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Avoided crossing

2014-01-24T23:52:35Z

14.97.89.152:

In [[mathematics]], a '''Sylvester matrix''' is a [[matrix (mathematics)|matrix]] associated to two [[univariate polynomial]]s with coefficients in a [[field (mathematics)|field]] or a [[commutative ring]]. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their [[resultant]], which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an [[integral domain]]).

Sylvester matrix is named after [[James Joseph Sylvester]].

==Definition==
Formally, let ''p'' and ''q'' be two nonzero polynomials, respectively of degree ''m'' and ''n''. Thus:
:<math>p(z)=p_0+p_1 z+p_2 z^2+\cdots+p_m z^m,\;q(z)=q_0+q_1 z+q_2 z^2+\cdots+q_n z^n.</math>
The '''Sylvester matrix''' associated to ''p'' and ''q'' is then the <math>(n+m)\times(n+m)</math> matrix obtained as follows:
* the first row is:
:<math>\begin{pmatrix} p_m & p_{m-1} & \cdots & p_1 & p_0 & 0 & \cdots & 0 \end{pmatrix}.</math>
* the second row is the first row, shifted one column to the right; the first element of the row is zero.
* the following ''n'' − 2 rows are obtained the same way, still filling the first column with a zero.
* the (''n'' + 1)th row is:
:<math>\begin{pmatrix} q_n & q_{n-1} & \cdots & q_1 & q_0 & 0 & \cdots & 0 \end{pmatrix}.</math>
* the following rows are obtained the same way as before.

Thus, if ''m'' = 4 and ''n'' = 3, the matrix is:
:<math>S_{p,q}=\begin{pmatrix}
p_4 & p_3 & p_2 & p_1 & p_0 & 0 & 0 \\
0 & p_4 & p_3 & p_2 & p_1 & p_0 & 0 \\
0 & 0 & p_4 & p_3 & p_2 & p_1 & p_0 \\
q_3 & q_2 & q_1 & q_0 & 0 & 0 & 0 \\
0 & q_3 & q_2 & q_1 & q_0 & 0 & 0 \\
0 & 0 & q_3 & q_2 & q_1 & q_0 & 0 \\
0 & 0 & 0 & q_3 & q_2 & q_1 & q_0
\end{pmatrix}.</math>

==Applications==
These matrices are used in [[commutative algebra]], e.g. to test if two polynomials have a (non constant) common factor. In such a case, the [[determinant]] of the associated '''Sylvester matrix''' (which is named the [[resultant]] of the two polynomials) equals zero. The converse is also true.

The solutions of the simultaneous linear equations
:<math>{S_{p,q}}^\mathrm{T}\cdot\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}</math>
where <math>x</math> is a vector of size <math>n</math> and <math>y</math> has size <math>m</math>, comprise the coefficient vectors of those and only those pairs <math>x, y</math> of polynomials (of degrees <math>n-1</math> and <math>m-1</math>, respectively) which fulfill
:<math>x \cdot p + y \cdot q = 0</math>
(where polynomial multiplication and addition is used in this last line).
This means the [[Null space|kernel]] of the transposed Sylvester matrix gives all solutions of the [[Bézout's identity|Bézout equation]] where <math>\deg x < \deg q</math> and <math>\deg y < \deg p</math>.

Consequently the [[Rank_(linear_algebra)|rank]] of the Sylvester matrix determines the degree of the [[Polynomial greatest common divisor|greatest common divisor]] of ''p'' and ''q'':
:<math>\deg(\gcd(p,q)) = m+n-\mathrm{rank}~S_{p,q}</math>
Moreover, the coefficients of this greatest common divisor may be expressed as [[determinant]]s of submatrices of the Sylvester matrix (see [[Subresultant]]).

==See also==
* [[Transfer matrix]]

==References==
* {{mathworld|urlname=SylvesterMatrix|title = Sylvester Matrix}}

==External links==
*[http://aix1.uottawa.ca/~jkhoury/elimination.htm Additional overview]

[[Category:Matrices]]
[[Category:Polynomials]]

Capital account

2014-01-17T12:00:04Z

14.97.153.235: /* Sterilization */

{{Unreferenced|date=June 2007}}

The '''forward price''' (or sometimes [[forward rate]]) is the agreed upon price of an [[asset]] in a [[forward contract]]. Using the [[rational pricing]] assumption, for a forward contract on an underlying asset that is ''tradeable'', we can express the forward price in terms of the [[spot price]] and any dividends etc. For forwards on non-tradeables, pricing the forward may be a complex task.

== Forward Price Formula ==
If the underlying asset is tradeable and a dividend exists, the forward price is given by:

:<math> F = S_0 e^{(r-q)T} - \sum_{i=1}^N D_i e^{(r-q)(T-t_i)} \,</math>

where

:<math>F</math> is the forward price to be paid at time <math>T</math>
:<math>e^x</math> is the [[exponential function]] (used for calculating continuous compounding interests)
:<math>r</math> is the [[risk-free interest rate]]
:<math>q</math> is the [[cost-of-carry]]
:<math>S_0</math> is the [[spot price]] of the asset (i.e. what it would sell for at time 0)
:<math>D_i</math> is a [[dividend]] which is guaranteed to be paid at time <math>t_i</math> where <math>0< t_i < T.</math>

==Proof of the forward price formula==

The two questions here are what price the short position (the seller of the asset) should offer to maximize his gain, and what price the long position (the buyer of the asset) should accept to maximize his gain?

At the very least we know that both do not want to lose any money in the deal.

The short position knows as much as the long position knows: the short/long positions are both aware of any schemes that they could partake on to gain a profit given some forward price.

So of course they will have to settle on a fair price or else the transaction cannot occur.

An economic articulation would be:

(fair price + future value of asset's dividends) - spot price of asset = cost of capital

Forward price = Spot Price - cost of carry

The future value of that asset's dividends (this could also be coupons from bonds, monthly rent from a house, fruit from a crop, etc.) is calculated using the risk-free force of interest. This is because we are in a risk-free situation (the whole point of the forward contract is to get rid of risk or to at least reduce it) so why would the owner of the asset take any chances? He would reinvest at the risk-free rate (i.e. U.S. T-bills which are considered risk-free). The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into.
So OUT - IN = NET GAIN and his net gain can only come from the opportunity cost of keeping the asset for that time period (he could have sold it and invested the money at the risk-free rate).

let:
:''K'' = fair price
:''C'' = cost of capital
:''S'' = spot price of asset
:''F'' = future value of asset's dividend
:''I'' = present value of ''F'' (discounted using ''r'' )
:''r'' = risk-free interest rate compounded continuously
:''T'' = length of time from when the contract was entered into

Solving for fair price and substituting mathematics we get:

:<math> K = C + S - F \,</math>

where:

:<math>C = S(e^{rT} - 1) \,</math>
(since <math> e^{rT} = 1 + j \,</math> where ''j'' is the effective rate of interest per time period of ''T'' )

:<math> F = c_1 e^{r(T - t_1)} + \cdots + c_n e^{r(T - t_n)} </math>
where ''ci'' is the ''i th '' dividend paid at time ''t i''.

Doing some reduction we end up with:

:<math> K = (S - I)e^{rT}. \,</math>

Notice that implicit in the above derivation is the assumption that the underlying can be traded. This assumption does not hold for certain kinds of forwards.

==Forward versus Futures prices==

There is a difference between forward and futures prices when interest rates are [[stochastic]]. This difference disappears when interest rates are deterministic.

In the language of [[stochastic processes]], the forward price is a [[Martingale (probability theory)|martingale]] under the [[forward measure]], whereas the futures price is a martingale under the [[risk-neutral measure]]. The forward measure and the risk neutral measure are the same when interest rates are deterministic.

See Musiela and Rutkowski's book on Martingale Methods in Financial Markets for a continuous-time proof of this result. See van der Hoek and Elliott's book on Binomial Models in Finance for the discrete-time version of this result.

==See also==

*[[Forward rate]]
*[[Forward measure]]
*[[Convenience yield]]
*[[Cost of carry]]
*[[Backwardation]]
*[[Contango]]

{{Derivatives market}}
[[Category:Derivatives (finance)]]
[[Category:Financial economics]]
[[Category:Financial terminology]]

Marconi's law

2012-09-02T06:50:28Z

14.97.115.40:

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Hotelling's rule

2012-04-02T06:52:20Z

14.97.233.229: /* Framework */

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