By Ewald C.-O.

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**Extra resources for Discrete time finance**

**Sample text**

Then we can rewrite the minimum variance problem as follows : Problem 2 minimize F ⊤ CF subject to (1 − F1 − ... − Fn )r + n i Fi R = ρ i=1 Fi ≥ 0. This is a quadratic programming problem which can now be solved by the Lagrange multiplier method for example. In order to obtain the actual trading strategy from the Fi one has set φi = FSi ·x i , where x denotes 0 the initial investment. 1. Let us consider a model with two risky assets and a money market account. For simplicity we assume that the interest rate r is equal to zero.

A continuously differentiable function u : R+ → R is called a risk averse utility function if it has the following three properties : 1. limx→0 u′ (x) = +∞ and limx→∞ u′ (x) = 0 2. u is strictly increasing in the way that u′ (x) > 0 for all x ∈ R 3. u is strictly concave in the way that u(λx + (1 − λ)y) > λu(x) + (1 − λ)u(y). 1 is equivalent to u′′ (x) < 0 for all x ∈ R+ . Let us now consider a random variable X which in interpreted as a random payoff. Fixing a utility function u, we will measure the performance of this payoff by E(u(X)) = k i=1 P(ωi )u(X(ωi )).

M = ∅. Assume first that there is only one risk neutral measure. 6 is trivially satisfied for all contingent claims X and so the market model is complete. On the other side, assume the market model is complete and consider two risk neutral measures Q1 and Q2 in M. , k consider the contingent claim X i given by X i (ω) = 1+r 0 36 if ω = ωi otherwise Since the model is complete, X i is an attainable contingent claim. 2 that Q1 (ωi ) = EQ1 1 Xi 1+r = EQ2 1 Xi 1+r = Q2 (ωi ). Therefore Q1 = Q2 and we have hence shown, that M consists of only one risk neutral measure.