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On a finite probability space, we consider the problem of indifference pricing of contingent claims, where the preferences of an economic agent are modeled by an Inada utility stochastic field — the interior of its effective domain being (a,∞) — for some a∈R∪{−∞}. This allows for including utilities on both R and R+. We consider arbitrary contingent claims and show that, for replicable ones, the indifference price equals the initial value of the replicating strategy and thus depends neither on the agent’s initial wealth, for which the indifference pricing problem is well-posed, nor the utility stochastic field. This, in particular, shows the consistency of the indifference and arbitrage-free pricing methodologies for complete models. For nonreplicable claims, we show that the indifference price is equal to the expectation of the discounted payoff under the dual-optimal measure, which is equivalent to the reference probability measure. In particular, we demonstrate that the indifference price is unique for every choice of a smooth Inada utility stochastic field and initial wealth in (a,∞). Our proofs rely on the change of numéraire technique and a reformulation of the indifference pricing problem. The advantages of the settings of this paper and the approach allow for bypassing the technicalities and issues related to choosing the notion of admissibility and for including a wide range of utilities, including stochastic ones. We augment the results with examples.
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This content will become publicly available on December 6, 2025
Pricing of contingent claims in large markets
We consider the problem of pricing in a large market, which arises as a limit of small markets within which there are finitely many traded assets. We show that this framework allows accommodating both marginal-utility-based prices (for stochastic utilities) and arbitrage-free prices. Adopting a stochastic integration theory with respect to a sequence of semimartingales, we introduce the notion of marginal-utility-based prices for the large (post-limit) market and establish their existence, uniqueness, and relation to arbitrage-free prices. These results rely on a theorem of independent interest on utility maximisation with a random endowment in a large market that we state and prove first. Further, we provide approximation results for the marginal utility-based and arbitrage-free prices in the large market by those in small markets. In particular, our framework allows pricing asymptotically replicable claims, where we also show consistency in the pricing methodologies and provide positive examples.
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- Award ID(s):
- 1848339
- PAR ID:
- 10565446
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Finance and Stochastics
- Volume:
- 29
- Issue:
- 1
- ISSN:
- 0949-2984
- Page Range / eLocation ID:
- 177 to 217
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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