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

and the EU-equilibrium become transparent when we assume HARA-based
risk functions.

5.2 HARA-Based Risk Functions

We assume that all investors have HARA-based risk functions with the same
7. We will show that more heterogeneity of investors raises the convexity of
the pricing kernel c(ε):

The convexity of the pricing kernel is essential for option pricing. Consider
two economies with pricing kernels π_i (ε) and π₂ (ε) such that the convexity
of the first pricing kernel is higher everywhere and the forward price of the
aggregate payoff-distribution is the same, E[ε π₁ (ε)] = E[επ₂(ε)].Then based
upon a result of Franke, Stapleton and Subrahmanyam (1999) proposition 8
shows that all European options on ε are more expensive under the more
convex pricing kernel πι (ε).

Proposition 8 : Let π₁ (ε) and π₂ (ε) be two forward pricing kernels with
the underlying asset’s forward price E[επ₁ (ε)] = E[επ₂(ε)] being the same.
Assume that the convexity of π_i (ε) exceeds that of π₂ (ε) for every ε. Then
all European options on ε are more expensive under π₁ (ε) than under π₂ (ε).

Proof. See Appendix F.

Given the importance of the pricing kernel we investigate it in the risk-
value equilibrium. Two investors are said to be heterogeneous if their sharing
constants differ. The more they differ, the more convex or concave are the
relative sharing rules, the higher is the convexity of the pricing kernel as will

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