1 Basic Terminology and Concepts: Asset Prices, States, Returns, and Pay-Offs
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316
CHAPTER 5
. Project: Arbitrage theory
Next we introduce the concept of states of the world. That is, we assume that each possible
outcome or scenario corresponds to an elementary event, or state of the world, i , where there
is only a finite number M of them: i = 1
M. These states are mutually exclusive, with
at least one of them occurring with nonzero probability. All possible states are represented
by the set = 1
M .
Financial assets will attain different values and give rise to differing payouts corresponding
to the different states i . Shortly we discuss in detail an instructive example. Before that,
however, we recall a couple of other concepts. One is that of payoffs Di j , which represent
the number of units of account paid out per unit of security i in the state j. Generally, for
an N -asset and M-state system we can represent all single-period pay-offs by an N × M
dividend matrix for an interval [t t + 1]:
⎛
D11 · · · D1M
⎜
D=⎝
⎞
⎟
⎠
(5.2)
DN 1 · · · DNM
This payoff matrix can be interpreted in two different ways. The first is that each ith row
of the matrix corresponds to pay-offs for one unit of a given ith security in all the different
states of the world. In the second interpretation, each jth column represents pay-offs for all
the different assets within a given jth state of the world.
The other concept of importance is that of a portfolio. Recall from Chapter 1 that a portfolio
is defined as a linear combination of assets or securities. That is, one can generally have
positions given by i in the ith asset and by specifying all such N positions i i = 1
N,
we have uniquely specified a portfolio as a vector,
⎛
⎜
⎜
=⎜
⎝
⎞
1
2
⎟
⎟
⎟
⎠
(5.3)
N
Positive i correspond to long positions and negative values correspond to short positions.
A zero position i = 0 implies that the ith asset is not included in the portfolio. A portfolio
that delivers the same pay-off regardless of any possible state of the world is defined as
riskless. By taking the dot product of with the asset price vector pt ≡ p(t) we obtain the
value of the portfolio at time t:
N
N
Vt = · p t =
i pi
t =
i=1
i
i At
(5.4)
i=1
The payoff Vt+1 j , denoted here by j , for the portfolio given by in a given jth state
is then expressible as a sum over all asset pay-offs weighted by their respective positions,
where Dij = Ait+1 j ,
N
j
=
N
Dij
i=1
i
=
D
i=1
ji i
(5.5)
5.2 Arbitrage Portfolios and the Arbitrage Theorem
317
The superscript stands for matrix transpose. We can therefore express the payoff vector with
components j , j = 1
M, in matrix form, ≡ D ,
⎞
⎛
⎜
⎜
⎜
⎝
1
⎛
⎟
⎟ ⎜
⎟=⎝
⎠
2
⎞
D11 · · · DN 1
D1M · · · DNM
⎞
⎛
⎜
⎟⎜
⎠⎜
⎝
M
1
2
⎟
⎟
⎟
⎠
(5.6)
N
5.2 Arbitrage Portfolios and the Arbitrage Theorem
As in Chapter 1, we define to be an arbitrage portfolio, or sometimes simply called an
arbitrage, if either one of the following conditions applies:
(i) pt · = 0
(ii) pt · < 0
D · ≥ 0, where D
D · ≥ 0.
and
and
j
> 0 for some j.
Note that these vector inequalities are meant to be applicable component by component.
In case (i) the portfolio guarantees a positive return in some states with no possible loss, yet
costs nothing to purchase. In case (ii) the portfolio will guarantee a nonnegative return and
has a negative cost to purchase.
Finally, we can state the arbitrage theorem as follows:
1. If there are no arbitrage opportunities then there exist positive constants i > 0, i =
1
M (in vector notation we write simply > 0, where
is the vector of i
components), such that
pt = D
(5.7)
2. If condition 1 is true, then there is no arbitrage.
One notes that, apart from a positive constant [i.e., the inverse of the discount factor
as shown in upcoming equation (5.10)], the i correspond to certain nonzero probabilities
of occurence for all the states i = 1
M. In fact, these coefficients give the risk-neutral
probabilities for the correct pricing of financial securities, as explained in the following
section and as was observed in the discrete case of the fundamental theorem of asset pricing
given in Chapter 1. In matrix form, equation (5.7) reads
⎛
⎜
⎝
p1
pN
⎞
⎛
⎟ ⎜
⎠=⎝
D11 · · · D1M
DN 1 · · · DNM
⎞⎛
⎟⎜
⎠⎝
⎞
1
⎟
⎠
(5.8)
M
In the arb spreadsheet assignment we consider the case M = N and the special type of payoff
matrix
⎛
⎞
1+R ··· 1+R
⎜
D21 · · · D2M ⎟
⎜
⎟
(5.9)
D=⎜
⎟
⎝
⎠
DN 1 · · · DNM
318
CHAPTER 5
. Project: Arbitrage theory
The first row has all equal payoff values and corresponds to the riskless return on a moneymarket or bond; i.e., p1 = p1 t = 1, with single-period rate of return R. Without loss of
generality, here we have simply set the bond’s present value to one unit of worth. The first
row in equation (5.8) of the arbitrage theorem then gives
M
M
1+R
i=1
i
≡
˜i = 1
(5.10)
i=1
The coefficients ˜ i defined here correspond to the risk-neutral probabilities for all possible
states. In fact, ˜ i are recognized as being the qi probabilities used to define the pricing
measure in the fundamental theorem of asset pricing discussed in Chapter 1. They sum up to
unity as required and also satisfy the condition 0 < ˜ i < 1. As noted earlier, these probabilities
are very different from the real-world probabilities, which provide no information on the
risk-neutral probabilities used for pricing. The risk-neutral probabilities therefore exist with
the correct properties mentioned if, and only if, there is no arbitrage.
5.3 An Example of Single-Period Asset Pricing: Risk-Neutral
Probabilities and Arbitrage
The single-period setting assumes that time consists of the present time t and a later time
T = t + 1 and that there is a finite time separation. We consider here a portfolio consisting of
just one bond with present value of unity, B t = 1, one asset (or stock) S, and a call option
C on the underlying stock S. Moreover, we assume only two possible states of the world.
In this situation the stock, which has present value S t , can attain either of two values:
S1 t + 1 or S2 t + 1 at time t + 1. Accordingly, the option with present value C t can take
on the values given by C1 t + 1 or C2 t + 1 in state 1 and 2 , respectively. No matter
what the outcome, however, the bond has a fixed (riskless) return of 1 + R, with R being the
single-period rate of return. In this situation we have a 3 × 2 payoff matrix and the foregoing
arbitrage theorem gives
⎞
⎛
⎞ ⎛
1
1+R
1+R
1
⎝ S t ⎠ = ⎝ S1 t + 1 S2 t + 1 ⎠
(5.11)
2
Ct
C1 t + 1 C2 t + 1
which implies a linear system of three equations:
˜1+ ˜2 = 1
(5.12)
˜ 1 S1 t + 1 + ˜ 2 S2 t + 1 = 1 + R S t
(5.13)
˜ 1 C1 t + 1 + ˜ 2 C2 t + 1 = 1 + R C t
(5.14)
Here we have used the same definition as before for the risk-neutral probabilities ˜ i ≡
1 + R i . These equations have the familiar form of the binomial pricing equations for
options, as discussed in the project that deals with binomial lattice pricing. That is, the price
today of a security is given as the discounted sum of the risk-neutral expected payoff values
for all possible future values of the security. We also note that if we allow for three states of
the world, we then obtain pricing equations that resemble the trinomial pricing equations.
To demonstrate an example of arbitrage, let us consider R = 7% and the two possible
values at time t + 1: S1 t + 1 = 50 dollars , S2 t + 1 = 150 dollars, where S t = 100 dollars.
5.4 Arbitrage Detection and the Formation of Arbitrage Portfolios
319
Say the call option C has strike price of 100 dollars and expires exactly at time t + 1. This
option then has pay-off of zero and 50 dollars, respectively. If we denote the price of the call
option today as C, then equations (5.12) to (5.14) give
˜1+ ˜2 = 1
0 5 ˜ 1 + 1 5 ˜ 2 = 1 07
50 ˜ 2 = 1 07C
(5.15)
(5.16)
(5.17)
By satisfying the first two equations we actually obtain the arbitrage-free price for C by
substituing the resulting risk-neutral values ˜ 1 = 0 43, ˜ 2 = 0 57 into the third equation.
The correct (no-arbitrage) price is therefore C = 26 6355 dollars. If, however, we are given
a market price for C = 25 dollars and we wish to answer the question of whether there
is arbitrage or not in this case, then we solve equation (5.17), giving ˜ 2 = 0 535, and
then equation (5.16) gives ˜ 1 = 0 535. These values, however, do not satisfy probability
conservation equation (5.15), therefore, one concludes that there is indeed arbitrage at that
market price.
5.4 Arbitrage Detection and the Formation of Arbitrage Portfolios
in the N-Dimensional Case
The preceding example involves an overdetermined system of linear equations. Now, however,
we shall consider a uniquely specified system where the number of unknowns is equal to the
number of equations. Hence, we consider the case of N states and N assets, i.e., M = N .
We shall assume that one of the assets always corresponds to a bond with fixed rate of
return R. The payoff matrix has the form given in equation (5.9), where the first row has
all equal elements of value 1 + R . The corresponding system of N equations is given in
equation (5.18). The problem is then the following. Generate an arbitrary price vector in
one of two fashions: Set p1 t = 1 and then generate independent components pi t (i ≥ 2)
distributed either (i) uniformly as integers lying within some given minimum and maximum
integer values or (ii) continuously using some standard normal distribtuion, say, pi t N 0 1
(i ≥ 2). Similarly, generate N N − 1 arbitrary payoff matrix elements Dij (i ≥ 2) in the
discrete or continuous cases, respectively. The numerical library called MFRangen is useful
for random-number and random-matrix generation. For a given generated pair of price vector
p t and payoff matrix D one obtains the vector of risk-neutral probabilities ˜ = 1 + R
by solving the linear system
⎛ ⎞ ⎛
⎞⎛ ⎞
1+R ··· 1+R
1
1
⎜ p2 ⎟ ⎜
⎟⎜ ⎟
D
·
·
·
D
21
2N
⎜ ⎟ ⎜
⎟⎜ ⎟
⎟⎜ ⎟
⎜ ⎟=⎜
·
·
(5.18)
⎟⎜ ⎟
⎜ ⎟ ⎜
⎠⎝ ⎠
⎝ ⎠ ⎝
·
·
pN
DN 1 · · · DNN
N
In practice, one can solve this system numerically by inverting the payoff matrix using a
routine based on the singular value decomposition. Note that the first equation in the system is
that of probability conservation [this is equation (5.10) with M = N ]. Arbitrage then exists if
the solution gives at least one nonpositive component, that is, if for any given i, i ≤ 0 (since
we are enforcing probability conservation). For every such i we then have a corresponding ith
state, which we can use to form an arbitrage portfolio that we denote by i with components