price P ,

E[U ′(x∗)d] = λP

where x∗ is the (random) payoff of the optimal portfolio and λ > 0

is the Lagrange multiplier.

If we expand this equation to show the details of the expected value

operation, we find

P =1

λ

S∑

s=1

psU′(x∗)sds

where U ′(x∗) is the value of U ′(x∗) in state s.

28

Now we define

ψs =psU ′(x∗)s

λ.

We see that ψs > 0 because ps > 0, U ′(x∗)s > 0, and λ > 0. We also

have

P =S∑

s=1

ψsds

showing that the ψs’s are state prices. They are all positive.

29

Example (The plain film venture)

Consider again the original film venture. There are three states,

but only two securities: the venture itself and the riskless security.

Hence state prices are not unique.

We can find a set of positive state prices and the values of the θi’s

and λ = 1 found in earlier Example (with W = 1). We have

ψ1 =0.3

3θ1 + 1.2θ2= 0.221

ψ2 =0.4

θ1 + 1.2θ2= 0.338

ψ3 =0.3

1.2θ3= 0.274.

These state prices can be used only to price combinations of the

original two securities. They could not be applied, for example, to

the purchase of residual rights. To check the price of the original

venture we have P = 3 × 0.221 + 0.338 = 1, as it should be.

30

5.4 Risk neutral pricing

Suppose there are positive state prices ψs, s = 1,2, · · · , S. Then the

price of any security d = 〈d1, d2, · · · , dS〉 can be found from

P =S∑

s=1

dsψs.

We now normalize these state prices so that they sum to 1. Hence,

we let ψ0 =S∑

s=1

ψs, and let qs = ψs/ψ0.

31

We can then write the pricing formula as

P = ψ0

S∑

s=1

qsds.

The quantities qs, s = 1,2, · · · , S, can be though of as (artificial)

probabilities, since they are positive and sum to 1. Using these as

probabilities, we can write the pricing formula as

P = ψ0E(d)

where E denotes expectation with respect to the artificial probabil-

ities qs.

32

• Since ψ0 =S∑

s=1

ψs, we see that ψ0 is the price of the security

〈1,1, · · · ,1〉 that pays 1 in every state — a risk-free bond.

• By definition, its price is 1/R, where R is the risk-free return.

Thus we can write the pricing formula as

P =1

RE(d).

• The price of a security is equal to the discounted expected value

of its payoff, under the artificial probabilities.

• We term this risk-neutral pricing since it is exactly the formula

that we would use if the qs’s were real probabilities and we used

a risk-neutral utility function (that is, the linear utility function).

We also refer to the qs’ as risk-neutral probabilities.

33

Here are three ways to find the risk-neutral probabilities qs:

(a) The risk-neutral probabilities can be found from positive state

prices by multiplying those prices by the risk-free rate.

(b) If the positive state prices were found from a portfolio problem

and there is a risk-free asset, we define

qs =psU ′(x∗)s

∑St=1 ptU

′(x∗)t.

(c) If there are n states and at least n independent securities with

known prices, and no arbitrage possibility, then the risk-neutral

probabilities can be found directly by solving the system of equa-

tions

pi =1

R

S∑

s=1

qsdsi , i = 1,2, · · · , n

for the n unknown qs’s.

34

Example (The film venture)

We found the state prices of the full film venture (with three secu-

rities) to be

ψ1 =1

6, ψ2 =

1

2, ψ3 =

1

6.

Multiplying these by the risk-free rate 1.2, we obtain the risk-neutral

probabilities

q1 = 0.2, q2 = 0.6, q3 = 0.2.

Hence the price of a security with payoff 〈d1, d2, d3〉 is

P =0.2d1 + 0.6d2 + 0.2d3

1.2.

This pricing formula is valid only for the original securities or linear

combinations of those securities. The risk-neutral probabilities were

derived explicitly to price the original securities.

35

Pricing alternatives

• Suppose that there is an environment of n securities for which

prices are known, and then a new security is introduced, defined

by the (random) cash flow d to be obtained at the end of the

period. What is the correct price of that new security?

• List here are five alternative ways we might assign it a price.

• In each case R is the one-period risk-free return.

36

1. Discounted expected value:

P =E(d)

R.

2. CAPM pricing:

P =E(d)

R+ β(RM −R)

where β is the beta of the asset with respect to the market, and

RM is the return on the market portfolio. We assume that the

market portfolio is equal to the Markowitz fund of risky assets.

37

(3) Certainty equivalent from of CAPM:

P =E(d) − cov(RM , d)(RM −R)/σ2

M

R.

(4) Log-optimal pricing:

P = E

(d

R∗

)

where R∗ is the return on the log-optimal portfolio.

(5) Risk-neutral pricing:

P =E(d)

R

where the expectation E is taken with respect to the risk-neutral

probabilities.

38

• Method 1 is the simplest extension of what is true for the de-

terministic case. In general, however, the price determined this

way is too large (at least for assets that are positively correlated

with all others). The price usually must be reduced.

• Method 2 reduces the answer obtained in 1 by increasing the

denominator. This method essentially increases the discount

rate.

• Method 3 reduces the answer obtained in 1 by decreasing the

numerator, replacing it with a certainty equivalent.

• Method 4 reduces the answer obtained in 1 by putting the re-

turn R∗ inside the expectation. Although E(1/R∗) = 1/R, the

resulting price usually will be smaller than that of method 1.

• Method 5 reduces the answer obtained in 1 by changing the

probabilities used to calculate the expected value.

39

1. Methods 2–5 represent four different ways to modify Method

1 to get a more appropriate result. What are the differences

between these four modified methods? If the new security is

a linear combination of the original n securities, all four of the

modified methods give identical prices. Each method is a way

of expressing linear pricing.

2. If d is not a linear combination of these n securities, the prices

assigned by the different formulas may differ, for these formu-

las are then being applied outside the domain for which they

were derived. Methods 2 and 3 will always yield identical values.

Methods 3 and 4 will yield identical values if the log-optimal

formula is used to calculate the risk-neutral probabilities. Oth-

erwise they will differ as well.

3. If the cash flow d is completely independent of the n original

securities, then all five methods, including the first, will produce

the identical price.

40

Summary

1. Two types of arbitrage type A, which rules out the possibility of

obtaining something for nothing — right now; and type B, which

rules out he possibility of obtaining a change for something later

— at no cost now.

Ruling out type A arbitrage leads to linear pricing. Ruling out

both type A and B implies that the problem of finding the port-

folio that maximizes the expected utility has a well-defined so-

lution.

2. The optimal portfolio problem can be used to solve realistic

investment problems.

41

• The necessary conditions of this general problem can be used

in a backward fashion to express a security price as an expected

value.

• Different choices of utility functions lead to different pricing for-

mulas, but all of them are equivalent when applied to securities

that are linear combinations of those considered in the original

optimal portfolio problem.

• Utility functions that lead to especially convenient pricing equa-

tions include quadratic functions (which lead to the CAPM for-

mula) and the logarithmic utility function.

42

3. Insight and practical advantage can be derived from the use of

finite state models. In these models it is useful to introduce the

concept of state prices. A set of positive state prices consistent

with the securities under consideration exists if and if there are

no arbitrage opportunities. One way to find a set of positive

state prices is to solve the optimal portfolio problem. The state

prices are determined directly by the resulting optimal portfolio.

4. A concept of major significance is that of risk-neutral pricing.

By introducing artificial probabilities, the pricing formula can be

written as P = E(d)/R, where R is the return of the riskless asset

and E denotes expectation with respect to the artificial (risk-

neutral) probabilities. A set of risk-neutral probabilities can be

found by multiplying the state prices by the total return R of

the risk-free asset.

43

5. The pricing process can be visualized in a special space. Starting

with a set of n securities defined by their (random) outcomes di,

define the space S of all linear combinations of these securities.

• A major consequence of the no-arbitrage condition is that there

exists another random variable v, not necessarily in S, such that

the price of any security d in the space S is E(vd).

• In particular, for each i, we have Pi = E(vdi). Since v is not

required to be in S, there are many choices for it. One choice

is embodied in the CAPM; and in the case v is in the space S.

• Another choice is v = 1/R∗, where R∗ is the return on the log-

optimal portfolio, and in this case v is often not in S.

44

• The optimal portfolio problem can be solved using other utility

functions to find other v’s. If the formula P = E(vd) is applied

to a security d outside of S, the result will generally be different

for different choices of v.

6. If the securities are defined by a finite state model and if there

are as many (independent) securities as states, then the market

is said to be complete. In this case the space S contains all

possible random vectors (in this model), and hence v must be

in S as well. Indeed, v is unique. It may be found by solving an

optimal portfolio problem; all utility functions will produce the

same v.

45