Market Making and Portfolio Liquidation Under Uncertainty

Abstract

Market making and optimal portfolio liquidation in the context of electronic limit order books are of considerably practical importance for high frequency (HF) market makers as well as more traditional brokerage firms supplying optimal execution services for clients. In general the two problems are based on probabilistic models defined on certain reference probability spaces. However, due to uncertainty in model parameters or in periods of extreme market turmoil, ambiguity concerning the correct underlying probability measure may appear and an assessment of model risk, as well as the uncertainty on the choice of the model itself, becomes important, as for a market maker or a trader attempting to liquidate large positions, the uncertainty may result in unexpected consequences due to severe mispricing. This paper focuses on the market making and the optimal liquidation problems using limit orders, accounting for model risk or uncertainty. Both are formulated as stochastic optimal control problems, with the controls being the spreads, relative to a reference price, at which orders are placed. The models consider uncertainty in both the drift and volatility of the underlying reference price, for the study of the effect of the uncertainty on the behavior of the market maker, accounting also for inventory restriction, as well as on the optimal liquidation using limit orders.

Optimal market making

“…In the single-asset case, closed-form approximations were obtained in Section 4, in the special case where Λ b = Λ a =: Λ and H ξ (0) > 0. 21 In the multi-asset case, if we assume that ∀i ∈ {1, . .…”

“…Cartea and Jaimungal, with their coauthors, have proposed models with price impact, the possibility to consider short-term alpha, the existence of an adverse selection effect, 6 etc. Recently, new models have emerged to deal with ambiguity aversion: see for instance the paper [4] by Cartea et al and the paper [21] by Nyström et al -see also the PhD dissertation of Donnelly [8].…”

Market making under a weakly consistent limit order book model

“…We also require that price updates due to aggressive orders to be instantaneous to prevent phantom opportunities arising from stale prices, which would otherwise create a very profitable trading strategy 10 . For example, one may buy at the stale price right after a large buy market order and wait for the price to fully reflect the order book status, assuming the direction consistency will be observed eventually.…”

“…+ t , M − t ) and noninfluential ( M + t , M − t ) with (M + t , M − t , M + t , M − t )being a multivariate Hawkes process. The mid price S m t is a diffusion coupled with the market orders via an unobservable mean-reverting process α t as follows:dS m t = (ν + α t )dt + σ dW t (24) dα t = −ζ α t dt + σ α dB t + ε + dM + t − ε − dM − t(25)where W t and B t are independent Brownian motions and ν ∈ R, ζ , σ , σ α , ε + , ε − 0 are all strictly positive constants 10. It is not an arbitrage in the classical sense, since there can be a major market sell order right after such a buy 11.…”

A Stochastic Control Approach to Option Market Making

“…The issue of model ambiguity has been addressed in Cartea et al (2013) or Nystr€ om et al (2014, to account for the fact that the dynamics followed by the stock price and/or the arrival rate of market orders may be misspeci¯ed. The conclusion is that, as the uncertainty increases, the market maker has to adapt her quotes in order to reduce her inventory.…”