Scale-invariant asset pricing and consumption/portfolio choice with general attitudes toward risk and uncertainty

Motivated by notions of aversion to Knightian uncertainty, this paper develops the theory of competitive asset pricing and consumption/portfolio choice with homothetic recursive preferences that allow essentially any homothetic uncertainty averse certainty-equivalent form. The market structure is scale invariant but otherwise general, allowing any trading constraints that scale with wealth. Technicalities are minimized by assuming a finite information tree. Pricing restrictions in terms of consumption growth and market returns are derived and a simple recursive method for solving the corresponding optimal consumption/portfolio choice problem is established.     

一致风险度量和锥优化分析

一致风险理论的公理系统为风险分析建立了坚实的基础,然而它背后的数学却和凸优化理论思想密切相关,特别是对偶理论. 本文在有限维空间中,利用锥优化的对偶定理给出了一致风险度量的一般表达式的简单证明. 分析了可接受集的概念在一致风险度量中的中心作用,根据锥优化的对偶关系,探索了常用风险度量的性质. 尽管可接受集的大小能够表达风险控制的强弱,但是我们不知道如何定量地表示. 本文提出用相对熵控制风险度量松紧度的方法和意义. 另外,根据一致风险度量的灵活的结构,给出了无套利条件的一种放松,这一结果可用于不完全市场中的期权定价问题.     

基于不确定理论的风险中性测度及其在欧式期权定价中的应用

首先运用不确定理论推导了相应的不确定风险中性测度,修正了已有文献中涨跌期权不满足无套利原则的问题.然后将所得的风险中性测度用于欧式看涨和看跌期权的定价,并验证了涨跌期权价格之间的平价关系.最后研究了一类利差期权的定价问题,结合定义的风险中性测度给出了期权的定价公式.所推导的不确定风险中性测度与经典的无套利原则相吻合,而且考虑到了问题描述过程中存在的不精确性,弥补了单纯依赖随机理论的不足,可广泛地应用于金融衍生品的定价过程,为投资分析提供一定的理论依据.     

Valuing risky income streams in incomplete markets

A model for pricing and hedging in incomplete markets is proposed. This model is derived from expected utility theory, and a connection with the traditional no‐arbitrage framework is noted. It is shown that the CGM model can be implemented to value risky assets in incomplete markets.     

A note on arbitrage,approximate arbitrage and the fundamental theorem of asset pricing

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.     

不完全资产与现货市场的两期交换经济的非套利均衡的存在性

本文在第一章给出了经济模型及相关的基本概念,第二章中利用Grassmanian流形上的集值映射的不动点理论及非线性泛函分析中一类方程解的有限性定理,给出了不完全资产与现货市场的两期交换经济的非套利均衡存在的简捷证明.     

股票价格遵循几何分式Brown运动的期权定价

讨论了股票价格过程遵循几何分式B row n运动的欧式期权定价.由于该过程存在套利机会使得传统的期权定价方法(如资本资产定价模型(CAPM),套利定价模型(APT),动态均衡定价理论(DEPT))不可能对该期权定价.利用保险精算定价法,在对市场无其它任何假设条件下,获得了欧式期权的定价公式.并讨论了在有效期内股票支付已知红利和红利率的推广公式.     

股票价格服从分式Brown运动的股票期权保险精算定价

期权定价的保险精算方法由M ogens B ladt和H ina Hv iid R ydberg于1998年首次提出,由于无任何市场假设,所以它不光对无套利、均衡、完备的市场有效,且对有套利、非均衡、不完备的市场也有效.本文利用保险精算方法讨论了股票价格服从分式B row n运动的欧式期权定价问题.     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

Incomplete markets,Knightian uncertainty and high-water marks

This paper extends the pricing of the hedge fund compensation contracts to the case of ambiguity over the appropriate valuation approach originating from market incompleteness. It predicts that an increase in the level of Knightian uncertainty causes the erosion of the values of the fees and the claim, while an increase in the degree of market incompleteness (specified by the correlation between market asset and the fund’s asset or the volatility of non-diversifiable risk) has non-monotonic effects.     

No Arbitrage and the Growth Optimal Portfolio

Abstract

Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.     

Switching problem and related system of reflected backward SDEs

This paper studies a system of backward stochastic differential equations with oblique reflections (RBSDEs for short), motivated by the switching problem under Knightian uncertainty and recursive utilities. The main feature of our system is that its components are interconnected through both the generators and the obstacles. We prove existence, uniqueness, and stability of the solution of the RBSDE, and give the expression of the price and the optimal strategy for the original switching problem via a verification theorem.     

Arbitrage in skew Brownian motion models

Empirical skewness of asset returns can be reproduced by stochastic processes other than the Brownian motion with drift. Some authors have proposed the skew Brownian motion for pricing as well as interest rate modelling. Although the asymmetric feature of random return involved in the stock price process is driven by a parsimonious one-dimensional model, we will show how this is intrinsically incompatible with a modern theory of arbitrage in continuous time. Application to investment performance and to the Black-Scholes pricing model clearly emphasize how this process can provide some kind of arbitrage.     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

Confidence models of incomplete preferences

This paper introduces a new class of representations for incomplete preferences called confidence models. Confidence models describe decision makers who behave as if they have probabilistic uncertainty over their true preferences, and are only willing to express a binary preference if it is sufficiently likely to hold. Confidence models provide a natural way to connect incomplete preferences with stochastic choice. This connection is characterized by a simple joint condition on an incomplete preference relation and a random choice rule.     

Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing

We consider an incomplete market model where asset prices are modelled by Ito processes, and derive the first fundamental theorem of asset pricing using standard stochastic calculus techniques. This contrasts with the sophisticated functional analytic theorems required in the comprehensive works of F. Delbaen and W. Schachermayer (1993) No Arbitrage and the Fundamental Theorem of Asset Pricing, pp. 37–38; Math. Finance 4 (1994), pp. 343–348; Math. Ann. 300 (1994), pp. 464–520; Ann. Appl. Probab. 5 (1995), pp. 926–645 and Proc. Sympos. Appl. Math. 57 (1999), pp. 49–58, and the comparative lack of transparency of the associated technical conditions. An additional benefit is that a clear relationship between no arbitrage and the existence of equivalent local martingale measures is also presented.     

EXISTENCE OF STOCHASTIC EQUILIBRIUM WITH INCOMPLETE FINANCIAL MARKETS

This paper analyzes the aritrage-tree security markets and the general equilibrium ex-istence problem for a stochastic economy with incomplete financial markets. Information structure is given by an event tree. This paper restricts attention to puraly financial securities. It isassume that trading takes place in the sequence of spot markets and futures markets for securi-ties payable in units of account. Unlimited short-selling in securities is allowed. Financial markets may be incomplete, some consumption streams may be impossible to obtain by any tradingstrategy. Securities may be individually precluded from trade at arbitrary states and dates. Thesecurity price process is arbitrage-free the dividend process if and only if there exists a stochaticstate price (present value) process : the present value of the security prices at every vertex isthe present value of their dividend and capital values over the set of immediate successors ; thecurrent value of each security at every vertex is the present value of its future dividend streamover all succeeding vertices. The existence of such an equilibrium is proved under the followingcondition: continuous, weakly convex, strictly monotone and complete preferences, strictlypositive endowmenta and dividends processes.     

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.     

Expected gain–loss pricing and hedging of contingent claims in incomplete markets by linear programming

We analyze the problem of pricing and hedging contingent claims in the multi-period, discrete time, discrete state case using the concept of a “λ gain–loss ratio opportunity”. Pricing results somewhat different from, but reminiscent of, the arbitrage pricing theorems of mathematical finance are obtained. Our analysis provides tighter price bounds on the contingent claim in an incomplete market, which may converge to a unique price for a specific value of a gain–loss preference parameter imposed by the market while the hedging policies may be different for different sides of the same trade. The results are obtained in the simpler framework of stochastic linear programming in a multi-period setting, and have the appealing feature of being very simple to derive and to articulate even for the non-specialist. They also extend to markets with transaction costs.     

Asset prices in an ambiguous economy

Models with ambiguity averse preferences have the potential to explain some pricing anomalies on financial markets. However, the models used in applications make additional assumptions, beyond ambiguity aversion, on the structure of the investor’s preferences. Therefore, it is not clear how to disentangle the effect of ambiguity aversion from other features of preferences on equilibrium prices. This paper offers a general theory of asset pricing assuming only ambiguity aversion. Price indeterminacy may result in equilibrium when preferences are not smooth. A set of priors, which is identifiable in all the models used in applications, contains the relevant information to price assets. Ambiguity enriches the standard pricing formula by an additional stochastic discount factor and we calculate its explicit form for various models.