Abstract:      

Lacking a market for a divisible asset, a seller faces a stream of buyers who arrive at random times with random limit orders. This paper uses search theory to understand how this liquidation sale optimally proceeds. We solve the dynamic programming exercise characterizing his optimal trading behavior. His behavior changes as the asset position falls, reflecting the endogenous time-varying value of the asset position.

Using recursive methods and duality theory, we uncover a new “diminishing returns to optionality” property: The Bellman value function is increasing and concave in the position. The seller therefore takes greater advantage of more generous offers, but his marginal value shifts up as he unwinds his position, making him less willing to trade. Deducing a convex marginal value, we then offer new insights on transactional liquidity in finance for trade size, depth and spreads. We also explore a new dimension of liquidity, namely, the waiting time between trades. Finally, the model is tractable enough to allow for price-quantity bargaining. We find that greater buyer bargaining power is tantamount to greater frictions, and so increases supply, decreases price, and leads to a more liquid induced market.