The fundamental theory of asset pricing is at heart of the classical financial theory. According to the fundamental theorem of asset pricing in discrete time, there are no arbitrage
opportunities if and only if there is an equivalent martingale measure (Duffie 2001, p. 30).
Additional technical conditions have to be fulfilled for if and only if statement to apply in continuous time (Duffie 2001, ch. 6). An equivalent martingale measure to another probability measure is a measure that assigns a positive probability and a zero probability for the same states of the world as the other probability measure (Pennacchi p. 207). The equivalent martingale measure in this setting is the probability measure in a risk neutral world, i.e. the probability under which the price of an asset is the expected cash flows discounted with the risk free rate (Duffie 2001, p. 28 and p. 108). It is customary to denote the equivalent
martingale measure with . The transformation from a probability measure in a risk world to a probability measure in a risk neutral world is done by means of the Girsanov theorem.
Let a stochastic process:
4-17
Under the assumption that there exists a process so that:
4-18
Putting:
∫ ∫
4-19
And
4-20
Then:
̂ 4-21
by using the transformation:
34 ̂ ∫
The process should satisfy the Novikov condition: ( ∫ ) , which means that u is a square integrable function
4-22
Where
̂ is a Brownian motion with respect to the probability measure Q (Øksendal 2000, p.155) and is an event in the probability space Ω (Øksendal 2000, p.156).
Y and B depend on ; , and depend on and .
For an example of using Girsanov’s theorem for transforming a probability measure to so that the risk free rate can be used for finding the asset price (see appendix A - ix).
Assuming that an asset price follows a fractional Brownian motion, i.e. a Brownian motion which increments are correlated, one can construct arbitrage investment portfolios (Sottinen 2003). Hu and Øksendal (2003) expanded the fundamental theorem of asset pricing to include the fractional Brownian motion in a non-arbitrage fashion, using a mathematical operator called Wick product. This is the product of two square integrable random variables. Øksendal (2004) interprets the Wick product as a value process that becomes an asset price when observed by an economic agent like observations in quantum mechanics. The expansion of the fundamental asset pricing theorem to fractional Brownian motions has been met with counterarguments by Björk and Hult (2005) and Bender, Sottinen and Valkeila (2007) for either lacking a sound economic interpretation or producing arbitrage under some
observations.
The fundamental theorem of asset pricing supports the notion of efficient capital markets.
35 4.4.2 The Stochastic Discount Factor
Let a price functional that maps payoffs into prices of the form . It can be shown by Riesz Representation Theorem (Ødegaard 2013) that under certain conditions there exist a stochastic variable so that:
[ ]
Ødegaard lists up the following conditions:
- The set of payoffs is a linear space .
- The conditional expectation defines an inner product on this linear space. If are in the space , the conditional expectation [ ] is an inner product.
- The set of payoffs with the inner product of conditional expectation is a Hilbert Space.
In a Hilbert space every Cauchy sequence has a limit to converge (Bierens 2007, Borowski and Borwein 1989). ). A Cauchy sequence is a sequence which values can be brought arbitrarily closed together.
Let a bounded linear functional on a Hilbert space. Then, according to Riesz Representation Theorem, there exists a unique element in such that
〈 〉
Substituting the conditional expectation for the inner product we have:
[ ]
The stochastic variable used in finance is called:
- the stochastic discount factor (SDF) - the pricing kernel
- the intertemporal marginal rate of substitution of consumption
SDF can be equal to the equivalent martingale measure (the Radon-Nikodym derivative) under certain conditions. Let be the equivalent probability measure in a risk neutral world and the true probability measure. Then the equivalent martingale measure is which is also called the Radon-Nikodym derivative. Given a strictly positive stochastic process which satisfies the equation [ ] for all assets we can write [ ] where is the risk free gross return (Duffee 2012, pp. 21-22)
36 Another version of is the behavioral stochastic discount factor:
Where is the representative agent’s probability measure and is a sentiment risk (Shefrin 2007). If the sentiment risk disappears and collapses back to .
An example of a stochastic discount factor (SDF) is (Campbell et al. 1997, p. 294):
4-23
where is the subjective discount factor and is the derivative of the utility of consumption.
According to Campbell et al. (1997, p.296), a stochastic discount factor can be constructed for every pair of utility functions and . Given complete markets the stochastic discount factor is going to be unique. The unique stochastic discount factor is related to the equivalent martingale measure. The equivalent martingale measure transforms the investment from a world with risk to a risk neutral world. In incomplete markets there can be many stochastic discount factors due to idiosyncratic marginal utilities.
Cvitanic and Malamud (2010) ascertain that homogeneity in consumption preferences and beliefs is relevant for defining a unique stochastic discount factor when there are more than two types of agents. Given complete markets, homogeneous preferences and homogeneous beliefs, the stochastic discount factor and assets prices are uniquely defined. Bhamra and Uppal (2010) come up with a closed-form solution for the stochastic discount factor in an economy with two heterogeneous types of agents without assuming specific utility function values.
37 4.4.3 The Basic Asset Pricing Equation
Here we follow the presentation by Cochrane (2005, pp. 4-5).
Consider the following consumption utility maximization problem over two periods:
⏟
Subject to:
4-24
The notation is as follows:
is the amount of the asset the agent chooses to buy
is the payoff on an asset is the endowment
is the subjective discount factor.
The first order condition (F.O.C.) is:
( ) ( ) ( )
( )
4-25
Equation 4-25 is called the basic asset pricing equation.
38 4.4.4 The Capital Asset Pricing Model
The Capital Asset Pricing Model was developed independently by Sharpe (1964), Treynor (1962), Lintner (1965a, b) and Mossin (1966).
Let the expected return and variance of a portfolio be:
( )
4-26
( ) [ ] 4-27
where
is the rate of return of a portfolio, ( ) is the variance of the portfolio is the rate of return for the ith asset, is the variance of the risky asset
is the rate of the market return, is the variance of the market portfolio
is the covariance of the ith asset and the market portfolio The F.O.C. with respect to are:
( )
4-28
( )
[ ]
4-29
Setting the excess return : ( )
4-30
Then
39 ( )
( )
4-31
This has to be equal to the slope of the Capital Market Line which shows a linear relationship between expected return and the risk of a portfolio:
[ ] 4-32
The above equation is the CAPM. The main prediction of this model is that the return of the risky asset depends on the risk premium [ ( ̃ ) ] and asset's risk .
The sample equation is with . The systematic risk is and the unsystematic risk is .
4.4.5 The Lucas Asset Pricing Model and the Consumption Capital asset Pricing