A characterization of equilibrium strategies in continuous-time mean–variance problems for insurers

In this work, we study the equilibrium reinsurance/new business and investment strategy for mean–variance insurers with constant risk aversion. The insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where the surplus of the insurers is assumed to follow a jump–diffusion model and the financial market consists of one riskless asset and a multiple risky assets whose price processes are driven by Poisson random measures and independent Brownian motions. By using a version of the stochastic maximum principle approach, we characterize the open loop equilibrium strategies via a stochastic system which consists of a flow of forward–backward stochastic differential equations (FBSDEs in short) and an equilibrium condition. Then by decoupling the flow of FSBDEs, an explicit representation of an equilibrium solution is derived as well as its corresponding objective function value.     

A note on utility based pricing and asymptotic risk diversification

In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitrage-free replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically if we consider a sequence of agents whose absolute risk aversions go to zero and if the number of sold claims goes to infinity simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agents behave asymptotically as exponential utility maximizers with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.     

一般均衡下基于产品市场和资本市场的单名CDS定价

次贷危机呼吁新的信用衍生品定价模型, 因此为存在产品市场和资本市场的经济结构建立一般均衡的单名CDS定价模型, 使用最优化求解一般均衡下的商品价格和CDS价格. 可以发现一般均衡的CDS定价具有资本市场和产品市场的因素, 这表示CDS的价格不再是由单纯的资本市场因素决定的, 而是由无风险利率、资本产出弹性、违约率、回收率同时决定的. 通过数量约束用模拟的方式研究多个均衡的动态变化, 发现违约风险的增加使得价格剧烈波动且市场交易萎缩. 在为以中国工商银行为参考资产的CDS定价过程中, 发现各种因素在不同的时期都可能成为定价的主要影响因素. 可以发现, 次贷危机的定价体系存在着信用调整问题和定价与实体经济脱节的问题. 可以认为, 一般均衡下基于产品市场和资本市场的单名CDS定价可以囊括多个市场的交叉影响, 为衍生品定价提供一个新的方向.     

Liquidity risks on power exchanges: a generalized Nash equilibrium model

The extreme volatility of electricity prices makes their financial derivatives important instruments for asset managers. Even if the volume of derivative contracts traded on Power Exchanges has been growing since the inception of the restructuring of the sector, electricity remains considerably less liquid than other commodity markets. This paper assesses the effect of limited liquidity in power exchanges using an equilibrium model where agents cannot hedge up to their desired level. Mathematically, the problem is formulated as a two stage stochastic Generalized Nash Equilibrium with possibly multiple equilibria. Computing a large panel of solutions, we show how the risk premium and players profits are affected by illiquidity. We also show that the illiquidity in the FTR market affects the trades in the electricity futures market.     

Wealth optimization and dual problems for jump stock dynamics with stochastic factor

An incomplete financial market is considered with a risky asset and a bond. The risky asset price is a pure jump process whose dynamics depends on a jump-diffusion stochastic factor describing the activity of other markets, macroeconomics factors or microstructure rules that drive the market. With a stochastic control approach, maximization of the expected utility of terminal wealth is discussed for utility functions of constant relative risk aversion type. Under suitable assumptions, closed form solutions for the value functions and for the optimal strategy are provided and verification results are discussed. Moreover, the solution to the dual problems associated with the utility maximization problems is derived.     

Actuarial risk measures for financial derivative pricing

We present an axiomatic characterization of price measures that are superadditive and comonotonic additive for normally distributed random variables. The price representation derived involves a probability measure transform that is closely related to the Esscher transform, and we call it the Esscher-Girsanov transform. In a financial market in which the primary asset price is represented by a stochastic differential equation with respect to Brownian motion, the price mechanism based on the Esscher-Girsanov transform can generate approximate-arbitrage-free financial derivative prices.     

Risk-neutral economy and zero price of risk

The paper investigates the equilibrium in an economy in which all participants are indifferent to risk. The mechanism of asset and derivative pricing in such economy is identified. It is shown that no economy in equilibrium with stochastic interest rates can be simultaneously risk-neutral and have zero market price of risk. On the other hand, there exist equilibrium economies with risk-averse participants and zero prices of risk.     

Utility maximization under risk constraints and incomplete information for a market with a change point

In this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.     

Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps

This paper analyzes the equilibrium strategy of a robust optimal reinsurance-investment problem under the mean–variance criterion in a model with jumps for an ambiguity-averse insurer (AAI) who worries about model uncertainty. The AAI’s surplus process is assumed to follow the classical Cramér–Lundberg model, and the AAI is allowed to purchase proportional reinsurance or acquire new business and invest in a financial market to manage her risk. The financial market consists of a risk-free asset and a risky asset whose price process is described by a jump-diffusion model. By applying stochastic control theory, we establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations. Furthermore, we derive both the robust equilibrium reinsurance-investment strategy and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated, and utility losses from ignoring model uncertainty, jump risks and prohibiting reinsurance are analyzed using numerical examples.     

The Dynamic Interaction of Speculation and Diversification

A discrete time model of a financial market is developed, in which heterogeneous interacting groups of agents allocate their wealth between two risky assets and a riskless asset. In each period each group formulates its demand for the risky assets and the risk‐free asset according to myopic mean‐variance maximizazion. The market consists of two types of agents: fundamentalists, who hold an estimate of the fundamental values of the risky assets and whose demand for each asset is a function of the deviation of the current price from the fundamental, and chartists, a group basing their trading decisions on an analysis of past returns. The time evolution of the prices is modelled by assuming the existence of a market maker, who sets excess demand of each asset to zero at the end of each trading period by taking an offsetting long or short position, and who announces the next period prices as functions of the excess demand for each asset and with a view to long‐run market stability. The model is reduced to a seven‐dimensional nonlinear discrete‐time dynamical system, that describes the time evolution of prices and agents' beliefs about expected returns, variances and correlation. The unique steady state of the model is determined and the local asymptotic stability of the equilibrium is analysed, as a function of the key parameters that characterize agents' behaviour. In particular it is shown that when chartists update their expectations sufficiently fast, then the stability of the equilibrium is lost through a supercritical Neimark–Hopf bifurcation, and self‐sustained price fluctuations along an attracting limit cycle appear in one or both markets. Global analysis is also performed, by using numerical techniques, in order to understand the role played by the chartists' behaviour in the transition to a regime characterized by irregular oscillatory motion and coexistence of attractors. It is also shown how changes occurring in one market may affect the price dynamics of the alternative risky asset, as a consequence of the dynamic updating of agents' portfolios.     

Pricing derivatives in a regime switching market with time inhomogenous volatility

This paper studies pricing derivatives in a componentwise semi-Markov (CSM) modulated market. We consider a financial market where the asset price dynamics follows a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local Black–Scholes–Merton-type PDE. We establish existence and uniqueness of a classical solution to the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. An explicit expression of quadratic residual risk is also obtained.     

Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model

In this paper, we study an optimal investment problem under the mean–variance criterion for defined contribution pension plans during the accumulation phase. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump–diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) and Björk et al. (2014). Finally, some numerical simulations are provided to illustrate our derived results.     

基于双边市场理论的金融超市竞争定价策略研究

基于双边市场理论,重点分析金融超市在双寡头垄断情形下的竞争定价策略。即在在一般定价模型的基础上,构建起加入金融超市双边用户交易次数为歧视标准的价格歧视竞争模型。并且围绕金融超市追求长期利益和短期利益两种不同动机,对采取该策略均衡时最终用户的均衡进入价格、金融超市利润和市场份额进行比较分析。最后,给出金融超市实施价格歧视策略的对策和建议。     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Market Viability and Martingale Measures under Partial Information

We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the optimal portfolio problem has a solution up to a stopping time, if and only if the (normalised) marginal utility of the terminal wealth generates a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control problems under partial information. We then characterize a global notion of market viability in terms of partial information local martingale deflators (PILMDs). We illustrate our results by means of a simple example.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Heterogeneous trading agents

In this article, we present a multiagent system (MAS) simulation of a financial market and investigate the requirements to obtain realistic data. The model consists of autonomous, interactive agents that buy stock on a financial market. Transaction decisions are based on a number of individual and collective elements, the former being risk aversion and a set of decision rules reflecting their anticipation of the future evolution of prices and dividends and the latter the information arriving on the market influencing the decision making process of each trader. We specifically look at this process and the following observations hold: The market behavior is determined by the information arriving at the market and agent heterogeneity is required in order to obtain the right statistical properties of the price and return time series. The observed results are not sensitive to changes in the parameter values. © 2003 Wiley Periodicals, Inc.     

Option pricing for large agents

This paper considers arbitrage-free option pricing in the presence of large agents. These large agents have a significant market power, and their trading strategies influence the dynamics of the financial asset prices. First, a simple asset pricing model in the presence of large agents is presented. Then a nonlinear partial differential equation is found for the prices of European options in the model. The unit option price depends on the large agent's asset holdings. Finally, a game model is introduced for the interaction between different market players. In this game, the outstanding number of options, as well as the option price, is found as a Nash equilibrium.     

Robust consumption-investment problems with random market coefficients

We consider consumption-investment problems in a financial market with general random coefficients where the market price of risk process is unknown. The investor tries to maximize his expected utility under the worst-case parameter configuration. To solve robust consumption-investment problems, we make use of stochastic Bellman?CIsaac equations. These equations can be explicitly solved for power, exponential and logarithmic utility. This enables us to characterize a robust optimal consumption-investment strategy and a worst-case market price of risk process in terms of the solution of a backward stochastic differential equation.     

Utility maximization in markets with bid–ask spreads

A one-period financial market model with transaction costs is considered in this paper. Redefining the risky asset price process in a suitable way, we obtain an explicit solution to the utility maximization problem when the risk preferences of the investor are based on the exponential utility function and a liability can be included in her portfolio. The arbitrage-free interval price for a general liability, as well as its replication price, is characterized in terms of expectations with respect to equivalent martingale measures. The indifference price is derived and its asymptotic limit when the risk aversion is going to infinity is analysed.