Optimal investment and contingent claim valuation in illiquid markets

Abstract

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.

Shadow price of information in discrete time stochastic optimization

“…Several examples can be found in the above references. Applications to financial mathematics are given in [9][10][11].…”

Existence of solutions in non-convex dynamic programming and optimal investment

“…in limit order markets where the limit order books always have finite depth. Further conditions are given in [28,29].…”

“…In particular, we do not assume the existence of a cash account a priori. As in [29], we assume that trading costs are given by an adapted sequence S = (S t ) T t=0 of convex F t -normal integrands on R d such that S t (0, ω) = 0. We also allow for portfolio constraints given by an adapted sequence D = (D t ) T t=0 of closed convex sets in R d , each containing the origin.…”

“…Problem (9) can be interpreted as an asset-liability management problem where one looks for trading strategies z whose proceeds cover the liability c as well as possible as measured by EV . In the convex case, problem (9) was studied in [29], where existence of solutions was derived from the results of [30].…”

“…To our knowledge, all previous existence results on optimal investment under such market frictions assume a concave utility function; see e.g. [20,29,15] and the references therein.…”