TRANSIENT LINEAR PRICE IMPACT AND FREDHOLM INTEGRAL EQUATIONS

We consider the linear‐impact case in the continuous‐time market impact model with transient price impact proposed by Gatheral. In this model, the absence of price manipulation in the sense of Huberman and Stanzl can easily be characterized by means of Bochner’s theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure‐valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko on the nonexistence of transaction‐triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.     

OPTIMAL LIQUIDATION IN A LIMIT ORDER BOOK FOR A RISK‐AVERSE INVESTOR

In a limit order book model with exponential resilience, general shape function, and an unaffected stock price following the Bachelier model, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. We show that the problem can be reduced to a two‐dimensional deterministic problem which involves no buy orders. We derive an explicit expression for the value function and the optimal liquidation strategy. The analysis is complicated by the fact that the intervention boundary, which determines the optimal liquidation strategy, is discontinuous if there are levels in the limit order book with relatively little market depth. Despite this complication, the equation for the intervention boundary is fairly simple. We show that the optimal liquidation strategy possesses the natural properties one would expect, and provide an explicit example for the case where the limit order book has a constant shape function.     

Optimal trade execution in order books with stochastic liquidity

In financial markets, liquidity changes randomly over time. We consider such random variations of the depth of the order book and evaluate their influence on optimal trade execution strategies. If the stochastic structure of liquidity changes satisfies certain conditions, then the unique optimal trading strategy exhibits a conventional structure with a single wait region and a single buy region, and profitable round‐trip strategies do not exist. In other cases, optimal strategies can feature multiple wait regions and optimal trade sizes that can be decreasing in the size of the position to be liquidated. Furthermore, round‐trip strategies can be profitable depending on bid–ask spread assumptions. We illustrate our findings with several examples including the Cox–Ingersoll–Ross model for the evolution of liquidity.     

OPTIMAL STOCK SELLING/BUYING STRATEGY WITH REFERENCE TO THE ULTIMATE AVERAGE

We are concerned with the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price. More precisely, we aim to determine an optimal selling (buying) time to maximize (minimize) the expectation of the ratio of the selling (buying) price to the ultimate average price over the period. This is an optimal stopping time problem which can be formulated as a variational inequality problem. The problem gives rise to a free boundary that corresponds to the optimal selling (buying) strategy. We provide a partial differential equation approach to characterize the free boundary (or equivalently, the optimal selling (buying) region). It turns out that the optimal selling strategy is bang‐bang, which is the same as that obtained by Shiryaev, Xu, and Zhou taking the ultimate maximum of the stock price as benchmark, whereas the optimal buying strategy can be a feedback one subject to the type of averaging and parameter values. Moreover, by a thorough characterization of free boundary, we reveal that the bang‐bang optimal selling strategy heavily depends on the assumption that no time‐vesting restrictions are imposed. If a time‐vested stock is considered, then the optimal selling strategy can also be a feedback one. In terms of a similar analysis developed by the present paper, the same phenomenon can be proved when taking the ultimate maximum as benchmark.     

农产品价格上涨时的企业原材料订购策略研究

在农产品价格普遍上涨的情况下,对重庆市某火锅食品公司主要原材料(辣椒)2006年的采购现状进行调查分析,发现辣椒采购价呈指数时变递增规律,运用基本经济订购批量模型对企业的订购策略进行研究,结果表明改进后的订购策略能大幅度降低库存成本,初始采购价和价格影响因子对最优订购次数、最优订购批量以及库存系统各项成本有较大程度的影响。在实际工作中,必须根据这些因素来制定原材料订购策略。     

Pairs‐trading and spread persistence in the European stock market

In this paper, we adapt the demand and supply framework introduced by Figuerola‐Ferretti and Gonzalo (Journal of Econometrics, 2010) to illustrate the dynamics of Pairs‐trading. We underline the process by which a finite elasticity of demand for spread trading determines the speed of mean reversion and pairs‐trading profitability. A persistence‐dependent trading trigger is introduced accordingly. Applied to STOXX Europe 600–traded equities, our strategy exploits price leadership for portfolio replication purposes and delivers Sharpe ratios that outperform the benchmark rules used in the literature. Portfolio performance and mean reversion are enhanced after firm fundamental factor restrictions are imposed.     

Portfolio choice with small temporary and transient price impact

We study portfolio selection in a model with both temporary and transient price impact introduced by Garleanu and Pedersen. In the large‐liquidity limit where both frictions are small, we derive explicit formulas for the asymptotically optimal trading rate and the corresponding minimal leading‐order performance loss. We find that the losses are governed by the volatility of the frictionless target strategy, like in models with only temporary price impact. In contrast, the corresponding optimal portfolio not only tracks the frictionless optimizer, but also exploits the displacement of the market price from its unaffected level.     

LIQUIDATION IN LIMIT ORDER BOOKS WITH CONTROLLED INTENSITY

We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power‐law and exponential‐decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.     

The Impact of Liquidity Risk: A Fresh Look*

This paper takes a fresh look at the importance of liquidity risk using a comprehensive liquidity measure, weighted spread, in a Value‐at‐Risk (VaR) framework. The weighted spread measure extracts liquidity costs by order size from the limit order book. Using a unique, representative data set of 160 German stocks over 5.5 years, we show that liquidity risk is an important risk component. Actually, liquidity risk is increasing the total price risk by over 25%, even at 10‐day horizons and for liquid blue chip stocks and especially in larger, yet realistic order sizes beyond €1 million. When correcting for liquidity risk, it is commonly assumed that liquidity risk can be simply added to price risk. Our empirical results show that this is not correct, as the correlation between liquidity and price is non‐perfect and total risk is thus overestimated.     

Callable bull/bear contracts,call auction sessions,and price manipulations: Evidence from Hong Kong

Call auction sessions are widely adopted to improve the price discovery process. The suspension of the closing call auction session (CAS) of the Hong Kong Stock Exchange (HKEx) in 2009 and the reintroduction of an enhanced CAS in 2016 provide us a unique experimental environment to assess the effectiveness of the two different CAS models in reducing market manipulation. In examining the probability of mandatory call events (MCEs) of callable bull/bear contracts (CBBCs), we find the enhanced CAS model being more effective in price manipulation reduction. We also find the enhanced CAS reducing price manipulation in the preopening auction session.     

Nonlinear price impact and portfolio choice

In a market with price impact proportional to a power of the order flow, we find optimal trading policies and their implied performance for long‐term investors who have constant relative risk aversion and trade a safe asset and a risky asset following geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the curvature of the price impact function. Trading rates are finite as with linear impact, but are lower near the target portfolio, and higher away from the target. The model nests the square‐root impact law and, as extreme cases, linear impact and proportional transaction costs.     

Arrow–Debreu equilibria for rank‐dependent utilities with heterogeneous probability weighting

We study Arrow–Debreu equilibria for a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents having heterogeneous probability weighting and outcome utility functions. In particular, we allow the economy to have a mix of expected utility agents and rank‐dependent utility ones, with nonconvex probability weighting functions. The standard approach for convex economy equilibria fails due to the incompatibility with second‐order stochastic dominance. The representative agent approach devised in Xia and Zhou (2016) does not work either due to the heterogeneity of the weighting functions. We overcome these difficulties by considering the comonotone allocations, on which the rank‐dependent utilities become concave. Accordingly, we introduce the notion of comonotone Pareto optima, and derive their characterizing conditions. With the aid of the auxiliary problem of price equilibria with transfers, we provide a sufficient condition in terms of the model primitives under which an Arrow–Debreu equilibrium exists, along with the explicit expression of the state‐price density in equilibrium. This new, general sufficient condition distinguishes the paper from previous related studies with homogeneous and/or convex probability weightings.     

Trading co‐integrated assets with price impact

Executing a basket of co‐integrated assets is an important task facing investors. Here, we show how to do this accounting for the informational advantage gained from assets within and outside the basket, as well as for the permanent price impact of market orders (MOs) from all market participants, and the temporary impact that the agent's MOs have on prices. The execution problem is posed as an optimal stochastic control problem and we demonstrate that, under some mild conditions, the value function admits a closed‐form solution, and prove a verification theorem. Furthermore, we use data of five stocks traded in the Nasdaq exchange to estimate the model parameters and use simulations to illustrate the performance of the strategy. As an example, the agent liquidates a portfolio consisting of shares in Intel Corporation and Market Vectors Semiconductor ETF. We show that including the information provided by three additional assets (FARO Technologies, NetApp, Oracle Corporation) considerably improves the strategy's performance; for the portfolio we execute, it outperforms the multiasset version of Almgren–Chriss by approximately 4–4.5 basis points.     

OPTIMAL TRADE EXECUTION IN ILLIQUID MARKETS

We study optimal trade execution strategies in financial markets with discrete order flow. The agent has a finite liquidation horizon and must minimize price impact given a random number of incoming trade counterparties. Assuming that the order flow N is given by a Poisson process, we give a full analysis of the properties and computation of the optimal dynamic execution strategy. Extensions, whereby N is a Markov‐modulated compound Poisson process are also considered. We derive and compare the properties of the various cases and illustrate our results with computational examples.     

OPTIMAL HIGH‐FREQUENCY TRADING IN A PRO RATA MICROSTRUCTURE WITH PREDICTIVE INFORMATION

We propose a framework to study optimal trading policies in a one‐tick pro rata limit order book, as typically arises in short‐term interest rate futures contracts. The high‐frequency trader chooses to post either market orders or limit orders, which are represented, respectively, by impulse controls and regular controls. We discuss the consequences of the two main features of this microstructure: first, the limit orders are only partially executed, and therefore she has no control on the executed quantity. Second, the high‐frequency trader faces the overtrading risk, which is the risk of large variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, and we provide the associated numerical resolution procedure and prove its convergence. We propose dimension reduction techniques in several cases of practical interest. We also detail a high‐frequency trading strategy in the case where a (predictive) directional information on the price is available. Each of the resulting strategies is illustrated by numerical tests.     

THE COST OF ILLIQUIDITY AND ITS EFFECTS ON HEDGING

Though liquidity is commonly believed to be a major effect in financial markets, there appears to be no consensus definition of what it is or how it is to be measured. In this paper, we understand liquidity as a nonlinear transaction cost incurred as a function of rate of change of portfolio. Using this definition, we obtain the optimal hedging policy for the hedging of a call option in a Black‐Scholes model. This is a more challenging question than the more common studies of optimal strategy for liquidating an initial position, because our goal requires us to match a random final value. The solution we obtain reduces in the case of quadratic loss to the solution of three partial differential equations of Black‐Scholes type, one of them nonlinear.     

Optimal pricing in social networks considering reference price effect

We study the optimal monopoly pricing strategies in a social network, in which consumers experience a network effect that is dependent on their neighbors' consumptions and a reference price which is the average price received by their neighbors. We establish a two-stage game model for any social network. Utilizing the backward induction, we derive the equilibrium price by maximizing the monopolist's profit. In addition, we apply this model to the two most commonly used network structures: the star network and the bipartite network. We find that both the network effect and the reference price effect play a critical role in deciding pricing strategies in social networks. Moreover, our numerical results demonstrate that whether to implement discriminatory pricing depends critically on the network structure. This work provides monopoly firms a useful guideline for optimal pricing decisions in social network marketing.     

Optimal investment,derivative demand,and arbitrage under price impact

This paper studies the optimal investment problem with random endowment in an inventory‐based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading and nonlinear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, nonlinear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage‐inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.     

The information content of an open limit‐order book

Using data from the Australian Stock Exchange, the authors assess the information content of an open limit‐order book with a particular focus on the incremental information contained in the limit orders behind the best bid and offer. The authors find that the order book is moderately informative—its contribution to price discovery is approximately 22%. The remaining 78% is from the best bid and offer prices on the book and the last transaction price. Furthermore, the authors find that order imbalances between the demand and supply schedules along the book are significantly related to future short‐term returns, even after controlling for the autocorrelations in return, the inside spread, and the trade imbalance. ©2008 Wiley Periodicals, Inc. Jrl Fut Mark 29:16–41, 2009     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.