Discusión de resultados

The financial market is said to becompleteif_M=RS, i.e. any state-contingent payoff can be generated by forming portfolios of the traded assets. Otherwise, the market is called incomplete.

4.2 A one-period model 81

The market is complete, if and only if for anyx_∈RS, we can findθ_∈RN such that

D>_θ=x.

Hence, we have the following result:

Theorem 4.4 The market is complete if and only if the rank of the N×S dividend matrixD is equal to S.

Clearly, a necessary (but not sufficient) condition for a complete market is thatN ≥S, i.e. that there are at least as many assets as states. The “pruned” dividend matrix ˆD will in this case be a non-singularS_×S matrix.

We can always define theS-dimensional vector

(4.7) ψ∗= ˆD>DˆDˆ>−1P,ˆ

which will satisfy

ˆ

Dψ∗= ˆDDˆ>DˆDˆ>−1Pˆ = ˆP.

The redundant assets are uniquely priced by no-arbitrage so that we have P=Dψ∗. It it clear that ψ∗ is in fact exactly equal to the dividend generated by a portfolio ˆθ∗ =DˆDˆ>−1Pˆ, i.e. it is a state-price vector in the set of attainable dividend vectors, at least if it has only strictly positive elements.

In the special case of a complete and arbitrage-free market the elements ofψ∗ will be strictly positive. Why? Form a portfolio that pays off 1 in statejand 0 in all other states. This is possible since the market is complete. The price of this portfolio is ψ∗

j. To avoid arbitrage, ψj∗ must be strictly positive. This argument works for all j = 1, . . . , S. Hence, ψ∗ is a state-price vector. In fact Exercise 4.2 shows thatψ∗ is theonly state-price vector when the market is complete. In the case of a complete market the matrix ˆD is non-singular, as noted above. Using matrix algebra we may then rewriteψ∗ as

(4.8) ψ∗= ˆD−1P.ˆ

Next, define theS-dimensional vectorζ∗ by

(4.9) ζ∗= ˆD>EhDˆDˆ>i−1_P.ˆ

To see the meaning of this, let us for simplicity assume that none of the basic assets are redundant so that ζ∗ =D>_{(E [DD}>_])−1_{P. Recall that D is theN-dimensional random variable for which}

thej’th component is given by the random dividend of assetj. Hence,DD> _{is anN×N matrix}

of random variables with the (i, j)’th entry given byDiDj, i.e. the product of the random dividend of asset i and the random dividend of assetj. The expectation of a matrix of random variables is equal to the matrix of expectations of the individual random variables. So E [DD>_{] is also an}

N_×Nmatrix. For the general case we see from the definition thatζ∗is in fact the dividend vector generated by the portfolio ˆθ =EhDˆDˆ>i

−1

ˆ

Pof the non-redundant assets. We can think ofζ∗ as a random variableζ∗ _{given by}

4.2 A one-period model 82

We can see that

EhDζˆ ∗i= E ˆ DDˆ>_Eh_Dˆ_Dˆ>i−1_Pˆ = EhDˆDˆ>i _Eh_Dˆ_Dˆ>i−1_Pˆ _{= ˆP.}

It follows thatζ∗_{is in fact a state-price deflator if it takes strictly positive values. It can be shown}

that no other state-price deflator can be written as the dividend of a portfolio of traded assets. In a complete market,ζ∗ _{will be a state-price deflator and it will be unique.}

Recall that there is a one-to-one relation between state-price vectors and state-price deflators. In general ζ∗ _{is not the state-price deflator associated with ψ}∗_{. However, this will be so if the}

market is complete.

In a complete market it is possible to generate a riskless dividend as a portfolio of the traded assets. Therefore, we can talk of risk-neutral probability measures. Due to the one-to-one cor- respondence between state-price vectors/deflators and risk-neutral probability measures, we can conclude that a complete market will have a unique risk-neutral probability measure. This is the probability measure Q∗ _{given by the state probabilities q}∗

j =erψ∗j, where ψ∗j is the j’th element of the unique state-price vector ψ∗_{. If the market is incomplete, there will be more than one}

risk-neutral measure. This is the case in Example 4.2.

Theorem 4.5 Suppose that prices admit no arbitrage. Then the market is complete if and only if there is a unique state-price vector (or deflator). Equivalently, the market is complete if and only if there is a unique risk-neutral probability measure.