Heterogeneity of Investors and Asset Pricing in a Risk-Value World

converges to zero. Yet the convexity of the pricing kernel approaches 0 and,

thus, becomes minimal since for ε → ∞,

Similarly, it can be shown in the case of exponential risk functions (1 = -∞)

for investor h having the lowest sharing constant that e_hε∕ε approaches 1 for
ε → ∞ and that V (ε) → 0 for ε → ∞ implying c (ε) →

It is interesting to compare the convexity in this risk-value equilibrium to
that in an EU-equilibrium in which all investors use a HARA-utility function

with the same exponent i Assume that 1 > -∞ and the sum Y Ai in the
i

in the risk-value world. If 1 = -∞,

∕B_i) to be the same in both worlds. Then the convexity of

the EU-pricing kernel is the same as that of the risk-value pricing kernel if all
sharing rules are linear in the risk-value equilibrium. Since sharing rules are
linear in the EU-equilibrium anyway, this finding reinforces the importance
of heterogeneity of investors: If in both worlds there exists a representative
investor, i.e. if all sharing rules are linear, then the convexity of the pricing
kernel is the same in both worlds. Otherwise the convexity is higher in the
risk-value world because a representative investor does not exist.

The important result is that all European options are more expensive rela-
tive to the underlying asset, the more convex the pricing kernel is. Since con-
vexity in the HARA-based risk-value equilibrium increases with investor het-
erogeneity, relative option prices increase with investor heterogeneity. Stated
differently, the more the sharing rules deviate from linearity, the higher is the
investors’ need for option trading, the more expensive are European options
relative to the underlying asset. Hence investor heterogeneity might pro-
vide an explanation for the observation that stock index options appear to

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