Risk-Averse Dynamic Arbitrage in Illiquid Markets

Arguments on the existence of dynamic arbitrage and price manipulation strategies are often invoked to guide modeling price impacts of large trades. We revisit the concept of dynamic arbitrage in illiquid markets in the presence of time-varying stochastic price impact functions and a broad class of market price dynamics. We first establish a sufficient condition under which searching in the space of $\mathcal F_{0}$-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage certificate. This result simplifies identifying price impact structures that rule out dynamic arbitrage and supports the analysis in some existing literature, where its assessment is limited to the search in the set of $\mathcal F_{0}$-measurable round-trip trades. For time-varying stochastic linear price impact functions, we show that this condition is necessary and sufficient for the absence of dynamic arbitrage. The present quantitative analysis implies that a trader's opinion on the existence of dynamic arbitrage opportunities for a price impact model depends on his belief about expected future price changes and expected future price impacts, which can be revised over time by the collection of new information. This motivates us to let the existence of such arbitrage opportunities depend not only on the trader's belief about expected price movements but also on his risk attitude. We thus introduce the concept of risk-averse dynamic arbitrage using a general time-consistent dynamic risk measure and a risk-aversion threshold level. Similar sufficient conditions are studied under which searching in the space of static round-trip trades enables us to conclude on no-risk-averse dynamic arbitrage