b2) Finally we prove statement b2). The strict convexity of R1 (R3) implies
that Ri(R3) and R3-(R3) can intersect at most twice. R* = R* requires at
least one intersection. Suppose there exists one intersection only. As π(ε) is
monotonically decreasing, purchasing claims in some low states ε < ε+ and
selling claims in some high states ε > ε+ such that this transaction is self-
financing implies a lower expected return. Thus, one intersection contradicts
R* = R* so that two intersections are implied. ■
Appendix F: Proof of Proposition 8
Franke/Stapleton/Subrahmanyan (1999) have shown the following. Con-
sider two pricing kernels πi(ε) and π2(ε) with E[πi(ε)] = E[π2(ε)] = 1 and
E[επ1 (ε)] = E[επ2(ε)]. Suppose that these kernels intersect twice such that
πj. (ε) > π2 (ε) for low and high of levels of ε and πi (ε) < π2 (ε) in between.
Then all European options on ε are more expensive under πχ (ε).
Hence in order to prove Proposition 8 it suffices to show: If π1 (ε) is more
convex than π2 (ε) everywhere, then these pricing kernels must intersect twice
such that for low and high ε-levels π1 (ε) > π2 (ε) and vice versa in between.
Franke/Stapleton/Subrahmanyan (1999) have shown that the pricing kernels
must intersect at least twice to produce the same forward price of the un-
derlying asset. More than two intersections will be shown to be impossible.
Ci (ε) > c2 (ε) Vε implies
dZn [—πz1 (ε)] dZn [—πz2(ε)]
dε dε
or — ^»—H (ε)/ — πz2 (ε)] > θ
dε
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