Portfolio selection with qualitative input

We formulate a mean-variance portfolio selection problem that accommodates qualitative input about expected returns and provide an algorithm that solves the problem. This model and algorithm can be used, for example, when a portfolio manager determines that one industry will benefit more from a regulatory change than another but is unable to quantify the degree of difference. Qualitative views are expressed in terms of linear inequalities among expected returns. Our formulation builds on the Black-Litterman model for portfolio selection. The algorithm makes use of an adaptation of the hit-and-run method for Markov chain Monte Carlo simulation. We also present computational results that illustrate advantages of our approach over alternative heuristic methods for incorporating qualitative input.     

基于copula的投资组合选择模型的研究

本文首先介绍了投资组合理论与copula,然后给出基于概率的收益率等定义,建立基于概率的收益率的投资组合选择模型并给出具体解法,接着通过选取上证领先指数与深证领先指数2004年9月1日至2006年5月26日的日收盘数据进行实证分析,发现在收益率(基于概率的收益率)一定的情况下,通过投资组合可以降低风险。pápápá     

Portfolio optimization with a defaultable security

In this paper we derive a closed-form solution for a representative investor who optimally allocates her wealth among the following securities: a credit-risky asset, a default-free bank account, and a stock. Although the inclusion of a credit-related financial product in the portfolio selection is more realistic, no closed-form solutions to date are given in the literature when a recovery value is considered in the event of a default. While most authors have assumed some recovery scheme in their initial model set up, they do not address the portfolio problem with a recovery when a default actually occurs. Given the tractability of the recovery of market value, we solved the optimal portfolio problem for the representative investor whose utility function is a Constant Relative Risk Aversion utility function. We find that the investor will allocate larger fraction of wealth to the defaultable security as long as the default-event risk is priced. These results are very intuitive and reasonable since it indicates that if the default risk premium is not priced properly the investor purchases less defaultable securities.     

Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility

In this paper, we show that if asset returns follow a generalized hyperbolic skewed t distribution, the investor has an exponential utility function and a riskless asset is available, the optimal portfolio weights can be found either in closed form or using a successive approximation scheme. We also derive lower bounds for the certainty equivalent return generated by the optimal portfolios. Finally, we present a study of the performance of mean–variance analysis and Taylor’s series expected utility expansion (up to the fourth moment) to compute optimal portfolios in this framework.     

Portfolio construction incorporating asymmetric dependence structures: a user's guide

We outline a method of portfolio selection incorporating asymmetric dependency structures using copula functions. Assuming normally distributed marginal returns, we illustrate how asymmetric return correlations affect the efficient frontier and subsequent portfolio performance under a dynamic rebalancing framework. Implementing this methodology within the context of tactically allocating a small set of market indices, we demonstrate several key findings. First, we establish the manner by which the efficient frontier constructed under asymmetric dependence differs from a mean‐variance frontier. By establishing a paper portfolio based on these differences, we find that asymmetric correlation structures do have real economic value. The primary source of this economic value is the ability to better protect portfolio value and reduce the size of any erosion in return relative to the normal portfolio when asymmetric return correlations are accounted for.     

Long-only equal risk contribution portfolios for CVaR under discrete distributions

Portfolios in which all assets contribute equally to the conditional value-at-risk (CVaR) represent an interesting variation of the popular risk parity investment strategy. This paper considers the use of convex optimization to find long-only equal risk contribution (ERC) portfolios for CVaR given a set of equally likely scenarios of asset returns. We provide second-order conic and non-linear formulations of the problem, which yields an ERC portfolio when CVaR is both positive and differentiable at the optimal solution. We identify sufficient conditions for differentiability and develop a heuristic that obtains an approximate ERC portfolio when the conditions are not satisfied. Computational tests show that the approach performs well compared to non-convex formulations that have been proposed in the literature.     

Portfolio choice under dynamic investment performance criteria

A new dynamic criterion for measuring the performance of self-financing investment strategies is introduced. To this aim, a family of stochastic processes defined on [0, ∞) and indexed by a wealth argument is used. Optimality is associated with their martingale property along the optimal wealth trajectory. The optimal portfolios are constructed via stochastic feedback controls that are functionally related to differential constraints of fast diffusion type. A multi-asset Ito-type incomplete market model is used.     

Portfolio Insurance with Liquidity Risk

This paper studies a portfolio insurance problem with liquidity risk. We consider an investor who wants to maximize the expected growth rate of wealth in a low liquid market. The investor can trade assets only at random times and his wealth must not fall below a predetermined floor. We find the optimal expected growth rate and an optimal strategy. The optimal strategy is closely related with a traditional constant proportion portfolio insurance strategy. Also we show that the same strategy maximizes the growth rate almost surely. Further we study the floor effect on the growth rate.     

期货投资组合交易保证金设置研究——基于Copula模型和极值理论的分析

本文以棉花、铜、天然橡胶三个期货合约为研究对象,基于t-Copula模型,利用Monte Carlo模拟法计算在一定权重下由三个品种构成的期货投资组合的VaR和ES值作为投资组合的保证金数值。Kupiec回溯测试结果表明,t-Copula模型结合极值理论计算出的期货投资组合保证金相比其他方法能够在较好覆盖极端风险的同时降低投资成本。     

Portfolio Optimization in Discontinuous Markets under Incomplete Information

We consider the problem of maximization of expected utility from terminal wealth for log and power utility functions in a market model that leads to purely discontinuous processes. We study this problem as a stochastic control problem both under complete as well as incomplete information. Our contribution consists in showing that the optimal strategy can be obtained by solving a system of equations that in some cases is linear and that a certainty equivalence property holds not only for log-utility but also for a power utility function. For the case of a power utility under incomplete information we also present an independent direct approach based on a Zakai-type equation.      

Additive and multiplicative duals for American option pricing

We investigate and compare two dual formulations of the American option pricing problem based on two decompositions of supermartingales: the additive dual of Haugh and Kogan (Oper. Res. 52:258–270, 2004) and Rogers (Math. Finance 12:271–286, 2002) and the multiplicative dual of Jamshidian (Minimax optimality of Bermudan and American claims and their Monte- Carlo upper bound approximation. NIB Capital, The Hague, 2003). Both provide upper bounds on American option prices; we show how to improve these bounds iteratively and use this to show that any multiplicative dual can be improved by an additive dual and vice versa. This iterative improvement converges to the optimal value function. We also compare bias and variance under the two dual formulations as the time horizon grows; either method may have smaller bias, but the variance of the multiplicative method typically grows much faster than that of the additive method. We show that in the case of a discrete state space, the additive dual coincides with the dual of the optimal stopping problem in the sense of linear programming duality and the multiplicative method arises through a nonlinear duality.      

Robust portfolio optimization with a generalized expected utility model under ambiguity

This paper proposes a robust approach maximizing worst-case utility when both the distributions underlying the uncertain vector of returns are exactly unknown and the estimates of the structure of returns are unreliable. We introduce concave convex utility function measuring the utility of investors under model uncertainty and uncertainty structure describing the moments of returns and all possible distributions and show that the robust portfolio optimization problem corresponding to the uncertainty structure can be reformulated as a parametric quadratic programming problem, enabling to obtain explicit formula solutions, an efficient frontier and equilibrium price system. We would like to thank Prof. Zengjing Chen from School of Mathematics and System Sciences, Shandong University for helpful suggestions, and to thank the anonymous referee for valuable comments.     

Optimal importance sampling with explicit formulas in continuous time

In the Black–Scholes model, consider the problem of selecting a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options. Employing large deviations techniques, the asymptotically optimal change of drift is identified as the solution to a one-dimensional variational problem, which may be reduced to the associated Euler–Lagrange differential equation. Closed-form solutions for geometric and arithmetic average Asian options are provided. The authors acknowledge the support of the National Science Foundation under grants DMS-0532390 (Guasoni) and DGE-0221680 (Robertson) at Boston University.     

Backward simulation methods for pricing American options under the CIR process

In this paper, we focus on backward simulation of the CIR process. The purpose is to solve the memory requirement issue of the Least Squares Monte Carlo method when pricing American options by simulation. The concept of backward simulation is presented and it is classified into two types. Under the framework of the second type backward simulation, we seek the solutions for the existing CIR schemes. Specifically, we propose forward–backward simulation approaches for Alfonsi’s two implicit schemes, the fixed Euler schemes and the exact scheme. The proposed schemes are numerically tested and compared in pricing American options under the Heston model and the stochastic interest rate model. Some numerical properties such as the convergence order of the explicit–implicit Euler schemes, the storage requirement estimation of the forward–backward exact scheme and its computing time comparison with the squared Bessel bridge are also tested. Finally, the pros and cons of the related backward simulation schemes are summarized.     

Quasi-Monte Carlo methods with applications in finance

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.      

基于GARCH-EVT-Copula的社保基金投资组合风险测度研究

社保基金是社会保障事业健康发展的物质基础,安全性是其投资的首要原则。文章基于GARCH-EVT-Copula方法测度了社保基金投资组合的VaR。首先,基于GARCH、EVT对投资组合中各金融资产收益的边缘分布建模,然后,采用极大似然估计法和Bootstrap方法估计尾部的分布函数,接着,基于Copula方法研究组合中金融资产间的相关结构,最后,运用Monte Carlo方法测度投资组合的VaR。Kupiec检验表明,基于GARCH-EVT-Copula模型测度社保基金投资组合的风险是合适的。     

A Benchmark Approach to Portfolio Optimization under Partial Information

This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.      

Portfolio optimization in discrete time with proportional transaction costs under stochastic volatility

This paper is devoted to evaluating the optimal self-financing strategy and the optimal trading frequency for a portfolio with a risky asset and a risk-free asset. The objective is to maximize the expected future utility of the terminal wealth in a stochastic volatility setting, when transaction costs are incurred at each discrete trading time. A HARA utility function is used, allowing a simple approximation of the optimization problem, which is implementable forward in time. For each of various transaction cost rates, we find the optimal trading frequency, i.e. the one that attains the maximum of the expected utility at time zero. We study the relation between transaction cost rate and optimal trading frequency. The numerical method used is based on a stochastic volatility particle filtering algorithm, combined with a Monte-Carlo method. The filtering algorithm updates the estimate of the volatility distribution forward in time, as new stock observations arrive; these updates are used at each of these discrete times to compute the new portfolio allocation.     

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.