The LIX: a model-independent liquidity index
Florence Guillaume (University of Antwerp, Belgium)

Tuesday June 3, 11:00-11:30 | session 1.9 | Liquidity | room H

This paper provides a new model-free indicator of liquidity, the so-called LIX index. The computation of the LIX combines the recently developed conic finance theory which is built upon the concept of indices of acceptability with the option payoff spanning formula allowing for the extraction of the model-free asset distribution. The conic finance theory drops the law of one equilibrium price in favor of a two-price economy where market participants buy from the market at the ask price and sell to the market at the lower bid price. Under the framework of indices of acceptability, parametric expressions of the bid and ask prices of financial instruments can be obtained in terms of the distribution function of the zero cost cash-flow only. Matching the conic bid and ask prices of the stock with those observed in the market allows us to derive a model-free and unit-less indicator of spot liquidity. Just as the VIX and the SKEW index issued by the CBOE quantify the volatility and the tail risk perceived by today's investors, the resulting LIX index measures, in a similar market-implied fashion, the liquidity risk. Besides, we extend the methodology to build the so-called implied liquidity surfaces, one for the call and one for the put options, summarizing the dependency of the option model-free liquidity across moneyness and time to maturity. As numerical study, we provide a maximum likelihood estimation of popular mean-reverting processes applied to the time series of the new model-free liquidity proxy for vanilla options and their underlying, allowing for a comparison of the dynamics of liquidity in the underlying and derivative markets, but also during periods characterized by different market fear levels.

Implied liquidity application for the high frequency liquidity data study on prime trading time
Monika Forys (KU Leuven, Belgium)

Tuesday June 3, 11:30-12:00 | session 1.9 | Liquidity | room H

Recently growing interest of investors and media in liquidity measure lead to increased attention not only of the financial world but also of the academic financial researchers. The most attention is paid to research on relationships between the liquidity and characteristics of the underlying stock or option, omitting the issue of the prime trading time with respect to liquidity and causing a gap. Moreover, most of the studies focus only on bid-ask spread as a measure of the liquidity. However, option price can move, causing illiquidity, without change in the spread, thus it is very difficult to measure the liquidity in an isolate manner. Hence, the concept of implied liquidity is proposed as a proper measure, that isolates and quantifies in a fundamental way liquidity level of investors’ positions. The idea of implied liquidity has its basis in recently developed two-way pricing theory (conic finance), where the traditional one-price model was replaced by a two-price model, yielding bid and ask prices for traded assets. Pricing is performed using distortion function and distorted expectation. Further the measure is used to fill in the gap and establish the prime trading time for the Dow Jones Index and its underlyings, yielding exact time for the most liquid trading. Due to the complex nature of implied liquidity formulas and the corresponding time consuming calculations, we will (also) predict implied liquidity based on several explanatory variables (e.g. time to maturity, moneyness and implied volatility) using multiple linear regression.

Moral Hazard, Informed Trading, and Stock Prices
Pierre Collin-Dufresne (SFI@EPFL, Switzerland)
Joint work with Slava Fos

Tuesday June 3, 12:00-12:30 | session 1.9 | Liquidity | room H

We analyze a model of informed trading where an activist shareholder accumulates shares in an anonymous market and then expends costly effort to increase the firm value. We find that equilibrium prices are affected by the position accumulated by the activist, because the level of effort undertaken is increasing in the size of his acquired position. In equilibrium, price impact has two components: one due to asymmetric information (as in the seminal \cite{kyle:85} model) and one due to moral hazard (a new source of illiquidity). Price impact is higher the more severe the moral hazard problem, which corresponds to a more productive activist. We thus obtain a trade-off: with more noise trading (less `price efficiency') the activist can build up a larger stake, which leads to more effort expenditure and higher firm value (more `economic efficiency').

Optimal Investment with Illiquid Assets
Sascha Desmettre (University of Kaiserslautern, Germany)

Tuesday June 3, 12:30-13:00 | session 1.9 | Liquidity | room H

We study the optimal portfolio problem of an investor who has the option to invest in an illiquid asset that is only traded at time 0. We use a generalized martingale method to solve the optimal investment problem and derive optimal strategies via Clark's formula and by introducing a liquidity derivative. As application, we study the optimal investment into a fixed-term deposit earning an excess return over the risk-free interest rate. We demonstrate that the presence of such an investment opportunity has a significant impact on optimum asset allocation: crra agents with realistic values of relative risk aversion can be expected to allocate more than 40\% of initial wealth to the illiquid investment opportunity if it yields an excess return of 100 basis points over the money market account. Furthermore, we look at an investor who gains utility by owning a housing good and solve the corresponding optimal investment problem in our setting.