Asymptotic arbitrage and large deviations

Typical models of mathematical finance admit equivalent martingale measures up to any finite time horizon but not globally, and this means that arbitrage opportunities arise in the long run. In this paper, we derive explicit estimates for asymptotic arbitrage, and we show how they are related to large deviation estimates for the market price of risk. As a case study we consider a geometric Ornstein–Uhlenbeck process. In this setting we also compute the optimal trading strategies and the resulting optimal growth rates of expected utility for all HARA utilities.     

Pricing and hedging in a single period market with random interval valued assets

In this article, a new financial market model, in which securities have random interval valued payoffs, is proposed. As an extension of traditional random market model, some concepts, such as robust arbitrage opportunities, risk-neutral pricing measures and robust replicative strategies, are given and discussed parallel to those in traditional market analysis. With these new concepts, problems of pricing and hedging are analyzed. It is shown that the requirement of no robust arbitrage opportunities is equivalent to the existence of risk-neutral pricing measures. Taking no robust arbitrage as the valuation principle, the problem of pricing a contingent claim with random interval valued payoff is discussed. All no robust arbitrage prices of the claim form an interval, whose endpoints can be got from the risk-neutral pricing measures or from robust replicative strategies.     

Optimal Consumption Portfolio and No-Arbitrage with Nonproportional Transaction Costs

In this paper we consider a finite-state financial market with non-proportional transaction cost and bid-ask spreads. The transaction cost consists of two parts: a fixed cost and a proportional cost to the size of transaction. We show that the existence of an optimal consumption policy implies that the market has no strong arbitrage; the opposite, however, is not true, i.e., no strong arbitrage does not imply the existence of an optimal consumption policy. This is in sharp contrast with the case of proportional transaction cost and other cases reported in the literature, where no strong arbitrage is equivalent to the existence of an optimal consumption policy. We also study the relationship between weak arbitrage and strong arbitrage. Different from the market with proportional transaction cost, we find that these two forms of arbitrage are equivalent unless the fixed cost is zero. A necessary and sufficient condition for the existence of an optimal consumption policy is also obtained. Supported by CAS, NSFC, RGC of Hong Kong and NSF under Grant No. DMI-0196084 and DMI-0200306.     

Arbitrage and utility maximization in market models with an insider

We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an \(\mathscr {F}_T\)-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails.     

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.     

Arbitrage and the tax code

We provide a detailed characterization of arbitrage-free asset prices in the presence of capital gains and income taxes. The distinguishing feature of our analysis is that we impose on the model two important features of the tax code: the limited use of capital losses and the inability to wash sell. We show that under remarkably mild conditions, the lack of pre-tax arbitrage implies the lack of post-tax arbitrage with the limited use of capital losses. The conditions are that the risk free interest rate be positive and that tax rates on interest income exceed capital gains tax rates. The result also holds when only a wash sale constraint is imposed and no investor holds a portfolio with a large capital loss. We allow investors to face different tax rates and have different bases for the calculation of capital gains taxes. The characterizations we provide have important implications for both asset pricing and portfolio choice. Our results imply that models that use arbitrage-free pre-tax models continue for derivative pricing and hedging are also arbitrage free in a world with taxes. Similarly, portfolio choice models with taxes typically specify pre-tax arbitrage free price processes and then analyze portfolio choice in the presence of taxes. In these models, it is unclear if portfolio recommendations are based on risk-return tradeoffs or on the arbitrage opportunities present in the model. Our results imply that if the above features of the tax code are modeled explicitly, then we can isolate the post-tax risk-return tradeoffs.     

A General Benchmark Model for Stochastic Jump Sizes

Abstract

Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market prices of risk as markets become approximately complete and consider the limits of traded securities, characterizing explicitly the growth optimal portfolio and investigating arbitrage and diversification in such markets.     

Arbitrage theory for non convex financial market models

We propose a unified approach where a security market is described by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a larger class of financial markets with transaction costs including non proportional trading costs. The usual tools from convex analysis however become inadequate to characterize the absence of arbitrage opportunities in non-convex financial market models. The natural question is to which extent the results of the classical arbitrage theory are still valid. Our contribution is a first attempt to characterize the absence of arbitrage opportunities in non convex financial market models.     

Admissible investment strategies in continuous trading

We consider a situation where relative prices of assets may change continuously and also have discrete jumps at random time points. The problem is the one of portfolio optimization. If the utility function used is the logarithm, we first argue that an optimal investment plan exists. Secondly, we show that any such plan has a certain optimality property known to hold also in discrete time models. Moreover, we show that this optimality criterion can be simplified significantly. In particular we show how admissibility can be related directly to observable characteristics of the investment strategy.     

Average shadow price and equilibrium price: A case study of tradable pollution permit markets

Shadow prices indicate implicit values of limited resources at the margin and provide important information in decision making for resource management. In continuous economic models, shadow prices associated with demand-supply balance constraints represent consumers’ willingness to pay and producers’ marginal cost, hence they correspond to market equilibrium prices. As well known, however, marginal analysis fails in the case of discrete optimization, such as mixed integer programming. An alternative concept has been introduced in the literature to measure the value of an extra unit of a limited resource in such cases. This concept is based on average rather than marginal values, thus called the average shadow price, and interpreted in the same way as conventional shadow prices. Whether average shadow prices in a discrete economic model can serve as market equilibrium prices has not been addressed in the related literature. The present paper addresses this issue in an empirical setting. Using a tradable pollution permit market as an example, where firms’ YES/NO type technology adoption decisions are represented by binary variables, we show that the average shadow price of tradable permits can be interpreted as the equilibrium price only when certain conditions related to the cost structure and emission levels hold. On the other hand, we show that an iterative procedure based on individual firms’ cost minimizing behavior presents a better approach for finding a price that can eliminate or reduce the gap between demand and supply of permits in the market.     

Illiquid financial market models and absence of arbitrage

We study financial market models with different liquidity effects. In the first part of this paper, we extend the short-term price impact model introduced by Rogers and Singh (2007) to a general semimartingale setup. We show the convergence of the discrete-time into the continuous-time modeling framework when trading times approach each other. In the second part, arbitrage opportunities in illiquid economies are considered, in particular a modification of the feedback effect model of Bank and Baum (2004). We demonstrate that a large trader cannot create wealth at no risk within this framework. Here we have to assume that the price process is described by a continuous semimartingale.     

Stochastic measures of arbitrage

Empirical research has provided evidence supporting the existence of arbitrage opportunities in real financial markets although market imperfections are often the main reason to explain these empirical deviations. Consequently, recent literature has turned the attention to imperfect markets in order to extend the most significant results on asset pricing. This paper develops several stochastic measures providing relative arbitrage earnings available in a financial market. The measures allow us to take into account different type of frictions. They are introduced by means of several dual pairs of vector optimization problems. Primal problems permit us to characterize the arbitrage absence even in an imperfect market and they also provide optimal arbitrage portfolios if the arbitrage absence fails. Dual ones allow us to extend the risk-neutral valuation methodology for imperfect and noarbitrage free markets and provide new interpretations for the measures in terms of “frictions effect” or “committed errors” in the valuation process. Partially funded by Comunidad Autónoma de Madrid (ref: CAM 07T/0027/2000) and Spanish Ministry of Science and Technology (ref: BEC2000-1388-C04)     

Asymptotic arbitrage in large financial markets with friction

In the modern version of arbitrage pricing theory suggested by Kabanov and Kramkov the fundamental financially meaningful concept is an asymptotic arbitrage. The ??real world?? large market is represented by a sequence of ??models?? and, though each of them is arbitrage free, investors may obtain non-risky profits in the limit. Mathematically, absence of the asymptotic arbitrage is expressed as contiguity of envelopes of the sets of equivalent martingale measures and objective probabilities. The classical theory deals with frictionless markets. In the present paper we extend it to markets with transaction costs. Assuming that each model admits consistent price systems, we relate them with families of probability measures and consider their upper and lower envelopes. The main result concerns the necessary and sufficient conditions for absence of asymptotic arbitrage opportunities of the first and second kinds expressed in terms of contiguity. We provide also more specific conditions involving Hellinger processes and give applications to particular models of large financial markets.     

考虑海外代购的品牌商定价模型研究

随着海外代购体量的日趋增大,代购带来的低价威胁对于在不同国家不同市场销售产品的公司来说已经成为一个日益严重的问题。同时,代购渠道中假货的问题也愈发严重。因此,在海外代购背景下探究产品定价模型具有必要性。以往研究普遍认为这种未经授权的销售会削减品牌方的利润,但实则不然。基于这一发现,本研究为在两个不同市场销售相同产品但面临代购低价威胁的公司制定考虑代购的市场定价模型。由公司制定两个市场的价格,消费者选择是否从包括代购在内的三个渠道购买产品。推出两个授权市场的最优价格,分析各参数变化对最优价格的影响,并校验最优价格对消费者需求和总利润的影响。模型分析表明,高价市场中有部分消费者需求转向海外代购,同时低价市场的消费者需求也受到了影响,且在一定条件下,提高高价市场的产品定价能够扩大低价市场的需求,从代购的角度解释了现实中需求曲线向上倾斜的现象。此外,两个独立市场之间的价格差距对代购市场的销售也产生了积极影响,并且在某些条件下,增大价格差距可以提高公司的收益水平。随后讨论了一种极端模型和三种扩展模型,通过模型分析表示,扩展后的定价模型也显示出与基础市场模型相似的灵敏度分析结果,同样得到两个市场的价差扩大会导致代购市场的销售额增加的结论,并且在一定条件下,公司的利润更高,增加了结论的可信度。     

Investment in electric energy storage under uncertainty: a real options approach

In this paper we develop a real options approach to evaluate the profitability of investing in a battery bank. The approach determines the optimal investment timing under conditions of uncertain future revenues and investment cost. It includes time arbitrage of the spot price and profits by providing ancillary services. Current studies of battery banks are limited, because they do not consider the uncertainty and the possibility of operating in both markets at the same time. We confirm previous research in the sense that when a battery bank participates in the spot market alone, the revenues are not sufficient to cover the initial investment cost. However, under the condition that the battery bank also can receive revenues from the balancing market, both the net present value (NPV) and the real options value are positive. The real options value is higher than the NPV, confirming the value of flexible investment timing when both revenues and investment cost are uncertain.     

Deep Reinforcement Learning for Market Making in Corporate Bonds: Beating the Curse of Dimensionality

ABSTRACT

In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for bonds to asset managers. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. The existing models, mostly inspired by the Avellaneda-Stoikov model, describe the complex optimization problem faced by market makers: proposing bid and ask prices for making money out of the difference between them while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving the equations characterizing the optimal bid and ask quotes numerically is seldom tackled in the literature, especially in high dimension. In this paper, we propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model à la Avellaneda–Stoikov. As classical finite difference methods cannot be used in high dimension, we present a discrete-time method inspired by reinforcement learning techniques, namely, a model-based deep actor-critic algorithm.     

Equivalent martingale measures for large financial markets in discrete time

We show that in a discrete-time large financial market the absence of certain asymptotic arbitrage opportunities is equivalent to the existence of martingale measures in a strong sense. We also consider the Arbitrage Pricing Model with stable random variables where we are able to give explicit necessary and sufficient conditions using market parameters.     

Mean‐Reverting Market Model: Speculative Opportunities and Non‐Arbitrage

The paper studies arbitrage opportunities and possible speculative opportunities for diffusion mean‐reverting market models. It is shown that the Novikov condition is satisfied for any time interval and for any set of parameters. It is non‐trivial because the appreciation rate has Gaussian distribution converging to a stationary limit. It follows that the mean‐reverting model is arbitrage‐free for any finite time interval. Further, it is shown that this model still allows some speculative opportunities: a gain for a wide enough set of expected utilities can be achieved for a strategy that does not require any hypothesis on market parameters and does not use estimation of these parameters.     

市场接纳不确定下短周期新产品产能投资研究 ——考虑竞争的投资时机与规模决策

新产品的市场接纳具有很大不确定性,传统投资理论并不适用于新产品投资。针对新产品投资中的产能投资,研究了垄断企业和有成本差异的竞争企业制定短周期新产品的产能投资时机与规模策略。给定企业“早”和“晚”两个投资时机可供选择,定义“早”投资时,企业只知道新产品市场规模的期望和方差;“晚”投资时,企业知道新产品真实的市场规模。垄断企业进入市场之前无法进行销售信息的收集,只会选择“早”投资或者不投资,给出其选择“早”投资的条件、最优产能投资规模及最大期望利润。有成本差异企业竞争的情形可以分为四种,分别给出四种情形下的最优产能投资规模及最大期望利润,并通过比较各情形下两企业的最大期望利润给出最优的产能投资时机策略。     

The impact of delivery lags on irreversible investment under uncertainty

We consider the valuation and rational exercise of irreversible investment opportunities in the presence of revenue uncertainty and delivery lags. In order to capture supply side market imperfections in the markets for investment goods, we assume that the lag depends on the revenue process faced by the investor. We show that such imperfections have a pronounced decelerating impact on rational investment demand as they may increase the value of waiting in excess of the exercise payoff even for projects which otherwise would be perceived as highly remunerative. We also consider the comparative static properties of the optimal investment policy and its value, and demonstrate that typically increased uncertainty decreases the investment incentives by increasing the value of waiting.