From Dynamic Linear Evaluation Rule to Dynamic CAPM in a Fractional Brownian Motion Environment ()
1. Introduction
In the 1960s, Sharpe (1964), Lintner (1965) and Mossin (1966) established the famous Capital Asset Pricing Model (CAPM for short). The CAPM has been used and cited in the literature over the past several decades. Some efforts have been made to extend this model. Dybvig and Ingersoll [1] investigated the relationship between the linear evaluation rule and the CAPM. They proved that the standard mean-variance separation theorem obtained in a complete market only if all investors had quadratic utility. In addition, the familiar CAPM pricing relation could hold for all assets in a complete market only if arbitrage opportunities existed. A description of the relationship between the linear evaluation rule and the theory of Markowitz portfolio choice can be found in [2], which they derived a general representation for asset prices that displayed the role of conditioning information. This representation was then used to examine restrictions implied by asset pricing models on the unconditional moments of asset payoffs and prices. An exhaustive discussion of the equivalence of these three theories (the linear evaluation rule, the CAPM and the theory of Markowitz portfolio choice) was presented in [3]. Shi [4] gave a fundamental probability model in the two-period security market. Under some conditions, if the linear evaluation rule holds, then there would be a stochastic discount factor. If this is true, all three theories (CAPM, linear evaluation rule and Markowitz portfolio choice) are equivalent. They are mainly deduced by the method of Hilbert space and stochastic discount factor. Particularly, CAPM could be deduced from the linear evaluation rule in the intertemporal market.
Since nowadays the market fluctuates promptly and dealings in securities require extremely high speed, no discrete-time model could adapt to the market well. However, the continuous-time model is regarded as a good approximation to real scenarios. If we assume that the model is continuous, then it facilitates the use of stochastic differential equations, stochastic analysis, and so on, to obtain some profound and concise conclusions. The famous Black-Scholes option pricing model is a classic issue of continuous-time finance. The fundamental theorem of asset pricing, the portfolio choice of securities and the CAPM all have their continuous-time version. Zhou and Wu [5] deduced the dynamic CAPM from the dynamic linear evaluation rule in the market driven by the Levy processes. They mainly used the predictable representation property in weak form and the Girsanov theorem of the Levy processes to obtain the results.
Ever since the pioneering work of Hurst [6,7] and Mandelbrot [8], the fractional Brownian motion has played an increasingly important role in various fields such as hydrology, economics, and telecommunications [9-12]. In this paper, we study the dynamic CAPM in the fractional Brownian motion environment, which represents a new perspective.
The remaining sections of this article are organized as follows: Some preliminaries of fractional Brownian motion are presented in Section 2. Section 3 presents the fundamental framework of the evaluation problem under which the evaluation operator satisfying some axioms is linear. In Section 4, we investigate the relationship between the dynamic linear evaluation rule and the dynamic CAPM in the market driven by fractional Brownian motions. Section 5 provides the conclusions.
2. Preliminaries of Fractional Brownian Motion
As preparation, collecting some important results concerning fractional Brownian motion is essential in this section. Also, it is necessary to introduce notation for further use.
Recall that if 0 < H < 1, then the fractional Brownian motion with Hurst parameter H is a Gaussian process 
with mean 
and covariance
where 
and 
denotes the expectation with respect to the probability law for 
Assume that
is defined on the 
of subsets of 
generated by the random variables
. For simplicity we assume 
If
, then 
coincides with the standard Brownian motion
, which has independent increments. If 
then 
has a long-range dependence, in the sense that if we put:
then
.
For any 
the process 
is self-similar in the sense that 
has the same law as 
for any 
See [8,12] for more information about fractional Brownian motion.
Due to these properties, 
with Hurst parameter 
has been suggested as a useful tool in many applications [11], including finance.
Fix a Hurst constant
, 
Since H is fixedthe probability measure is denoted by P and the filtration is denoted by
. In this case we have the integral representation [13] and the references therein):
where 
is a standard Brownian motion (Wiener process) and
with 
being a constant such that
With this 
we associate an operator
Recently, stochastic calculus for fractional Brownian motion has been developed by many researchers [13,14].
2.1. Quasi-Conditional Expectation and Fractional Girsanov Theorem
The quasi-conditional expectation is important to obtain the main results. It was initially introduced to find the hedging strategy in an application to finance [9]. Let f and g be two continuous functions on [0, T], where 
is a fixed time horizon. Define
where 
when
, denote: 
Apparently, for any 
is a Hilbert scalar product. Let 
be the completion of the continuous functions under this Hilbert norm. The elements in 
may be distributions [15].
For any 
let 
denote the set of all real symmetric functions 
variables on 
such that
It is known [15] that 
is a subspace of 
and they are not identical. Let 
denote the set of 
such that F has the following chaos expansion: 
where 
when restricted to 
is in 
for all
and
is the multiple stochastic integral (for the multiple integrals and the chaos expansion, [14,16].
Definition 2.1 If 
then the quasiconditional expectation [7] is defined as
where
The following Lemma 2.1 (resp. Lemma 2.2) is from [17] Theorem 3.9 (resp. Theorem 3.11).
Lemma 2.1 Let 
be continuous such that 
is an increasing function. Denote 
where g is a measurable real valued function of polynomial growth and 
Then 
The following lemma is an analogue of the StriebelKallianpur formula. It is called a form of fractional Girsanov theorem.
Lemma 2.2 Let 
and 
be continuous functions of s in [0, T] and 
Consider the translation of 
Let Q denote the probability measure given by
Then 
is a fractional Brownian motion under Q. If 
satisfies the integral equation
then for any measurable function g of exponential growth,
where 
The above two lemmas are indispensable to the main results.
2.2. A fractional Clark-Haussmann-Ocone (CHO) Theorem
Finally let us review a fractional version of the ClarkHaussmann-Ocone (CHO) representation obtained in Theorem 4.5 in [9].
Lemma 2.3 Let 
be 
-measurable, then 
exists. Define
Here 
is the stochastic gradient (Malliavin derivative) of G at t. Then,
Refer to Section 4 in [9] for details.
3. Mechanism of Evaluation of Contingent Claims
The mathematical formulation to the evaluation problem is provided below.
Let 
be a complete probability space defined in Section 2, and 
a filtration satisfying the usual conditions, 
Fix a time interval 
and set
For all 
define the inner product of these two random variables 
Then 
is a Hilbert space, which denotes the subspace of all contingent claims. 
is the space of 
-measurable and square-integrable random variables.
For all contingent claims
, denote the evaluated value by 
At each time t,
is an evaluation operator. We will present the following axiomatic hypotheses of the evaluation operator:
(H1) 
(H2) 
The following lemma is from [5] Lemma 2.1.
Lemma 3.1 For 
hypotheses (H1) and (H2) hold if and only if it is a continuous linear function defined on 
(H3) For each 
if 
, then 
, if in addition 
then
.
Remark 3.1 The financial meaning of hypothesis (H1) is self evident. Hypothesis (H3) is similar to that there is no arbitrage in the market. Hypotheses (H1)-(H3) are the static properties of the linear evaluation operator.
Remark 3.2 From Lemma 3.1, we know that 
is a continuous linear function defined on the Hilbert space
. It then follows from (H3) and the Riesz representative theorem that: there exists 
, such that
Since
, by [16] Theorem 3.1 and 3.2, without loss of generality, we may assume that there exists a Borel measurable (deterministic) function 
such that
(H4) For each
,
(3.1)
Remark 3.3 Hypothesis (H4) is the dynamic characteristic of the linear evaluation rules, which is shown uniquely in this paper. For the financial meaning of equation 3.1, you may see [5] Remark 2.2.
Now comes the explicit form of this evaluation operator in the market driven by fractional Brownian motion.
4. Deduce the Dynamic CAPM from the Dynamic Linear Evaluation Rule
4.1. The Explicit Form of the Evaluation Operator
Theorem 4.1 Let 
be continuous such that 
is an increasing function. Denote 
where g is measurable real valued function of polynomial growth such that 
If there exists a continuous function 
satisfying the integral equation
then let Q denote the probability measure given by
We have
(4.1)
where 
Here 
is the stochastic gradient (Malliavin derivative) of G at s.
Proof. Let 
be continuous function such that 
is an increasing function. Denote 
where g is measurable real valued function of polynomial growth such that 
By Lemma 2.2, we know that if there exists a continuous function 
satisfying the integral equation
then consider the translation of 
Let Q denote the probability measure given by
Then 
is a fractional Brownian motion under Q. It follows from (3.1) and Lemma 2.2 that
Let
. Define
where 
is the stochastic gradient (Malliavin derivative) of G at s. By Lemma 2.1 and 2.3, we obtain that
Thus,
The theorem is proved.
Suppose that 
Remark Equation (4.1) can be formally expressed as
(4.2)
4.2. Deduce the Dynamic CAPM from the Dynamic Linear Evaluation Rule
In fact, the CAPM attempts to relate 
the one-period rate of return of a specified security i, to 
the oneperiod rate of return of the entire market (as measured, say, by the Standard and Poor’s index of 500 stocks). If 
is the risk-free interest rate (usually taken to be the current rate of a US Treasury bill) then the model assumes that, for some constant 
where 
is a normal random variable with mean 0 that s values of
the CAPM model (which treats 
as a constant) implies that
or, equivalently, that 
where
That is, the difference between the expected rate of return of the security and the risk-free interest rate is assumed to equal 
times the difference between the expected rate of return of the market and the risk-free interest rate.
From the above formula, we know that the rate of return of a single security is determined by the relationship between this single security and the market portfolio.
Assume that the non-risk interest rate is 0. Equation (4.2) indicates that the instantaneous return of the contingent claim can be decomposed into two parts. Containing fractional Brownian motion, the first term to the right side denotes the stochastic volatility and 
denotes the volatility rate. The volatility rate changes in compliance with the change of the contingent claim
. In the other term, f is determined by the evaluation operator 
itself and it reflects the mechanism of the market. 
reflects the extent to which the instantaneous return of the contingent claim and the return of the market portfolio are related. Therefore, the instantaneous return of the contingent claim is mainly determined by this dependence. Equation (4.2) can be regarded as another version of CAPM. Accordingly, Equation (4.2) indicates that the instantaneous return of the contingent claim is mainly determined by the extent to which the instantaneous return of the contingent claim and the return of the market portfolio are related.
From the discussion above, we deduce the dynamic CAPM from the dynamic linear evaluation rule in the market driven by fractional Brownian motion.
5. Conclusion
In this paper, we first give some preliminaries of fractional Brownian motion. Then, we present the fundamental framework of the evaluation problem under which the evaluation operator satisfying some axioms is linear. Based on the dynamic linear evaluation mechanism of contingent claims, studying this evaluation rule in the market driven by fractional Brownian motions has led to a dynamic capital asset pricing model. It is deduced here mainly with the fractional Girsanov theorem and the Clark-Haussmann-Ocone theorem.
6. Acknowledgements
The authors would like to thank the referees for the careful reading of the paper and helpful suggestions. Part of this work was completed when author Zhou was visiting the University of Kansas. We would like to thank Professor Yaozhong Hu and David Nualart, and Dr. Alexander Uhl for their help.
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