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The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage. A positive payoff with a strictly negative cost is called an arbitrage. If the state price density (stochastic discount factor) is positive, then one cannot construct an arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist stochastic discount factorexists a state price vector $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage. A positive payoff with a strictly negative cost is called an arbitrage. If the state price density (stochastic discount factor) is positive, then one cannot construct an arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist stochastic discount factor $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage. A positive payoff with a strictly negative cost is called an arbitrage. If the state price density (stochastic discount factor) is positive, then one cannot construct an arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exists a state price vector $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.
6 added 197 characters in body
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The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage. A positive payoff with a strictly negative cost is called an arbitrage. If the state price density (stochastic discount factor) is positive, then one cannot construct an arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist state pricesstochastic discount factor $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist state prices $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage. A positive payoff with a strictly negative cost is called an arbitrage. If the state price density (stochastic discount factor) is positive, then one cannot construct an arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist stochastic discount factor $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.
5 added 2 characters in body
source | link

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas'Farkas's Lemma and No Arbitrage

By Farkas'Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist state prices $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas' Lemma and No Arbitrage

By Farkas' Lemma, exactly one of the following conditions holds:

  1. There exist state prices $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.

The 1 you are referring to is a vector of ones

The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:

s = (ones(n,k) + R)' \ ones(n, 1)

where R is an n by k matrix (i.e. specifying returns for $n$ periods of $k$ assets).

1 + R adds 1 to each element of the matrix R (eg. makes it a return like 1.02 instead of .02). The MATLAB command \ would solve the below system for $\mathbf{s}$ in the least squares sense:

$$ ( 1 + R)^T \mathbf{s} = \begin{bmatrix} 1 \\ 1 \\ \ldots \\ 1 \end{bmatrix} $$

A non-negative solution $\mathbf{s}$ guarantees the absence of arbitrage

Any solution $\mathbf{s}$ to this linear system will be a vector of state prices for the $n$ states that satisfies the Law of One Price (LOOP). For any asset $i$, the inner product of the return series and the state price vector gives the value 1 which is the price of a return.

$$ (1 + \mathbf{r}_i)^T \mathbf{s} = 1$$

If $\mathbf{s}$ correctly prices each asset, the Law of One Price is satisfied. No arbitrage is a somewhat different condition though.

The existence of a non-negative state price vector $\mathbf{s}$ implies the absence of arbitrage.

(Note: I'm using bold letters for vectors.)

Farkas's Lemma and No Arbitrage

By Farkas's Lemma, exactly one of the following conditions holds:

  1. There exist state prices $\mathbf{s} \in \mathbb{R}^n$ such that $(1+R)^T \mathbf{s} = \mathbf{1}$ and $\mathbf{s} \geq 0$
  2. There exists a $\mathbf{w} \in \mathbb{R}^k$ (giving investments in the $k$ assets) such that $(1+R)\mathbf{w} \geq 0$ and $\mathbf{1}^T \mathbf{w} < 0$

Condition (1) is the existence of a non-negative state price vector $\mathbf{s}$.Condition (2) implies it's possible to construct an arbitrage:

  • Investing $w_i$ in asset $i$ gives the non-negative payoff $(1 + R) \mathbf{w} \geq 0$
  • The portfolio has a negative cost since $ \mathbf{w}^T \mathbf{1} < 0$.
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