The capital asset pricing model (CAPM), developed
independently by various authors in the sixties, is a classical
equilibrium model for the valuation of risky securities (stocks).
It is based on the following assumptions:
- There exists homogenous information among the market
participants. This assumption can be weakened by assuming that
under homogenous information a rational equilibrium is fully
revealing (see the strong version of Definition
10.17).
- The market has no frictions,
i.e., there are no transaction costs, no taxes, no restrictions on
short selling or on the divisibility of stocks.
- There is
complete competition.
- There are no arbitrage opportunities.
- There are a finite number of stocks (
) and a riskless
security with return
.
- Every investor has a strictly
concave utility function as a function of the risky future cash
flows. This means that every investor is risk averse.
- Every investor maximizes his expected utility, which is
dependent only on the expectation and the variance of the risky
future cash flows. This is the crucial assumption of the CAPM.
Sufficient conditions for this (
)-criterion are either
of the following:
- Every investor has a quadratic utility function.
- The stock returns are normally distributed.
In the following
and
represent the price
and the number of
-th stock supplied in equilibrium at time
. We define the market portfolio
as
 |
(11.8) |
The relative weight
of the
-th stock in this
portfolio is as follows
Most of the well known stock indices are such value weighted
indices, nevertheless often only the largest stocks in the market
are included in the index (DAX, for example, contains only the 30
largest stocks). As in Definition 10.15, we define the
stock return as
and the
market return as
. We assume
that the underlying process of the return is covariance
stationary. In equilibrium according to the CAPM it holds for
every stock
that
![$\displaystyle {\mathop{\text{\rm\sf E}}}[R_{i,t}] = r + \beta_i({\mathop{\text{\rm\sf E}}}[R_{m,t}]-r),$](sfehtmlimg1768.gif) |
(11.9) |
with the `beta' factor
Equation (10.9) says that in equilibrium the expected return
of the
-th stock is comprised of two components: the return of
the riskless security and a risk premium which is specifically
determined for each stock through the beta factor. Stocks that are
positively correlated with the market have a positive risk
premium. The larger the correlation of a stock with the market
portfolio is, the larger is the premium in CAPM for portfolio
risk.
Since the CAPM can be derived using theories on utilities, it is
sometimes described as a demand oriented equilibrium model. In
contrast to this there are other models that explain the stock
returns in terms of various aggregate variables, so called factors, and are referred to as being supply oriented. In Section
12.3 we will relax the assumptions of time constant
variance and covariance implicit in equation (10.9).
For stocks one can find a large variety of econometric models and
for exchange rates there are even more. There are two standard and
quite simple theories. However they are not sufficient to explain
the considerable price movements in currency markets, especially
in the short-run. The purchasing power
parity (PPP) assumes that
identical goods in different countries must have the same relative
price, i.e., a relative price given in units of currency. It has
been empirically determined that in the long-run this theory
describes reality well, but in the short-run the price movements
could not be explained. The second simple theory, the theory
of interest rate parity, performs better as capital flows faster
than goods. The difference in interest rates can thus resemble the
exchange of capital in other currencies. So does the exchange
rate. The theory of the interest rate parity assumes that domestic
and foreign securities are perfect substitutes with respect to
duration and risk structure.
Assume that along with forward and futures markets currency can be
traded over time. The spot price is calculated by
,
the forward and future price by
, each is given in units of
the foreign currency, i.e., EUR/USD. An internationally
acting investor has two choices. Either he holds a domestic
capital investment with the domestic interest rate
or he
chooses a foreign investment with the foreign interest rate
. If he chooses the foreign investment, he must first
exchange his capital into foreign currency at the spot price and
at the end, exchange back again. The uncertainty about the future
developments of the exchange rate can be avoided by purchasing a
forward or future contract. In this case the return on the foreign
investment is
. If this return is not
equal to the domestic interest rate, then an equilibrium has not
been reached. Through immediate price adjustments the interest
rate arbitrage disappeares and then equilibrium is reached. Thus
in equilibrium it must hold that
 |
(11.10) |
i.e., the relationship between forward and future markets and spot
markets corresponds exactly to the relationship between domestic
and foreign gross interest rates. The relationship in (10.10)
is also called the covered interest rate parity, since at
the time of investment it deals with riskless exchange and
interest rates.
In addition to the interest rate arbitrageur there are the so
called forward and future speculators that compare the expected
future exchange rate with the forward and future price and the
corresponding risk of purchasing (selling) currency below or above
the equilibrium. Consider a simple case where forward and future
speculators are risk neutral. Then in equilibrium the expected
exchange rate is equal to the forward and future price, i.e.,
![$\displaystyle W_t^T = {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}],$](sfehtmlimg1776.gif) |
(11.11) |
with the information set
which contains all
relevant and available information. Here we assume that the
speculators have rational expectations, i.e., the true underlying
process is known and is used to build the optimal forecast by the
speculators. This can also be written as the relationship
![$\displaystyle W_{t+1}^K = {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}] + \varepsilon_{t+1}$](sfehtmlimg1778.gif) |
(11.12) |
which says that the deviations of the speculator's forecast
from the realized exchange
rates is white noise
(see Definition
10.8). The market is inefficient when the
speculators actually are risk neutral and
is not
white noise. In this case the set
does not reflect
all of the relevant information in the expectations of the
speculators - they do not have rational expectations. In order to
test for market efficiency (that is, in order to test whether
is white noise) we first need a model for
. This can be formulated from
(10.11) and (10.10).
Substituting (10.11) into (10.10) we obtain the so
called uncovered interest rate parity,
![$\displaystyle \frac{{\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}]}{W_t^K} = \frac{1+r_t^i}{1+r_t^a}.$](sfehtmlimg1780.gif) |
(11.13) |
This interest rate parity is risky because the future exchange
rates are uncertain and enter the relationship as expectations.
Together with (10.12) the risky interest rate parity
(10.13) implies that the following holds
 |
(11.14) |
When the difference in the long-run interest rates is zero on
average, then (10.14) is a random walk (see Definition
10.9). The random walk is the first model to describe
exchange rates.
It should be emphasized that the derivation of this simple model
occurred under the assumption of risk neutrality of the
speculators. In the case of risk aversion a risk premium must be
included. If, for example, we want to test the efficiency of the
currency markets, we could then test the combined hypothesis of
efficiency and uncovered interest rate parity using risk
neutrality. A rejection of this hypothesis indicates the market
inefficiency or that the interest rate parity model is a poor
model for currency markets.
Term structure models are applied to model the chronological
development of bond returns with respect to time to maturity. The
classical starting point is to identify one or more factors which
are believed to determine the term structure. Through
specification of the dynamic structure and using specific
expectation hypotheses, an explicit solution can be obtained for
the returns.
As a typical example we briefly introduce the Cox, Ingersoll and
Ross (CIR) model, which has been mentioned in Section
9.2.2. The price of a Zero Coupon Bond with a
nominal value of 1 EUR is given by
at time
, i.e.,
a security with no dividend payments that pays exactly one EUR at
maturity date
. The log return of the zero coupon bond is given
by
. We assume that continuous compounding holds. The
process
is frequently referred to as the yield to
maturity . The relationship between the
price and the return of the zero coupon bond is
with the remaining time to maturity
. This can be easily
seen from the definition of a log return (Definition
10.15). For very short time intervals the short
rate
is defined as
In practice the short rate corresponds to the spot rate, i.e., the
interest rate for the shortest possible investment (see Section
9.2.2). Consider, intuitively, the choice between an
investment in a zero bond with the return
and repeatedly
investing at a (risky) short-term interest rate in future periods.
An important expectation hypothesis says that the following holds
![$\displaystyle P_T(t) = {\mathop{\text{\rm\sf E}}}\left[\exp(-\int_t^T r(s) ds) \vert {\cal F}_t \right]$](sfehtmlimg1786.gif) |
(11.15) |
(also see equation (9.1) for variable but deterministic
interest). The short rate is frequently seen as the most important
predicting factor of the term structure. As the CIR model, most
one factor models use the short rate as factor. The CIR model
specifies the dynamic of the short rate as a continuous stochastic
process
 |
(11.16) |
with a Wiener process
and constant parameters
and
- see also Section 9.2.2. The process
(10.16) has a so called mean
reversion behavior, i.e., once deviations
from the stationary mean
occurs, the process is brought back
to the mean value again through a positive
. The volatility,
written as
, is larger whenever the interest
level is higher, which can also be shown empirically.
Since in the equation (10.16)
is specified as a
Markov process,
is, as a consequence of equation
(10.15), a function of the actual short rate, i.e.,
With Itô's lemma (5.10) and (9.7) we
obtain from (10.16) the differential equation
With the bounding constraint
the following
solution is obtained
 |
(11.17) |
where (see Section 9.2.4)
For increasing time periods
the term structure curve
converges to the value
If the short-term interest lies above
, then the term structure
is decreasing, see Figure 10.1; if it lies below
,
then the term structure is increasing, see Figure 10.2. If
the short-term interest rate lies between
and
, then
the curve could first rise and then fall.
Fig.:
Term structure curve according to the Cox-Ingersoll-Ross model with a short rate of
=0.2,
and
(dotted line).
SFEcir.xpl
|
Fig.:
Term structure curve according to the Cox-Ingersoll-Ross model with a short rate of
=0.01,
and
(dotted line).
SFEcir.xpl
|
Since we have thoroughly covered the Black-Scholes model on option
pricing in the first part of this book, here only a brief summary
of the model is given. Options are not only theoretically
interesting for financial markets, but also from an empirical
point of view. Just recently there have been indications of a
systematic deviation of actual market prices from the
Black-Scholes prices. These deviations will be discussed in more
detail in later chapters, specifically in dealing with ARCH
models.
As an example let's consider a European call option on a stock
which receives no dividends in the considered time periods and has
the spot price
at time
.
is the option price at
time
, when the actual price is
. The payoff at the
time to maturity
is
, where
is the strike price. The option price is determined from general
no arbitrage conditions as
where expectations are built on an appropriate risk neutral
distribution - see also (6.23).
is the fixed riskless
interest rate.
Special results can only be derived when the dynamics of the stock
prices are known. The assumptions made by Black and Scholes are
that the stock prices
are geometric Brownian motion, i.e.,
 |
(11.18) |
The option price
thus satisfies the Black-Scholes
differential equation (6.3) as a function of time and
stock prices
Black and Scholes derive the following solutions (see Section
6.2):
 |
(11.19) |
where
is the time to maturity for the option and
is an abbreviation for
In a risk neutral world the market price of risk, see Section
9.2.3, is equal to zero. In the following section, we
will consider the market price of risk and derive once again the
Black-Scholes formula. To do this we will consider derivatives of
financial instruments that are determined by a single random
process
We will assume that the process
is
geometric Brownian motion :
 |
(11.20) |
The variable
does not necessarily represent a financial
value. It could be the state of the market, a measure for the
popularity of a politician or the frequency of an ad-hoc
announcement at time
. Assume that
and
are
the prices for two derivatives of financial instruments that
depend only on
and
. As a simplification, no
payments are allowed during the observed time period. This process
,
also follows the schema
(10.20) with the same Wiener process
 |
(11.21) |
where
could be functions of
and
. The random process
in (10.20) and (10.21) is
always the same since we assume that this is the only source that
creates uncertainty.
The observation of (10.21) in discrete time leads to:
We could ``eliminate the random variable
'' by
constructing a riskless portfolio that continually changes. To do
this we take
units of the first instrument
and
of the second instrument, i.e., we
short sell the second instrument. Letting
be the total
value of the portfolio at time
we have:
 |
(11.24) |
and
 |
(11.25) |
Substituting in (10.22) and (10.23) we have:
 |
(11.26) |
This portfolio should be riskless, thus in time period
it must produce the riskless profit
:
 |
(11.27) |
Substituting (10.24) and(10.26) into this equation
produces:
Equating this as in (9.2.3) to
we see that
the price
of a derivative instrument, an instrument that
depends only on
and
, follows the dynamics
 |
(11.28) |
the value
 |
(11.29) |
represents the market price of risk . This market price of risk can depend on
(using
), but not on the actual price of the instrument
! We can rewrite equation (10.29) as:
 |
(11.30) |
Furthermore we can interpret
, which in this
interpretation can also be negative, as the level of the
-risk in
. Equation (10.30) has strong ties to
the CAPM model, which
we discussed in Section (10.4.1) - for further details
see also Hafner and Herwartz (1998).
Example 11.2
Assume that there are two objects, both are dependent on a 90 day
interest rate. The first instrument has an expected return of

per year and a volatility of

per year. For the second
instrument a volatility of

per year is assumed.
Furthermore,

per year. The market price of risk for the
first instrument according to (
10.29) is:
 |
(11.31) |
By substituting into equation (
10.30) for the second object we
obtain:
 |
(11.32) |
or

expected value.
Since
is a function of
and
, we can determine
the dependence on
using Itô's lemma. The direct
application of Itô's lemma (5.10) on
gives, in comparison to (10.28), the parameters
in this equation
Due to equation (10.30) we have
, so that we obtain the following
differential equation for
:
 |
(11.33) |
This equation (10.33) is very similar to the Black-Scholes
differential equation and is in fact identical to (6.3)
for
, where
denotes the stock price with no
dividends. In this case
itself is the price of the risk
bearing instrument and must therefore satisfy (10.30), like
the price
of any derivative based on the stock. Thus we
obtain
 |
(11.34) |
so that the second term in (10.33) is equal to
 |
(11.35) |
Thus we have a differential equation:
 |
(11.36) |
which is identical to (6.3) after renaming the variables.
More explicitly, let
(since there are no
dividends) and let
using the notation in Section
6.1.