Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 2

Interpret positivity of $Y$ as strict positivity almost surely. Suppose the stated bounded positive pricing variable exists. For a vector $H$ satisfying the two sign restrictions,A nonnegative random variable with zero expectation vanishes almost surely. Since $Y>0$, this gives $H\cdot P=0$ almost surely and $H\cdot p=0$. The supplied nondegeneracy condition now yields $\boxed{H=0}$. This proves the direction from a one-period martingale deflator to the separating-portfolio condition; no asset-pricing theorem has been invoked.

A direct exponential minimization construction of a bounded pricing kernel proves the reverse implication without using a course theorem. ConsiderBoundedness of $P$ makes $F$ finite, continuous, and differentiable. On every bounded set of $h$, the exponential and its derivatives have deterministic bounds, so differentiation under expectation is justified.

We first prove coercivity: $F(h)\to \infty$ as $|h|\to \infty$. Otherwise there would be a sequence $h_j$ with $|h_j|\to \infty$ along which $F(h_j)$ stays bounded above. Pass to a subsequence with $h_j/|h_j|\to u$, $|u|=1$. There are two cases.

If $\mathbb{P}(u\cdot P<0)>0$, choose $a>0$ with $\delta =\mathbb{P}(u\cdot P\le -a)>0$. The bound on $P$ gives $h_j\cdot P\le -a|h_j|/2$ on this event for all large $j$. ThereforeIf instead $u\cdot P\ge 0$ almost surely, the portfolio condition forces $u\cdot p>0$, since $u\ne 0$. Hence $h_j\cdot p\to \infty$, and the nonnegative exponential term again forces $F(h_j)\to \infty$. Both cases contradict the chosen sequence.

A minimizing sequence is consequently bounded. It has a convergent subsequence, and continuity of $F$ gives a global minimizer $h_{\ast }$. At a minimizer each directional derivative vanishes, soThusIf $|P|\le L$, then $e^{-|h_{\ast }|L}\le Y\le e^{|h_{\ast }|L}$: the constructed pricing kernel is bounded and even bounded away from zero. The compactness, coercivity, and differentiation arguments above supply the needed proof rather than appealing to the fundamental theorem of asset pricing.

Use the positive variable $Y$ from part (a). Since $0\le (S-B{)}^{+}\le S$ and $(S-B{)}^{+}\ge S-B$,Combining these inequalities gives the one-period call price boundsFor the strict assertion, both $\{S>B\}$ and $\{S<B\}$ have positive probability. Positivity of $Y$ givesIf $s-b\ge 0$, the second inequality is the required strict bound; if $s-b<0$, the first is. Hence $\boxed{c>(s-b{)}^{+}}$. The weighted expectations are finite because $P$ and $Y$ are bounded.