Paper:
Adaptive Market Making via Online Learning
We consider the design of strategies for market making in an exchange. A market maker generally seeks to profit from the difference between the buy and sell price of an asset, yet the market maker also takes exposure risk in the event of large price movements. Profit guarantees for market making strategies have typically required certain stochastic assumptions on the price fluctuations of the asset in question; for example, assuming a model in which the price process is mean reverting. We propose a class of "spread-based" market making strategies whose performance can be controlled even under worst-case (adversarial) settings. We prove structural properties of these strategies which allows us to design a master algorithm which obtains low regret relative to the best such strategy in hindsight. We run a set of experiments showing favorable performance on recent real-world stock price data.

Review:
The paper presents a nice algorithm for trading a 
      commodity (e.g. stock of certain type) for a market maker (i.e. an entity 
      that must buy from/sell to everybody). The paper nicely explains a 
      (somewhat simplified) mechanics of the stock exchange (limit orders and 
      market orders). The paper presents a natural set of base 
      strategies. The set of base strategies is parametrized by a parameter "b" 
      -- the spread between buy and sell price. On top of these strategies they 
      authors apply standard online algorithms (Randomized Weighted Majority 
      Algorithm, Follow the Perturbed Leader) for hedging/switching between the 
      base strategies. What makes the application non-trivial is that any 
      strategy (base or not) is stateful -- namely it maintains "cash" and 
      "inventory" (i.e. a number of underlying stocks). Since standard 
      algorithms "for learning with expert advice" are stateless, this makes 
      their application somewhat non-trivial. To overcome this authors prove 
      that certain interesting structural properties of set of base strategies 
      (e.g. the buy-sell price interval of a base strategy with smaller spread 
      is contained in the interval of a strategy with bigger spread). This 
      allows the authors to derive a bound on the regret (i.e. convergence) 
      relative to the best base strategy in hindsight. The paper is very 
      clearly written. I like it very much and I am happy to recommend it for 
      acceptance. I have to admit that I don't have time to go through 
      the proofs in detail, but all lemmas seem plausible. Also, I am not 
      familiar with the mathematical finance literature, so I am not sure how 
      interesting this result is within finance context. However, within context 
      of online learning this is a novel, and very interesting result. 
      1) Please fix the typo ("Marking" -> "Market") in the title of 
      the paper -- it's embarrassing :) 2) Please be consistent in 
      typesetting authors' names in the references. There are missing dots (e.g. 
      "Robert E Schapire"). Sometimes the first name of a person is present, 
      sometimes there's just the initial ("S. Das" vs "Sanmay Das"). 
      Couple of random ideas for future research: - It seems 
      that regret of your algorithm depends on the number of switches between 
      the underlying base strategies. There are few recent COLT papers with 
      variants of Randomized Weighted Majority Algorithm (i.e. Hedge/Exponential 
      Weights Algorithm) and Follow the Perturbed Leader that achieve small 
      number of switches between experts. (see e.g. "Regret Minimization for 
      Online Buffering Problems Using the Weighted Majority Algorithm", COLT 
      2010 and "Prediction by random-walk perturbation", COLT 2013). - 
      Would it be possibly to get rid of the delta-discretization? The 
      discretization seems to be just an ugly artifact of the model/algorithm. 
      Or to put it differently, what happens in the limit delta --> 0?