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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. 

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. 

We investigate the stability of the Epstein–Zin problem with respect to small distortions in the dynamics of the traded securities. We work in incomplete market model settings, where our parametrization of perturbations allows for joint distortions in returns and volatility of the risky assets and the interest rate. Considering empirically the most relevant specifications of risk aversion and elasticity of intertemporal substitution, we provide a condition that guarantees the convexity of the domain of the underlying problem and results in the existence and uniqueness of a solution to it. Then, we prove the convergence of the optimal consumption streams, the associated wealth processes, the indirect utility processes, and the value functions in the limit when the model perturbations vanish. 

We study the analyticity of the value function in optimal investment with expected utility from terminal wealth and the relation to stochastically dominant financial models. We identify both a class of utilities and a class of semimartingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in the sense of infinite-order stochastic dominance. We construct two counterexamples, themselves of independent interest, which show that analyticity fails if either the utility or the market model does not belong to the respective special class. We also provide explicit formulas for the derivatives of all orders of the value functions as well as their optimizers. Finally, we show that for the set of supermartingale deflators, stochastic dominance of infinite order is equivalent to the apparently stronger dominance of second order. 

We study the response of the optimal investment problem to small changes of the stock price dynamics. Starting with a multidimensional semimartingale setting of an incomplete market, we suppose that the perturbation process is also a general semimartingale. We obtain second-order expansions of the value functions, first-order corrections to the optimisers, and provide the adjustments to the optimal control that match the objective function up to the second order. We also give a characterisation in terms of the risk-tolerance wealth process, if it exists, by reducing the problem to the Kunita–Watanabe decomposition under a change of measure and numéraire. Finally, we illustrate the results by examples of base models that allow closed-form solutions, but where this structure is lost under perturbations of the model where our results allow an approximate solution. 

On a finite probability space, we consider the problem of fair pricing of contingent claims and its sensitivity to a distortion of information, where we follow the weak information modeling approach. We show that, in complete models, or more generally, for replicable contingent claims, the weak information does not affect the fair price. For incomplete models, this is not the case for non-replicable claims, where we obtain explicit formulas for the information premium and correction to an optimal trading strategy. We illustrate our results by an example, where we demon- strate that under weak information, the fair price can increase, stay the same, or decrease. Finally, we perform the stability analysis for the information premium and the correction of the optimal trading strategy to perturbations of the contingent claim payoff, stock price dynamics, and the reference probability measure.