We model a market with multiple liquidity takers and a single market maker maximizing his discounted consumption while keeping a prescribed probability of bankruptcy. We show that, given this setting, spread and price bias (a difference between the midpoint- and the expected fair price) depend solely on the MM's inventory and his uncertainty concerning the fair price. Tested on ten-second data from ten US electronic markets, our model gives significant results with the price bias decreasing in the inventory and increasing in the uncertainty and with the spread mostly increasing in the uncertainty.
Suppose that at any stage of a statistical experiment a control variable X that affects the distribution of the observed data Y at this stage can be used. The distribution of Y depends on some unknown parameter θ, and we consider the problem of testing multiple hypotheses H1:θ=θ1, H2:θ=θ2,…, Hk:θ=θk allowing the data to be controlled by X, in the following sequential context. The experiment starts with assigning a value X1 to the control variable and observing Y1 as a response. After some analysis, another value X2 for the control variable is chosen, and Y2 as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses H1,…, Hk is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations Y1,Y2,…,Yn are independent, given controls X1,X2,…,Xn, n=1,2,….