LARGE PORTFOLIO ASYMPTOTICS FOR LOSS FROM DEFAULT

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogeneous portfolios. The density of the limiting measure is shown to solve a nonlinear stochastic partial differential equation, and certain moments of the limiting measure are shown to satisfy an infinite system of stochastic differential equations. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.     

Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff—the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs.     

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.     

The risk of betting on risk: Conditional variance and correlation of bank credit default swaps

Credit default swaps (CDS) have been used to speculate on the default risk of the reference entity. The risk of CDS can be measured by their second moments. We apply a Glosten, Jagannathan, and Runkle (GJR)-t model for the conditional variance and a Dynamic Conditional Correlation (DCC)-t model for the conditional correlation. Based on the CDS of six large US banks from 2002 to 2018, we find that CDS conditional variance is asymmetric and leptokurtic. A positive innovation actually increases CDS conditional variance more than a negative innovation does. CDS conditional correlations have stayed elevated since the financial crisis, in contrast to the decreasing stock conditional correlations.     

VAR模型及其在投资组合中的应用

VAR是目前国际上金融风险管理的主流方法之一,本文在对其概念、VAR计算的基本思想和主要特点进行简要介绍、分析的基础上,鉴于VAR已在金融领域中得到了广泛的应用,与投资组合的管理和决策有着密切的联系,重点讨论了基于VAR的投资组合管理,及在VAR约束下的投资组合决策,最后提出了我国在应用VAR所面临的主要问题.     

TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON-NESTED MONTE CARLO

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic "true" stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator. Numerical experiments indicate the efficiency of the proposed algorithms.     

贝叶斯下的市场风险资产损失计量

目前VaR模型是国际上通行的综合衡量和管理风险的计量技术,但当分布是非正态分布或是不连续的时候VaR就没有稳定性。在应用VaR以及CVaR的基本原理,利用具有尖峰后尾特征的对数正态分布并结合贝叶斯风险,对参数通过先验分布和后验分布进行修正得到市场风险资产CVaR以及VaR的计算公式,对其进行比较分析,将其与pareto分布下的市场风险VaR值进行比较。研究表明:对数正态分布下市场风险资产的损失率比损失率呈pareto分布的风险损失显现出更好的拟合特征,能够描述市场风险资产的随机性最大损失,以及CvaR计量下的市场风险资产损失比VaR计量下的有更好的尾部特征,更加符合风险的充分性。     

金融资产日内跳跃检验比较及其股市跳跃特征分析

运用Monte Carlo模拟分析的方法，通过对现存几种日内跳跃检验方法进行比较后发现．就检测能力、错判率和跳跃方差偏差而言，ABD跳跃检验方法表现最优。利用ABD跳跃检验方法，通过对上证综指高频数据进行分析后发现，上证综指平均大约每四天发生一次跳跃。跳跃方差贡献约为19％，并且跳跃行为存在非对称性；同时发现跳跃行为是造成上证综指收益率尖峰厚尾特征的重要因素之一。     

中国开放式基金表现的动态能力研究

本文以中信指数与中信债券指数的加权超额收益率为市场基准 ,以GDP增长率和消费物价指数的增长率以及广义货币供给增长率为条件变量 ,构建了证券投资基金表现的条件评价模型 ,对中国开放式证券投资基金的选股能力和市场时机把握能力进行了测度。计量方法上采用混合数据模型的回归方法以提高回归结果的有效性。研究结论为 :中国开放式基金不但具有一定的选股能力 ,而且具有良好的市场时机把握能力 ;条件模型的解释能力高于无条件模型。     

MEAN–VARIANCE PORTFOLIO CHOICE: QUADRATIC PARTIAL HEDGING

In this paper we investigate the problem of mean–variance portfolio choice with bankruptcy prohibition. For incomplete markets with continuous assets' price processes and for complete markets, it is shown that the mean–variance efficient portfolios can be expressed as the optimal strategies of partial hedging for quadratic loss function. Thus, mean–variance portfolio choice, in these cases, can be viewed as expected utility maximization with non-negative marginal utility.     

MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.     

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity‐adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta‐normal VaR or RiskMetrics (J.P. Morgan, 1996).     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.     

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999) , as well as Ma and Zhang (2000) , using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.     

Price risk in the NYMEX energy complex: An extreme value approach

We estimate tail parameters and construct risk statistics for unconditional distributions of daily logarithmic price changes of the NYMEX energy complex and apply the conditional extreme value method proposed by A. J. McNeil and R. Frey (2000) for estimating VAR and related risk statistics from the tails of conditional distributions for these commodities. The unconditional distribution of spot market price declines is found to be fat tailed relative to the normal for all commodities examined. Backtesting of candidate conditional risk measurement methods indicates that the conditional extreme value method is significantly more accurate for measuring risk exposure due to price declines for 7 of the 10 price series examined. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:309–337, 2005     

基于COPULA-EVT的利率衍生品风险管理

监管层提出对＂系统重要性银行＂和＂非系统重要性银行＂进行分类管理的思路,表明在强化宏观审慎监管过程中,微观个体宏观审慎经营行为仍然起着重要的作用。新巴塞尔协议对于银行信用风险的监控和计量有了更加严格的规定,然而对于涉及到衍生品的市场风险只是强调银行要根据自身的交易业务进行合理评估,这样便使得衍生品的市场风险成为了银行整体风险中最不稳定的因素。本文基于极值分布、Copula连接函数和蒙特卡洛模拟理论,获得商业银行包括利率期货、利率期权、利率互换在内的单个利率衍生品的风险度量指标,如VaR,CVaR,EVA,RAROC,EC,并得到衍生品组合的风险度量指标,这些指标可以帮助商业银行更加清晰地了解自身的潜在风险。同时,商业银行在给定风险容忍度VaR下能得到各种衍生产品的最优配置,从而为银行的投资决策提供参考。     

基于信用风险修正的多阶段银行贷款组合优化决策模型

针对银行的信用风险和贷款的周期性等问题,建立一个基于信用风险修正的多阶段银行贷款组合优化决策模型,该模型在多阶段模型中考虑了信用风险修正问题,根据模型的特点给出了把Monte Carlo模拟的动态算法和差分进化的多阶段算法相结合的求解方法,前者求解银行各类贷款的期望收益率,后者求解每一阶段银行对各类贷款的最优投资比重。数值试验表明所建立的模型是合理的且符合商业银行的实际操作要求,给出的方法是有效的和可行的。     

Market Price of Risk: A Comparison among the United States,United Kingdom,Australia and Japan*

This study examines and compares the market price of risk of the S&P 500, FTSE 100, All Ordinaries, and Nikkei 225 markets from 1984 to 2009 in the framework of Intertemporal Capital Asset Pricing Model (ICAPM). We follow the Vector Autoregressive instrumental variable approach in identifying the risk and hedge components of market returns and argue that in the context of market integration, covariance with a world market portfolio is a better measure of market risk than conditional market variance. Evidence is documented in support of using covariance as a risk measure in explaining market risk premiums in the Australian and Japanese markets. CAY, the consumption wealth ratio from the US market is found to be a robust state variable that helps to explain both conditional variance and covariance processes in the four markets. The market prices of risk, after controlling for the hedging demands, are positive and significant with the United States having the highest price of risk. The results are confirmed using a series of robustness tests that include varying the sampling interval.     

Use of high-frequency data to evaluate the performance of dynamic hedging strategies

The hedging performance results of generalized autoregressive conditional heteroskedasticity models are mixed; we address this herein by adopting an asymptotic setting to determine the relative performance of competing hedge ratios. The proxy variable is constructed through precise realized measures rather than through noisy squared returns because the substitution of the latent true hedged portfolio variance with a noisy proxy renders the loss function incapable of ranking forecasts consistently. The merits of allowing some features in modeling the spot–futures distribution are assessed. Empirical comparisons suggest that hedgers may favor the wrong model when the quality of the proxy variable deteriorates.     

The demand for risky assets under uncertain inflation : An examination of some widely used assumptions

This paper examines the effect of the following commonly used methods of incorporating random inflation into discrete-time models of the demand for risky assets: 1) the use of a multivariate normal probability distribution for nominal asset returns and the random inflation rate, and 2) the approximation of real asset returns by the difference between nominal returns and the rate of inflation. The combination of these assumptions results in a deceptively simple version of the inflationary capital asset pricing model (CAPM). However, in an approximation-free version of this model the expected value of real wealth does not exist. While it is obvious that mean-variance analysis is not applicable in such models, we also find that the model does not satisfy Ohlson's weakened conditions for a quadratic approximation to the portfolio selection problem. Furthermore, this model is neither a member of the generalized Pareto-Levy nor log-stable class of portfolio models analyzed by Fama, Samuelson, Ohlson, and Struck.