Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 32 / 1 / a

Suppose a strictly positive pricing density exists. Its weighted payment for any fixed holdings is integrable, andIf the terminal payment is nonnegative and strictly positive on an event of positive probability, positivity of $\nu$ makes this expected value strictly positive. It cannot therefore have nonpositive initial cost. Thus the first form of arbitrage is incompatible with a positive pricing density. The existence argument in part (b) proves that this form occurs whenever separation within the payment span obstructs a density.

The same pricing relation gives $x^{\top }{A}_0\ge 0$ whenever the terminal payment is nonnegative. Therefore negative initial cost also contradicts a strictly positive state-price density, even if the terminal payment is identically zero.

This second form is needed for directions invisible in the terminal payments. Let $L$ be the linear span of the essential values of $A_1$, equivalentlyIf $A_0\notin L$, let $v$ be its nonzero orthogonal projection on $L^{\perp }$ and set $x=-v$. Then $x^{\top }{A}_1=0$ almost surely while $x^{\top }{A}_0=-\Vert v{\Vert }^2<0$. Hence the second form exists in this case. Part (b) handles all remaining cases.