In incomplete financial markets, the classical hedging argument for valuation of contingent claims has two natural generalizations. The first one has important applications in financial supervision and accounting while the second one is more relevant in trading of financial products. In the presence of illiquidity effects, these values become nonlinear functions of the underlying cash-flows.
This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. Explicit consideration of swap contracts is essential in illiquid markets where the valuation of swaps cannot be reduced to the valuation of cumulative claims at maturity. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of optimal investment under conditions that extend the no-arbitrage condition in the classical linear market model. All results are derived with the "direct method" without resorting to duality arguments.