A note on convergence of an approximate hedging portfolio with liquidity risk

When the underlying asset price depends on activities of traders, hedging errors include costs due to the illiquidity of the underlying asset and the size of this cost can be substantial. Cetin et al. (2004), Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8(3), 311-341, proposed a hedging strategy that approximates the classical Black–Scholes hedging strategy and produces zero liquidity costs. Here, we compute the rate of convergence of the final value of this hedging portfolio to the option payoff in case of a European call option; i.e. we see how fast its hedging error converges to zero. The hedging strategy studied here is meaningful due to its simple liquidity cost structure and its smoothness relative to the classical Black–Scholes delta.     

Concentrated Equilibrium and Intraday Patterns in Financial Markets

Abstract

We introduce endogenous participation of market makers into a Kyle-type model with long-lived asymmetric information. In our model with plausible parameter values, the trading volume and price volatility show a U-shaped intraday pattern, often observed in actual financial markets. It will be shown that the pattern is caused not only by the trading behaviour of liquidity traders but also by that of market makers. Our findings shed new light on the stylized fact of the trade concentration at the opening and closing periods.     

氏调制的几何布朗运动与布林带

在证券市场, 布林带作为流行的技术分析工具被广泛的运用\bd 到目前为止有许多模型被建立用来预测证券的价格, 因此研究这些模型是否具有布林带性质是一个重要的问题\bd Liu, Huang and Zheng (2006)和Liu and Zheng (2006)分别讨论了Black-Scholes模型和随机波动率模型作为真实的股票市场的布林带, 并且证明了相应的统计量的平稳性和大数定律成立\bd 本文我们将上述结果推广到马氏调制的几何布朗运动模型.     

High frequency trading,liquidity, and execution cost

We build a model under the framework of discrete optimization to explain how high frequency trading (HFT) can be applied to supply liquidity and reduce execution cost. We derive the analytical properties of our model in finding the optimal solution to minimize the overall execution cost of HFT. We show that the execution cost can be reduced after increasing trading frequency (i.e., the higher the trading frequency, the lower the execution cost) with a simulation study. In addition, we conduct an empirical investigation with tick level data from US equity market through January 2008 to October 2010 to verify our conclusion drawn from the simulation study. Based on the simulation and empirical results we collected, we show that the HFT can reduce the execution cost when supplying liquidity.     

Real options valuation of forest plantation investments in Brazil

In this paper, we consider investments in eucalyptus plantations in Brazil. For such projects, we discuss real options valuation in the place conventional methods such as IRR or NPV, possibly with CAPM. Traditionally, real options valuation assumes complete markets and neglects market imperfections. Yet, market frictions, such as transaction costs, interest rate spreads, and restricted short positions, can play an important role. We extend real options valuation to allow incomplete and imperfect markets. The value is obtained as a competitive price, given markets of competing investment opportunities, such as real and financial assets. Under perfect and complete markets, such valuation method is consistent with conventional real options theory. Stochastic programming and standard software is used for valuation of eucalyptus plantations. We estimate the underlying interdependent diffusion processes of stock market, interest rates, exchange rates and pulpwood price, and derive novel expressions of stochastic integrals to be employed in scenario generation for discrete time stochastic programming.     

中国新三板市场阶段性集合竞价制度的价格发现研究

市场微观结构理论表明交易机制对资产价格的形成过程具有重要影响。本文以中国新三板交易机制改革为背景,从理论上分析了阶段性集合竞价制度的市场出清过程。阶段性集合竞价制度的核心在于市场出清时间间隔的设定。本文构建了一个存在信息摩擦和知情交易者学习机制的集合竞价市场出清模型,讨论了市场出清时间间隔对价格发现效率、资产价值不确定性和流动性风险的影响。研究发现:(1)在完美信息条件下,如果对市场规模较大和价值波动率较高的资产设定较短的市场出清时间间隔,将会降低投资者的流动性风险,提升市场质量;(2)在不完美信息条件下,除市场规模和资产价值波动率之外,信息不对称程度和知情交易者比例也是影响最优市场出清频率的重要因素;(3)在不完美信息条件下,对价值波动率较低的资产缩短市场出清时间间隔才能降低流动性风险,这与完美信息条件下的结论相反。     

基于综合流动性度量指标的中国期货市场流动性溢价研究

本文从流动性成本、流动性波动和到期日三个角度出发构建了衡量期货市场的综合流动性度量指标,并利用该指标对中国期货市场的流动性溢价问题进行研究。实证结果表明,流动性水平的差异对不同到期日期货合约的收益差异的影响存在差异性,当期流动性水平差异及其滞后期对收益差额的波动影响显著,其中,期货铜和铝市场中流动性对收益差额的影响存在过度反应→适度矫正的过程。     

A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities

Incomplete financial markets are considered, defined by a multi-dimensional non-homogeneous diffusion process, being the direct sum of an Itô process (the price process), and another non-homogeneous diffusion process (the exogenous process, representing exogenous stochastic sources). The drift and the diffusion matrix of the price process are functions of the time, the price process itself and the exogenous process. In the context of such markets and for power utility functions, it is proved that the stochastic control problem consisting of optimizing the expected utility of the terminal wealth, has a classical solution (i.e. \(C^{1,2}\) ). This result paves the way to a study of the optimal portfolio problem in incomplete forward variance stochastic volatility models, along the lines of Ekeland and Taflin [7].     

Modelling Asset Prices for Algorithmic and High-Frequency Trading

ABSTRACT

Algorithmic trading (AT) and high-frequency (HF) trading, which are responsible for over 70% of US stocks trading volume, have greatly changed the microstructure dynamics of tick-by-tick stock data. In this article, we employ a hidden Markov model to examine how the intraday dynamics of the stock market have changed and how to use this information to develop trading strategies at high frequencies. In particular, we show how to employ our model to submit limit orders to profit from the bid–ask spread, and we also provide evidence of how HF traders may profit from liquidity incentives (liquidity rebates). We use data from February 2001 and February 2008 to show that while in 2001 the intraday states with the shortest average durations (waiting time between trades) were also the ones with very few trades, in 2008 the vast majority of trades took place in the states with the shortest average durations. Moreover, in 2008, the states with the shortest durations have the smallest price impact as measured by the volatility of price innovations.     

Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree

In this paper, we present a natural mathematical framework to model trader behavior as a continuous time discrete event process, and derive stochastic differential equations for aggregate behavior and price dynamics by passing to diffusion limits. In particular, we model extraneous, value, momentum and hedge traders. Through analysis and numerical simulation we explore some of the effects these trading strategies have on price dynamics.     

On the role of heterogeneous and imperfect information in a laboratory financial market

In this paper, we analyze the behavior of a group of heterogeneously informed investors in an laboratory asset market. Our experimental setting is inspired by Huber et al. (On the benefit of information in markets with heterogeneously informed traders: an experimental study, 2004). However, instead of their system of cumulative and exogenously given information structure, we introduce an information market where the traders can buy an imperfect prediction of the future value of the dividend with a maximum anticipation of four periods. The accuracy of the prediction decreases with the chosen time horizon, whereas its price remains constant. Our results confirm a non-strictly monotonic increasing value of the information.     

Optimal risk and liquidity management with costly refinancing opportunities

In this paper we study risk and liquidity management decisions within an insurance firm. Risk management corresponds to decisions regarding proportional reinsurance, whereas liquidity management has two components: distribution of dividends and costly equity issuance. Contingent on whether proportional or fixed costs of reinvestment are considered, singular stochastic control or stochastic impulse control techniques are used to seek strategies that maximize the firm value. We find that, in a proportional-costs setting, the optimal strategies are always mixed in terms of risk management and refinancing. In contrast, when fixed issuance costs are too high relative to the firm’s profitability, optimal management does not involve refinancing. We provide analytical specifications of the optimal strategies, as well as a qualitative analysis of the interaction between refinancing and risk management.     

Optimal portfolios of a small investor in a limit order market: a shadow price approach

We study Merton’s portfolio optimization problem in a limit order market. An investor trading in a limit order market has the choice between market orders that allow immediate transactions and limit orders that trade at more favorable prices but are executed only when another market participant places a corresponding market order. Assuming Poisson arrivals of market orders from other traders we use a shadow price approach, similar to Kallsen and Muhle-Karbe (Ann Appl Probab, forthcoming) for models with proportional transaction costs, to show that the optimal strategy consists of using market orders to keep the proportion of wealth invested in the risky asset within certain boundaries, similar to the result for proportional transaction costs, while within these boundaries limit orders are used to profit from the bid–ask spread. Although the given best-bid and best-ask price processes are geometric Brownian motions the resulting shadow price process possesses jumps.     

中国股指期货市场流动性非线性特征分析与波动趋势预测

针对中国股指期货市场的流动性问题,利用多重分形去趋势波动法研究股指期货市场流动性非线性特征及其成因,并对比分析不同期限下期货市场流动性多重分形程度的差异;进一步,采用趋势熵维数方法识别期货市场流动性的变化趋势,还利用识别正确率和随机识别正确率验证该方法有效性和准确性。研究发现,中国股指期货市场具有明显的多重分形特征;与合约期限较短的股指期货相比,合约期限较长的股指期货流动性多重分形程度更低;股指期货市场流动性多重分形特征主要是流动性时间序列的相关多重分形和分布多重分形造成的;趋势熵维数方法可有效预测期货市场流动性的变化趋势。     

Pricing and clearing combinatorial markets with singleton and swap orders

In this article we consider combinatorial markets with valuations only for singletons and pairs of buy/sell-orders for swapping two items in equal quantity. We provide an algorithm that permits polynomial time market-clearing and -pricing. The results are presented in the context of our main application: the futures opening auction problem. Futures contracts are an important tool to mitigate market risk and counterparty credit risk. In futures markets these contracts can be traded with varying expiration dates and underlyings. A common hedging strategy is to roll positions forward into the next expiration date, however this strategy comes with significant operational risk. To address this risk, exchanges started to offer so-called futures contract combinations, which allow the traders for swapping two futures contracts with different expiration dates or for swapping two futures contracts with different underlyings. In theory, the price is in both cases the difference of the two involved futures contracts. However, in particular in the opening auctions price inefficiencies often occur due to suboptimal clearing, leading to potential arbitrage opportunities. We present a minimum cost flow formulation of the futures opening auction problem that guarantees consistent prices. The core ideas are to model orders as arcs in a network, to enforce the equilibrium conditions with the help of two hierarchical objectives, and to combine these objectives into a single weighted objective while preserving the price information of dual optimal solutions. The resulting optimization problem can be solved in polynomial time and computational tests establish an empirical performance suitable for production environments.     

Multilevel marketing networks in less-developed countries

A mathematical dynamic economic model of a hierarchal periodic marketing system is constructed. Goods move from producers through many traders and many markets, first being bulked as they ascend through the several levels of the market hierarchy and then being distributed as they descend the hierarchy in another region. Such a complex system is typical for many Third World countries where a commodity is produced in one region and distributed to consumers in another. Unlike most models of marketing, transportation, and storage systems, our model is founded upon the individual transportation and storage behaviors of the traders, who, we assume, have nonlinear marginal operating costs and are provided with possibly incomplete and tardy market news. The traders' individual behaviors are then aggregated and combined with the supply schedules in the initial producers' markets and with the demand schedules in the final consumers' markets to obtain the overall model. It is a truly dynamic one and provides through recursive analyses time series in all prices and quantity flows. In general, the overall system remains in dynamic disequilibrium even though its individual markets keep clearing under perfect competition on every market day. This is a realistic result. It is also shown that the system has a unique equilibrium state, which may however never be attained. The paper concludes with some computer-simulation studies, which indicate the effects of varying the systems parameters.     

Hedging of game options in discrete markets with transaction costs

Game options introduced in [10] in 2000 were studied, by now, mostly in frictionless both complete and incomplete markets. In complete markets the fair price of a game option coincides with the value of an appropriate Dynkin's game, whereas in markets with friction and in incomplete ones there is a range of arbitrage free prices and superhedging comes into the picture. Here we consider game options in general discrete time markets with transaction costs and construct backward and forward induction algorithms for the computation of their prices and superhedging strategies from both seller's (upper arbitrage free price) and buyer's (lower arbitrage free price) points of view extending to the game options case most of the results from [12].     

具有两类交易费的欧式未定权益定价

本考虑了在具有成比例和固定两类交易费情形下欧式未定权量的定价问题．通过引入脉冲随机控制，定义欧式未定权益的销售价，利用渐近分析的方法，得到其销售价为理想市场的欧式未定价格摄动。     

A closed form solution for vulnerable options with Heston’s stochastic volatility

Over-the-counter stock markets in the world have been growing rapidly and vulnerability to default risks of option holders traded in the over-the-counter markets became an important issue, in particular, since the global finance crisis and Eurozone crisis. This paper studies the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics. In this paper, we obtain a closed form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula and find how the stochastic volatility effect on the Black–Scholes price as well as on the decreasing speed of the option price with credit risk depends on moneyness.     

Partially informed noise traders

The single auction equilibrium of Kyle??s (1985) is studied, in which noise traders may be partially informed, or alternatively they can be manipulated. Unlike Kyle??s assumption that the quantity traded by the noise traders is independent of the asset value, we assume that the noise traders are able to correlate their trade with the true price. This has several implications for the equilibrium, one being that the informed trader??s expected profits decrease as the noise traders?? ability to correlate positively improve. In the limit, the noise traders do not lose on average, and the informed trader makes zero expected profits. When the correlation is negative, we interpret this as manipulation. In this case the insider makes the highest expected profits, and the informativeness of prices is at its minimum.