Farkas’ lemma

Farkas’ lemma states,

let \(\textbf{A} \in \mathrm{R}^{m \times n}\) and \(\textbf{b} \in \mathrm{R}^{m}\). Then exactly one of the following two statements is true:

  1. There exists an \(\textbf{x} \in \mathrm{R}^{n}\) such that \(\textbf{Ax} = \textbf{b}\) and \(\textbf{x} \geq 0\).
  2. There exists a \(\textbf{y} \in \mathrm{R}^{m}\) such that \(\textbf{A}^T \textbf{y} \geq 0\) and \(\textbf{b}^T \textbf{y} < 0\).

Where, \(\textbf{x} \geq 0\) means \(x_i \geq 0\) \(\forall i\). (There’s a version of Farkas’ lemma called Stiemke’s lemma in which the relationship hold with strict inequality.)

If we imagine \(\textbf{A}\) to be a payoff matrix of an asset, \(\textbf{b}\) a corresponding price vector and \(\textbf{y}\) portfolio weights, we can think about the theorem in the following way. (1.) implies no arbitrage and (2.) implies arbitrage, and only one can be true.

Arrow-Debreu security

Suppose that the world exists of two time periods, today and tomorrow, in which tomorrow is stochastic. Tomorrow the world might be in \(S\) different states of nature. An asset that pays \(1\) tomorrow if state \(s \in S\) occurs is called an A-D security, and the price of this security is called state price \(q_s\).

Imagine an arbitrary asset \(i\) that pays \(X_{i,s}\) in state \(s\). The price of an arbitrary asset can be described in terms of A-D securities in the following way,

\begin{equation} P_i = \sum_s q_s X_{i,s} \tag{1} \end{equation}

The lemma in finance

In a complete market with no arbitrage, Stiemke’s lemma implies that the equation \(\textbf{Xq} = \textbf{P}\) has a solution \(\textbf{q} > 0\). We rewrite this as equation (1), \(P_i = \sum_s q_s X_{i,s}\).

This equation states that the price of asset \(i\), is equal to the sum of the assets state payoffs times the state prices. No arbitrage (Farkas’/Stieke’s lemma) implies (in both directions) a positive state-price vector, this is sometimes called the fundamental theorem of asset pricing.

Implications

Consider equations (1) again and multiply with the probability that state \(s\) occurs,

\begin{equation} P_i = \sum_s \pi_s \frac{q_s}{\pi_s} X_{i,s} \tag{2} \end{equation}

\(\frac{q_s}{\pi_s} \equiv M\) which is a random variable. \(M\) is in financial theory called the stochastic discount factor (SDF). Further,

\begin{equation} P_i = \sum_s \pi_s M X_{i,s} \iff \tag{3} \end{equation}

\begin{equation} P_i = \mathrm{E}[ M X_{i,s} ] \tag{4} \end{equation}

This tells us that the price of any asset \(i\) is equal to the expected value of the payoff times the SDF. Henceforth, we drop the s in the notation for convenience. Dividing both sides by the price \(P_i\) yields what is called the fundamental equation of asset pricing,

\begin{equation} 1 = \mathrm{E}[ M R_{i} ] \tag{5} \end{equation} where the return \(R_i\) is the payoff divided by the price.

Equation (5) hold for all assets. If we assume that the return, \(R_i\) is non-stochastic, i.e. risk-free (and can thus be taken out of the expectation),

\begin{equation} 1 = \mathrm{E}[ M R_{rf} ] \iff \tag{6} \end{equation}

\begin{equation} 1 = R_{rf} \mathrm{E}[ M ] \iff \tag{7} \end{equation}

\begin{equation} \frac{1}{\mathrm{E}[M]} = R_{rf} \tag{8} \end{equation}

Thus, the risk-free rate is equal to the inverse of the SDF. This can also be shown from our initial definition of SDF,

\begin{equation} \mathrm{E}[M] = \sum_s \pi_s \frac{q_s}{\pi_s} = \sum_s q_s \tag{9} \end{equation}

where \(\sum_s q_s\) implies that you have bought all A-D securities and thus have secured a payoff of \(1\) tomorrow, and thus the risk-free return is,

\begin{equation} R_{rf} = \frac{1}{\sum_s q_s} = \frac{1}{\mathrm{E}[M]} \tag{10} \end{equation}

Asset pricing models

Now we have the tools to formulate the core of “complete market no arbitrage asset pricing theory”. Consider equation (4),

\begin{equation} P = \mathrm{E}[ M X ] \end{equation}

By using the fact that \(Cov(M, X) = \mathrm{E}[M X] - \mathrm{E}[M] \mathrm{E}[X]\),

\begin{equation} P = Cov(M, X) + \mathrm{E}[M] \mathrm{E}[X] \iff \tag{11} \end{equation}

\begin{equation} P = \frac{1}{R_{rf}} \mathrm{E}[X] + Cov(M, X) \tag{12} \end{equation}

This, says that the price of any asset is equal to its dicounted future payoff plus a risk premium. We can now rearrage equation (12) and then divide each side by the price,

\begin{equation} \mathrm{E}[X] = P R_{rf} - R_{rf} Cov(M, X) \iff \tag{13} \end{equation}

\begin{equation} \mathrm{E}[R] = R_{rf} - R_{rf} Cov(M, R) \tag{14} \end{equation}

Equation (14) is a fundamental result and says that the return of any asset depends on the risk-free rate and the covaraince with the SDF. Rearranging this into excess return,

\begin{equation} \mathrm{E}[R] - R_{rf} = - R_{rf} Cov(M, R) \tag{15} \end{equation}

Furthermore, multiplying and dividing by \(Var(M)\) and substituting back \(R_{rf} = \frac{1}{\mathrm{E}[M]}\),

\begin{equation} \mathrm{E}[R] - R_{rf} = \frac{Cov(M, R)}{Var(M)} \cdot - \frac{Var(M)}{\mathrm{E}[M]} \tag{16} \end{equation}

Equation (16) is called a beta pricing model and sometimes written as,

\begin{equation} \mathrm{E}[R] - R_{rf} = \beta_{M,R} \cdot \lambda_{M} \tag{17} \end{equation}

\(\beta_{M,R}\) is the quantity of risk in the asset. Note that \(\lambda_{M}\) does not directly depend on the individual return but on the first and second moments of the SDF.

A large literature in theoretical asset pricing is about finding the SDF and its properties. The theory of SDF relates several independent asset pricing models, some commonly known examples are: