Efficient option risk measurement with reduced model risk

Options require risk measurement that is also computationally efficient as it is important to derivatives risk management. There are currently few methods that are specifically adapted for efficient option risk measurement. Moreover, current methods rely on series approximations and incur significant model risks, which inhibit their applicability for risk management.In this paper we propose a new approach to computationally efficient option risk measurement, using the idea of a replicating portfolio and coherent risk measurement. We find our approach to option risk measurement provides fast computation by practically eliminating nonlinear computational operations. We reduce model risk by eliminating calibration and implementation risks by using mostly observable data, we remove internal model risk for complex option portfolios by not admitting arbitrage opportunities, we are also able to incorporate liquidity or model misspecification risks. Additionally, our method enables tractable and convex optimisation of portfolios containing multiple options. We conduct numerical experiments to test our new approach and they validate it over a range of option pricing parameters.     

Risk measurement in the presence of background risk

A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolio’s re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measure’s aggregation properties are obtained.     

Fuzzy portfolio model with different investor risk attitudes

We propose a fuzzy portfolio model designed for efficient portfolio selection with respect to uncertain or vague returns. Although many researchers have studied the fuzzy portfolio model, no researcher has yet attempted a behavioral analysis of the investor in the fuzzy portfolio model. To address this problem, we examined investor risk attitudes—risk-averse, risk-neutral, or risk-seeking behaviors—to discover an efficient method for fuzzy portfolio selection. In this study, we relied on the advantages of possibilistic mean–standard deviation models that we believed would fit the risk attitudes of investors. Thus, we developed a fuzzy portfolio model that focuses on different investor risk attitudes so that fuzzy portfolio selection for investors who possess different risk attitudes can be achieved more easily. Finally, we presented a numerical example of a portfolio selection problem to illustrate ways to address problems presented by a variety of investor risk attitudes.     

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.     

Sample path large and moderate deviations for risk model with delayed claims

Sample path large and moderate deviation principles for Markov modulated risk models with delayed claims are proved by the exponential martingale method. As applications, asymptotic estimates and exponential bounds of the ruin probability are also studied.     

n-Monotone exact functionals

We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals.     

The optimal portfolio problem with coherent risk measure constraints

One of the basic problems of applied finance is the optimal selection of stocks, with the aim of maximizing future returns and constraining risks by an appropriate measure. Here, the problem is formulated by finding the portfolio that maximizes the expected return, with risks constrained by the worst conditional expectation. This model is a straightforward extension of the classic Markovitz mean–variance approach, where the original risk measure, variance, is replaced by the worst conditional expectation.The worst conditional expectation with a threshold α of a risk X, in brief WCE_α(X), is a function that belongs to the class of coherent risk measures. These are measures that satisfy a set of properties, such as subadditivity and monotonicity, that are introduced to prevent some of the drawbacks that affect some other common measures.This paper shows that the optimal portfolio selection problem can be formulated as a linear programming instance, but with an exponential number of constraints. It can be solved efficiently by an appropriate generation constraint subroutine, so that only a small number of inequalities are actually needed.This method is applied to the optimal selection of stocks in the Italian financial market and some computational results suggest that the optimal portfolios are better than the market index.     

Omega diffusion risk model with surplus-dependent tax and capital injections

In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). The study of this new risk model is closely related to the Azéma–Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma–Yor process below a constant level until the first passage time of another Azéma–Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of Li and Zhou (2013) and Zhang (2015). We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma–Yor process.     

Consistent modeling of risk averse behavior with spectral risk measures

This paper clarifies the relation between decisions of a risk-averse decision maker, based on expected utility theory on the one hand, and spectral risk measures on the other.     

On the Markov-dependent risk model with tax

In this paper we consider the Markov-dependent risk model with tax payments in which the claim occurrence, the claim amount as well as the tax rate are controlled by an irreducible discrete-time Markov chainSystems of integro-differential equations satisfied by the expected discounted tax payments and the non-ruin probability in terms of the ruin probabilities under the Markov-dependent risk model without tax are establishedThe analytical solutions of the systems of integro-differential equations are also obtained by the iteration method.     

A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.     

A multicriteria model for risk sorting of natural gas pipelines based on ELECTRE TRI integrating Utility Theory

This paper proposes a multicriteria model for assessing risk in natural gas pipelines, and for classifying sections of pipeline into risk categories. The model integrates Utility Theory and the ELECTRE TRI method. It aims to help transmission and distribution companies, when engaged in risk management and decision-making, to consider the multiple dimensions of risk that may arise from pipeline accidents. Pipeline hazard scenarios are presented, and it is argued that the assessment of risk in natural gas pipelines should not be based solely on probabilities of human fatalities, but should involve a wider perspective that simultaneously takes into consideration the human, environmental and financial dimensions of impacts of pipeline accidents. Finally, in order to verify the effectiveness of the model set out, a numerical application based on a real case study is presented.     

The asymptotic validity of invariant procedures for the repeated measures model and multivariate linear model

Fairly general sufficient conditions are given to guarantee that invariant tests about means in the multivariate linear model and the repeated measures model have the correct asymptotic size when the normal assumption under which the tests are derived is relaxed. These conditions are the same as Huber's condition which guarantees asymptotic validity of the size of the F-test for the univariate linear model.     

On a risk model with claim investigation

In this paper, a queue-based claims investigation mechanism is considered to model an insurer’s claim processing practices. The resulting risk model may be viewed as a first step in developing models with more realistic claim investigation mechanisms. Related to claim investigations, claim settlement delays and time dependent payments have been studied in a ruin context by, e.g. Taylor (1979), Cai and Dickson (2002), and Trufin et al. (2011). However, little has been done on queue-based investigation mechanisms. We first demonstrate the impact of a particular claim investigation system on some common ruin-related quantities when claims arrive according to a compound Poisson process, and investigation times are of a combination of exponential form. Probabilistic interpretations for the defective renewal equation components are also provided. Finally, via numerical examples, we explore various risk management questions related to this problem such as how claim investigation strategies can help an insurer control its activities within its risk appetite.     

A continuous covering location model with risk consideration

This paper proposes a continuous covering location model with risk consideration. The investigated model is an extension of the discrete covering location models in continuous space. The objective function consists of installation and risk costs. Because of uncertain covering radius, customer satisfaction degree of covering radius is introduced by fuzzy concept. Since, the uncertainty may cause risk of uncovering customers; the risk cost is added to the objective function. The installation cost is assigned to a zone with a predetermined radius from its center. The model is solved by a fuzzy method named α$\alpha$-cut. After solving the model based on different α$\alpha$-values, the zones with the largest possibilities are determined for locating new facilities and the best locations are calculated based on the obtained possibilities. Then, the model is solved to determine the best covering values. This paper, also introduces a risk analysis method based on Response Surface Methodology (RSM) to consider risk management in the location models. Finally, a numerical example is expressed to illustrate the proposed model.     

Coherent and convex risk measures for portfolios with applications

In this paper, we propose a framework of risk measures for portfolio vectors, which is an extension of the ones introduced by Burgert and Rüschendorf (2006) and Rüschendorf (2013). Representation results for coherent and convex risk measures for portfolio vectors are provided. Applications to the multi-period risk measures are also given.     

Credit risk model with contagious default dependencies affected by macro-economic condition

We consider a credit risk model with two industrial sectors, where defaults of corporations would be influenced by two factors. The first factor represents the macro economic condition which would affect the default intensities of the two industrial sectors differently. The second factor reflects the influences of the past defaults of corporations against other active corporations, where such influences would affect the two industrial sectors differently. A two-layer Markov chain model is developed, where the macro economic condition is described as a birth-death process, while another Markov chain represents the stochastic characteristics of defaults with default intensities dependent on the state of the birth-death process and the number of defaults in two sectors. Although the state space of the two-layer Markov chain is huge, the fundamental absorbing process with a reasonable state space size could capture the first passage time structure of the two-layer Markov chain, thereby enabling one to evaluate the joint probability of the number of defaults in two sectors via the uniformization procedure of Keilson. This in turn enables one to value a variety of derivatives defined on the underlying credit portfolios. In this paper, we focus on a financial product called CDO, and a related option.     

带干扰的理赔为一般到达的风险模型

本文应用M arkov骨架过程方法,研究了带干扰的理赔为一般到达的保险风险模型,得到了破产时间与破产时刻前后资产盈余的联合分布以及破产时间的分布.     

Constant dividend barrier in a risk model with interclaim-dependent claim sizes

The risk model with interclaim-dependent claim sizes proposed by Boudreault et al. [Boudreault, M., Cossette, H., Landriault, D., Marceau, E., 2006. On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actur. J., 265-285] is studied in the presence of a constant dividend barrier. An integro-differential equation for some Gerber-Shiu discounted penalty functions is derived. We show that its solution can be expressed as the solution to the Gerber-Shiu discounted penalty function in the same risk model with the absence of a barrier and a combination of two linearly independent solutions to the associated homogeneous integro-differential equation. Finally, we analyze the expected present value of dividend payments before ruin in the same class of risk models. An homogeneous integro-differential equation is derived and then solved. Its solution can be expressed as a different combination of the two fundamental solutions to the homogeneous integro-differential equation associated to the Gerber-Shiu discounted penalty function.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.