# Risk-averse dynamic arbitrage in illiquid markets

## Somayeh Moazeni, Ricardo A. Collado and Andy Zhang

#### Need to know

• To rule out dynamic arbitrage, trades adapted to the price process filtration should be searched.
• The concept of risk-averse dynamic arbitrage using a time-consistent dynamic risk measure is introduced.
• Sufficient conditions are established to certify no-dynamic arbitrage by only searching in the space of $\mathcal F_0$-measurable admissible round-trip trades.

#### Abstract

Arguments on the existence of dynamic arbitrage and price manipulation strategies are often invoked to guide the modeling of 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 F0-measurable admissible round-trip trades is enough to attain a no-dynamic arbitrage characterization. This result both simplifies the identification of price impact structures that rule out dynamic arbitrage and supports analyses in some of the existing literature, where its assessment has been limited to searching in a set of F0-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 concerning the existence of dynamic arbitrage opportunities for a price impact model depends on their belief about expected future price changes and expected future price impacts, which can be revised over time due to 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 their 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.

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