Download 5. Linear pricing and risk neutral pricing

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 + d2
were 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
X
θidi.
i=1
Under the assumption of no type A arbitrage, the price of the portfolio θ is found by linearity. Thus the total price is
P =
n
X
θiPi
i=1
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 negative payoff, that investment is said to be a type B arbitrage.
• In other words, a type B arbitrage is a situation where an individual 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.
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5.2
Portfolio choice under utility maximization
• Consider the portfolio problem of an investor who uses an expected 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
subject to
E[U (x)]
n
X
θidi = x
n
X
θiPi ≤ W.
i=1
x≥0
i=1
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 θ 0 such that
n
X
θi0di > 0.
i=1
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 θ 0 can be purchased.
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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 θ 0 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 probability 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∗ =
0.
n
X
θi∗di satisfies x∗ >
i=1
• We can immediately deduce that the inequality
n
X
θiPi ≤ W will
i=1
be met with equality at the solution; otherwise some positive
fraction of the portfolio θ 0 (or θ ∗) could be added to improve
the result.
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• To derive the equation satisfied by the solution, we substitute
x =
n
X
θidi in the objective and ignore the constraint x ≥ 0
i=1
since we have assumed that it is satisfied by strict inequality.
The problem therefore becomes


maximize E U 
subject to
n
X
n
X
i=1

θidi 
θiPi = W.
i=1
12
By introducing a Lagrange multiplier λ for the constraints, and using
x∗ =
n
X
θi∗di for the payoff of the optimal portfolio, the necessary
i=1
conditions are found by differentiating the Lagrangian


L = E U 
n
X
i=1


θidi − λ 
n
X
i=1
with respect to each θi. This gives

θiPi − W 
E[U ′(x∗)di ] = λPi
for i = 1, 2, · · · , n.
The original budget constraint
n
X
θiPi = W is one more equation.
i=1
Altogether, there are n+1 equations for the n+1 unknowns θ1, θ2, · · · , θn
and λ.
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These equations are very important because they serve two roles.
1. They give enough equations to actually solve the optimal portfolio problem.
2. Since these equations are valid if there are no arbitrage opportunities, 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 ]
= Pi.
′
∗
RE[U (x )]
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Portfolio pricing equation
If x∗ =
n
X
θi∗di is a solution to the optimal portfolio problem, then
i=1
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]
= Pi
′
∗
RE[U (x )]
for i = 1, 2, · · · , n.
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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
High success
3.0
Moderate success
1.0
Failure
0.0
Risk free
1.2
Probability
0.3
0.4
0.3
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.
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• 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.
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The necessary conditions are
0.9
0.4
+
= λ
3θ1 + 1.2θ2
θ1 + 1.2θ2
0.36
0.48
0.36
+
+
= λ.
3θ1 + 1.2θ2
θ1 + 1.2θ2
1.2θ2
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.
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Log-optimal pricing
• We use the optimal x∗ to recover the price. We shall choose
U (x) = ln x 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.
• Since
d
1
ln x = , the price equation becomes
dx
x
di
E
R∗
= λPi
for all i.
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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
!
∗
R
1=E
= λ.
R∗
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.
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Using the value of λ = 1, the pricing equation becomes
di
.
Pi = E
∗
R
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
d
P =E
R∗
where R∗ is the return on the log-optimal portfolio.
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Finite state models
• Suppose that there are a finite number of possible states that
describe the possible outcomes of a specific investment situation. 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 important 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 reference 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 .
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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 = h0, 0, · · · , 0, 1, 0, · · · , 0i, 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 combination of the elementary state securities as d =
S
X
dses, and hence
s=1
by the linearity of pricing, the price of d must be
P =
S
X
dsψs.
s=1
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• 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 h1, 1i and h1, −1i
exist, then one-half the sum of these two securities is equivalent
to the first elementary state security h1, 0i.
Question Whether a given set of securities can generate all elementary state securities.
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Positive state prices
If a complete set of elementary securities exists or can be constructed as a combination of existing securities, it is important that
their prices be positive. Otherwise there would be an arbitrage opportunity.
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.
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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 = hd1, d2, · · · , dS i with ds ≥ 0
for each s = 1, 2, · · · , S. The price of d is P =
S
X
ψsds, which since
s=1
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.
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To prove the converse, we assume that there are no arbitrage opportunities, and we make use of the result on the portfolio choice
problem. This proof requires some additional assumptions. We
assume there is a portfolio θ 0 such that
n
X
θi0di > 0. We assign
i=1
positive probabilities ps, s = 1, 2, · · · , S, to the state arbitrarily, with
S
X
ps = 1, and we select a strictly increasing utility function U .
i=1
Since there is no arbitrage, there is, by the portfolio choice theorem, a solution to the optimal portfolio choice problem. We assume
that the optimal payoff has x∗ > 0.
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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
S
1 X
ps U ′(x∗)s ds
P =
λ s=1
where U ′(x∗) is the value of U ′(x∗) in state s.
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Now we define
psU ′(x∗)s
ψs =
.
λ
We see that ψs > 0 because ps > 0, U ′(x∗)s > 0, and λ > 0. We also
have
P =
S
X
ψsds
s=1
showing that the ψs’s are state prices. They are all positive.
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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
0.3
= 0.221
3θ1 + 1.2θ2
0.4
ψ2 =
= 0.338
θ1 + 1.2θ2
0.3
ψ3 =
= 0.274.
1.2θ3
ψ1 =
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.
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5.4
Risk neutral pricing
Suppose there are positive state prices ψs, s = 1, 2, · · · , S. Then the
price of any security d = hd1, d2, · · · , dS i can be found from
P =
S
X
dsψs.
s=1
We now normalize these state prices so that they sum to 1. Hence,
we let ψ0 =
S
X
ψs, and let qs = ψs/ψ0.
s=1
31
We can then write the pricing formula as
P = ψ0
S
X
qs ds.
s=1
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
b
P = ψ0E(d)
b denotes expectation with respect to the artificial probabilwhere E
ities qs.
32
• Since ψ0 =
S
X
ψs, we see that ψ0 is the price of the security
s=1
h1, 1, · · · , 1i 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
1 b
P = E(d).
R
• 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.
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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
ps U ′(x∗)s
.
qs = PS
′
∗
t
t=1 pt U (x )
(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 equations
S
1 X
pi =
qs dsi,
R s=1
i = 1, 2, · · · , n
for the n unknown qs’s.
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Example (The film venture)
We found the state prices of the full film venture (with three securities) to be
1
1
1
ψ1 = , ψ2 = , ψ3 = .
6
2
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 hd1, d2, d3i is
0.2d1 + 0.6d2 + 0.2d3
P =
.
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.
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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:
E(d)
.
P =
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.
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(3) Certainty equivalent from of CAPM:
2
E(d) − cov(RM , d)(RM − R)/σM
P =
.
R
(4) Log-optimal pricing:
d
P =E
R∗
where R∗ is the return on the log-optimal portfolio.
(5) Risk-neutral pricing:
b
E(d)
P =
R
b is taken with respect to the risk-neutral
where the expectation E
probabilities.
38
• Method 1 is the simplest extension of what is true for the deterministic 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 return 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.
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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 formulas 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. Otherwise 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 portfolio that maximizes the expected utility has a well-defined solution.
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 formulas, 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 equations include quadratic functions (which lead to the CAPM formula) 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
b
written as P = E(d)/R,
where R is the return of the riskless asset
b denotes expectation with respect to the artificial (riskand E
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 logoptimal 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