不完备证券市场中博弈未定权益的保值

考虑不完备证券市场中博弈未定权益(GCC)的保值问题,通过Kramkov关于上鞅的可选分解定理给出未定权益的上保值价格和下保值价格。指出关于买卖双方都存在着一个最优保值策略。给出价格的一个无套利区间,并针对前面的结论,给出几个性质以及在限制投资组合方面的一个应用。     

Cooperative hedging with a higher interest rate for borrowing

The paper studies the cooperative hedging problem of contingent claims in an incomplete financial market. Firstly we give the characterization of the optimal cooperative hedging strategy for the Black-Scholes model and the Volatility Jump model explicitly, then we consider the problem of cooperative hedging for the multi-agent case in a market with a higher borrowing interest rate. By the results of concave and linear backward stochastic differential equations, we give the optimal cooperative hedging strategy in our model.     

具有变系数和红利的多维Black-Scholes模型

本文提出具有变系数和红利的多维Ｂｌａｃｈ－Ｓｃｈｏｌｅｓ模型,利用倒向随机微分方程和鞅方法,得到欧式未定权益的一般定价公式及套期保值策略,在具体金融市场,给出欧式期权的定价公式和套期保值策略,以及美式看涨期权价格的界。     

Time-consistent mean–variance hedging of longevity risk: Effect of cointegration

This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.     

跳跃扩散型汇率过程的外汇期权定价

在完全外汇市场环境下 ,讨论了外汇汇率过程受 Brown运动和 Poisson过程共同驱动时外汇欧式未定权益的定价问题 ,并在常系数情形下获得了欧式外汇期权 Black- Scholes定价公式及其套期保值策略 ,最后给出了一种多汇率过程的线性组合式未定权益的定价     

Quadratic hedging for sequential claims with random weights in discrete time

We study a quadratic hedging problem for a sequence of contingent claims with random weights in discrete time. We obtain the optimal hedging strategy explicitly in a recursive representation, without imposing the non-degeneracy (ND) condition on the model and square integrability on hedging strategies. We relate the general results to hedging under random horizon and fair pricing in the quadratic sense. We illustrate the significance of our results in an example in which the ND condition fails.     

最小方差准则下一个未定权益的近似对冲

本文研究了不完备的离散时间股票市场下未定权益的定价的对冲问题.利用在最小方差准则下选择概率测度Q或权重函数LN来求最优投资组合的方法,给出了离散时间情况下的鞅表示定理,在最小方差准则下提供一个简单的方法来近似对冲一个未定权益或一个欧氏期权.     

基于动态规划原理的平方套期保值策略研究

在自融资约束下研究了标的资产价格服从跳扩散过程时欧式未定权益的平方套期保值问题。假定套期保值者用与未定权益相关的风险资产和另一种无风险资产来进行套期保值,利用动态规划原理,得到了离散时间集上均方最优套期保值策略的显式解。文章最后通过对比分析不同期限、不同策略调整频率的欧式看涨期权的套期保值结果表明:(1)对冲头寸与期限具有相依关系,期限越长,头寸比例通常也高;(2)对冲头寸与标的资产价格呈同向变化,标的资产价格越高,可以持有的头寸比例也高;(3)对冲头寸与交割价格呈反向变化,交割价格越高,可以适当降低头寸比例。     

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.     

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.     

Optimal partial hedging of an American option: shifting the focus to the expiration date

As a main contribution we present a new approach for studying the problem of optimal partial hedging of an American contingent claim in a finite and complete discrete-time market. We assume that at an early exercise time the investor can borrow the amount she has to pay for the option holder by entering a short position in the numéraire asset and that this loan in turn will mature at the expiration date. We model and solve a partial hedging problem, where the investor’s purpose is to find a minimal amount at which she can hedge the above-mentioned loan with a given probability, while the potential shortfall is bounded above by a certain number of numéraire assets. A knapsack problem approach and greedy algorithm are used in solving the problem. To get a wider view of the subject and to make interesting comparisons, we treat also a closely related European case as well as an American case where a barrier condition is applied.     

Optimal partial hedging in a discrete-time market as a knapsack problem

We present a new approach for studying the problem of optimal hedging of a European option in a finite and complete discrete-time market model. We consider partial hedging strategies that maximize the success probability or minimize the expected shortfall under a cost constraint and show that these problems can be treated as so called knapsack problems, which are a widely researched subject in linear programming. This observation gives us better understanding of the problem of optimal hedging in discrete time.     

Cooperative Hedging in the Complete Market under g-expectation Constraint

The paper studies the muiti-agent cooperative hedging problem of contingent claims in the complete market when the g-expected shortfall risks are bounded. We give the optimal cooperative hedging strategy explicitly by the Neyman-Pearson lemma under g-probability.     

Risk Minimization for a Filtering Micromovement Model of Asset Price

Abstract

The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.     

Super-replication under proportional transaction costs: From discrete to continuous-time models

In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.     

Minimizing coherent risk measures of shortfall in discrete‐time models with cone constraints

The paper studies the problem of minimizing coherent risk measures of shortfall for general discrete‐time financial models with cone‐constrained trading strategies, as developed by Pham and Touzi. It is shown that the optimal strategy is obtained by super‐hedging a contingent claim, which is represented as a Neyman–Pearson‐type random variable.     

利率风险与或有权益定价

应用无差异方法研究不完全市场中或有权益的保值和定价问题,并证明了或有权益的价格不仅依赖于或有权益的不可复制部分,而且受利率风险的影响．在最优保值意义下利率风险分解为可控风险和不可控风险．利率的可控风险与资本市场波动有关,可通过套期保值方法避免,可能产生正、零或负的期望收益．利率的不可控风险与资本市场波动无关,无法对冲,而且带来正的期望收益．利率风险的分解有助于更准确地解释或有权益的价格-它受利率的不可控风险影响,而与可控风险无关．当利率的不可控收益与或有权益的不可复制部分正(负)相关时,或有权益的不可复制部分的风险越大导致或有权益的价格越高(低)．     

Discrete–time delta hedging and the Black–Scholes model with transaction costs

The paper deals with the problem of discrete–time delta hedging and discrete-time option valuation by the Black–Scholes model. Since in the Black–Scholes model the hedging is continuous, hedging errors appear when applied to discrete trading. The hedging error is considered and a discrete-time adjusted Black–Scholes–Merton equation is derived. By anticipating the time sensitivity of delta in many cases the discrete-time delta hedging can be improved and more accurate delta values dependent on the length of the rebalancing intervals can be obtained. As an application the discrete-time trading with transaction costs is considered. Explicit solution of the option valuation problem is given and a closed form delta value for a European call option with transaction costs is obtained.     

Asymptotic analysis of hedging errors in models with jumps

Most authors who studied the problem of option hedging in incomplete markets, and, in particular, in models with jumps, focused on finding the strategies that minimize the residual hedging error. However, the resulting strategies are usually unrealistic because they require a continuously rebalanced portfolio, which is impossible to achieve in practice due to transaction costs. In reality, the portfolios are rebalanced discretely, which leads to a ‘hedging error of the second type’, due to the difference between the optimal portfolio and its discretely rebalanced version. In this paper, we analyze this second hedging error and establish a limit theorem for the renormalized error, when the discretization step tends to zero, in the framework of general Itô processes with jumps. The results are applied to the problem of hedging an option with a discontinuous pay-off in a jump-diffusion model.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.