Indifference pricing for Contingent Claims: Large Deviations Effects

We study utility indifference prices and optimal purchasing quantities for a non-traded contingent claim in an incomplete semi-martingale market with vanishing hedging errors. We make connections with the theory of large deviations. We concentrate on sequences of semi-complete markets where in the $n^{th}$ market, the claim $B_n$ admits the decomposition $B_n = D_n+Y_n$. Here, $D_n$ is replicable by trading in the underlying assets $S_n$, but $Y_n$ is independent of $S_n$. Under broad conditions, we may assume that $Y_n$ vanishes in accordance with a large deviations principle as $n$ grows. In this setting, for an exponential investor, we identify the limit of the average indifference price $p_n(q_n)$, for $q_n$ units of $B_n$, as $n\rightarrow \infty$. We show that if $|q_n|\rightarrow\infty$, the limiting price typically differs from the price obtained by assuming bounded positions $\sup_n|q_n|