A market maker is a specialist who provides liquidity by continuously offering bid and ask quotes for a financial asset. The market maker’s objective is to maximize profit while avoiding the accumulation of a large position in the asset to control inventory risk. To achieve model-free results, online learning has been applied to design market-making strategies that make no assumptions on the dynamics of the limit order book and asset price. However, existing work primarily focuses on profit rather than inventory risk. To address this limitation, this paper develops market-making strategies with inventory constraints within the online learning framework. To manage inventory risk, we propose two classes of market-making strategies with fixed bid-ask spreads that serve as reference strategies. Each reference strategy can ensure that the inventory remains under control, which enables the online learning algorithms designed for each class of reference strategies to satisfy inventory constraints. Different from the standard online learning model where the gain in each period is assumed to lie within a fixed bounded interval, the gain in our model depends on a state variable (i.e., the inventory size). Thus, a key challenge in analyzing the regret bounds is to bound the difference between the gains of any two reference strategies, which becomes significantly more complicated compared with scenarios without inventory constraints. By tackling these difficulties, we show that these algorithms achieve low regrets. Experimental results illustrate the superior performance of our algorithms in inventory risk control.

Adaptive Market Making with Inventory Constraints via Online Learning

Classic option pricing models, such as the Black-Scholes formula, often depend on some rigid assumptions on the dynamics of the underlying asset prices. These assumptions are inevitably violated in practice and thus induce the model risk. To mitigate this, robust option pricing that only requires the no-arbitrage principle has attracted a great deal of attention among researchers. In this paper, we give new robust upper bounds for option prices based on a novel {\it $\eta$-momentum trading strategy}. Our bounds for European options are tighter for most common moneyness, volatility, and expiration date setups than those presented in (\citeauthor{DKM16} \citeyear{DKM16}). Our bounds for average strike Asian options are the first closed-form robust upper bounds for those options. Numerical simulations demonstrate that our bounds significantly outperform the benchmarks for both European and Asian options. 

Tighter Robust Upper Bounds for Options via No-Regret Learning

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Shan Xue

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Adaptive Market Making with Inventory Constraints via Online Learning

Tighter Robust Upper Bounds for Options via No-Regret Learning

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