= Minimal martingale measure in a one-period market
{title2=$Z_*=1-m^TC^{-1}(Y-m)$}
Let $m=\mathbb EY$ and $C=\operatorname{Cov}(Y)$ be an invertible <covariance matrix> of discounted gains. The displayed <signed martingale measure> leaves unchanged the mean-zero <random variables> orthogonal to the martingale part $Y-m$: if $\mathbb EL=0$ and $\mathbb E[(Y-m)L]=0$, then $\mathbb E[Z_*L]=0$. Conversely, preservation of all these orthogonal directions forces the density to lie in the span of $1,Y-m$, and its pricing constraints give the displayed formula. If $Z_*>0$ it is an <equivalent martingale measure>; if $Z_*\geq0$ it is a <dominated martingale measure>. Positivity is an extra condition, not a consequence of absence of <arbitrage> alone. In one period this density also minimizes $\mathbb EZ^2$ among square-integrable signed pricing densities, because every other such density differs by a vector orthogonal to $1,Y-m$. The initial capital in the unrestricted <one-period quadratic hedge> of a discounted payoff $H$ is $\mathbb E[Z_*H]$.
Back to article page
