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 , making connections with the theory of large deviations. We concentrate on sequences of semi-complete markets where for each n the claim B_n admits the decomposition B_n = D_n+Y_n where D_n is replicable and Y_n is completely unhedgeable in that the indifference price of Y_n for an exponential investor is its certainty equivalent . Under broad conditions, we may assume that Y_n vanishes in accordance with a large deviations principle as n grows. In this setting, we identify limiting indifference prices as the position size becomes large, and show the prices typically are not the unique arbitrage free price in the limiting market . Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions endogenously arise in this setting.
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 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|<\infty, and the difference is explicitly identifiable using large deviations theory . Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions arise endogenously in this setting.
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