**Finance and Economics**

I am interested generally in Asset Pricing and the concept of Liquidity - how to define it, and how to evaluate its impact on prices. Here is a list of published and working papers:

** Published and Accepted Papers**

- Evolution of Shares in a Proof-of-Stake Cryptocurrency, with Fahad Saleh

*Management Science***67**(2021), 661-672 (*lead article*).Do the rich always get richer by investing in a cryptocurrency for which new coins are issued according to a Proof-of-Stake (PoS) protocol? We answer this question in the negative: Without trading, the investor shares in the cryptocurrency are martingales that converge to a Dirichlet distribution, hence are stable in the long run. This result is robust to allowing trading when investors are risk-neutral. Then, investors have no incentive to accumulate coins and gamble on the PoS protocol, but weakly prefer not to trade. See the Management Science Review blog here.

- Option Prices and the Probability of Success of Cash Mergers, with Alan Bester and Victor Martinez

*Journal of Financial Econometrics,*forthcoming.We study both theoretically and empirically option prices on firms undergoing a cash merger offer. To estimate the merger's success probability, we use a Markov Chain Monte Carlo (MCMC) method using a state space representation of our model. Our estimated probability measure has significant predictive power for the merger outcome even after controlling for variables used in the merger literature. As predicted by the model, a graph of the target firm's implied volatility against the strike price has a kink at the offer price, and the kink's magnitude is proportional to the merger's success probability.

- Quoting Activity and the Cost of Capital (+ Internet Appendix), with Elvira Sojli and Wing Wah Tham

*Journal of Financial and Quantitative Analysis,*forthcoming.We study the quoting activity of market makers in relation to trading, liquidity, and expected returns. Empirically, we find larger quote-to-trade (QT) ratios in small, illiquid or neglected firms, yet large QT ratios are associated with low expected returns. The last result is driven by quotes, not by trades. We propose a model of quoting activity consistent with these facts. In equilibrium, market makers monitor the market faster (and thus increase the QT ratio) in neglected, difficult-to-understand stocks. They also monitor faster when their clients are more precisely informed, which reduces mispricing and lowers expected returns.

- The data on quote-to-trade ratios are available for download here. Wharton Research Data Services (WRDS) data was used in preparing this output, please cite accordingly.

- Liquidity and Information in Limit Order Markets (+ Internet Appendix)

*Journal of Financial and Quantitative Analysis***55**(2020), 1792-1839.How does informed trading affect liquidity in limit order markets, where traders can choose between market orders (demanding liquidity) and limit orders (providing liquidity)? In a dynamic model, informed trading overall helps liquidity: A higher share of informed traders (i) improves liquidity as proxied by the bid-ask spread and market resiliency, and (ii) has no effect on the price impact of orders. The model generates other testable implications, and suggests new measures of informed trading.

- Fast and Slow Informed Trading (+ Internet Appendix)

*Journal of Financial Markets***43**(2019), 1-30 (*lead article*).I develop a model in which traders receive a stream of private signals, and differ in their information processing speed. In equilibrium, the fast traders (FTs) quickly reveal a large fraction of their information. If a FT is averse to holding inventory, his optimal strategy changes considerably as his aversion crosses a threshold. He no longer takes long-term bets on the asset value, gets most of his profits in cash, and generates a "hot potato" effect: after trading on information, the FT quickly unloads part of his inventory to slower traders. The results match evidence about high-frequency traders.

- News Trading and Speed (+ Internet Appendix); with Thierry Foucault and Johan Hombert

*Journal of Finance***71**(2016), 335-382.We compare the optimal trading strategy of an informed speculator when he can trade ahead of incoming news (is "fast"), versus when he cannot (is "slow"). We find that speed matters: the fast speculator's trades account for a larger fraction of trading volume, and are more correlated with short-run price changes. Nevertheless, he realizes a large fraction of his profits from trading on long-term price changes. The fast speculator's behavior matches evidence about high frequency traders. We predict that stocks with more informative news are more liquid even though they attract more activity from informed high frequency traders.

- A Dynamic Model of the Limit Order Book

*Review of Financial Studies***22**(2009), 4601-4641This paper shows that waiting costs can satisfactorily explain various empirical facts about the limit order book, without relying on asymmetric information. In particular, the model can explain: (1) limit orders cluster away from the bid-ask spread, generating a hump-shaped limit order book; (2) prices overshoot, i.e., the temporary price impact is larger than the permanent one; (3) there is a comovement effect between bid and ask prices; (4) higher trading activity and trading competition cause smaller spreads; and (5) in some cases, trader may submit quick, or fleeting, limit orders. The equilibrium in this model is relatively simple: the only state variables are the numbers of buyers and sellers placing limit orders in the book.

- On the Derivation of the Black-Scholes formula, with Dan Stroock

*Séminaire de Probabilités***37**(2004), 399-414Methods of proving the Black-Scholes formula for the price of an European call option fall into two categories: the bond replication method (the original one by Black and Scholes), and the call replication method (originated by Merton). These two methods are not equivalent. While the call replication argument is simple and requires only continuity of the call price

*C*, the bond replication method puts more restrictions on*C*in order for the argument to work. Moreover, we show that the typical bond replication method fails if the call option delta is equal to one. That implies that at each point either*C*satisfies the Black-Scholes PDE, or it satisfies the linear PDE given by delta equal to one. We then show that these two PDEs cannot coexist if*C*is assumed continuously differentiable, which proves that the Black-Scholes PDE holds everywhere.

** Book Chapters**

- Order Choice and Information in Limit Order Markets

Chapter 2 in*Market Microstructure; Confronting Many Viewpoints*(2012), WileyThis surveys recent developments in the study of limit order markets. The focus is on the decisions of traders under symmetric or asymmetric information, and in particular on their choice between limit orders and market orders. We discuss also the price impact of different types of orders, and their information content.

** Working Papers**

- Dynamic Adverse Selection and Liquidity (+ Internet Appendix)

(4 May 2020)Does a larger fraction of informed trading generate more illiquidity, as measured by the bid-ask spread? We answer this question in the negative in the context of a dynamic dealer market where the fundamental value follows a random walk, provided we consider the long run (stationary) equilibrium. More informed traders tend to generate more adverse selection and hence larger spreads, but at the same time cause faster learning by the market makers and hence smaller spreads. This latter effect offsets the adverse selection effect when the trading frequency is equal to one, and dominates at larger frequencies.

- Weather and Time Series Determinants of Liquidity in a Limit Order Market, with Juhani Linnainmaa

(October 2009)When liquidity is measured by the bid-ask spread or price impact, markets with more trading activity are typically more liquid than markets with less trading activity. But showing a causal connection from trading activity to spreads is difficult because these variables are endogenous. In the case of Finland's fully electronic limit order market, we use deseasonalized sunshine as an instrument for trading activity, and find that indeed higher trading activity causes lower spreads in the time series. We introduce another instrument for spreads and show that causality runs the other way as well: lower bid-ask spreads invite more trading activity. By using the lagged CBOE Volatility Index as an instrument, we also find that an exogenous increase in intra-day volatility causes larger spreads.

- Multi-Stage Game Theory in Continuous Time

(January 2006)I define multi-stage stochastic games in continuous time. As in Bergin and MacLeod (1993), strategies have infinitesimal inertia, i.e., agents cannot change their strategies in an infinitesimal interval immediately after each time t. I extend the framework to allow for mixed strategies. As a novel feature in continuous time, mixing can be done both over actions, and over time (choosing the time of the action). I also define "layered times," which allow for stopping the clock and having various stages of the game be played at the same moment in time. I apply the theory to a trading game, where patient agents can choose whether to trade immediately or place a limit order and wait.

**List of notes** I wrote as I was studying some economics/finance topics. The notes are not very polished, but one might still find something useful there. Here is a partial list:

- Bellman principle of optimality;
- Conditional probability theory;
- Consumption-investment problems for CARA agents;
- Basic notions of asymmetric information: Walrasian equilibria, rational expectations equilibria, the Grossman-Stiglitz paradox, and the Milgrom-Stokey "no-trade" theorem. These notes were written for my market microstructure PhD course at Chicago in Spring 2006.