Abstract. We consider an agent who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost λ > 0. The utility function is of the form Up(c) = c p /p for p < 1, p = 0. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of λ 1/3 . We also obtain a "nearly optimal" strategy, whose utility asymptotically matches the leading terms in the value function.

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.

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We price a contingent claim liability using the utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε > 0 in two cases with and without a contingent claim liability. Using the computations from the heuristic argument in Whalley & Wilmott [37] we provide a rigorous derivation of the asymptotic expansion of the value function in powers of ε 1/3 in both cases with and without a contingent claim liability. Additionally, using utility indifference method we derive the price of the contingent claim liability up to order ε. In both cases, we also obtain a "nearly optimal" strategy, whose expected utility asymptotically matches the leading terms of the value function.