A method for determining risk aversion functions from uncertain market prices of risk

In Gzyl and Mayoral (2008) we developed a technique to solve the following type of problems: How to determine a risk aversion function equivalent to pricing a risk with a load, or equivalent to pricing different risks by means of the same risk distortion function. The information on which the procedure is based consists of the market prices of the risk. Here we extend that method to cover the case in which there may be uncertainties in the market prices of the risks.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

Pricing longevity risk with the parametric bootstrap: A maximum entropy approach

In recent years, there has been significant development in the securitization of longevity risk. Various methods for pricing longevity risk have been proposed. In this paper we present an alternative pricing method, which is based on the maximization of the Shannon entropy in physics. Specifically, we propose implementing this pricing method with the parametric bootstrap (Brouhns et al., 2005), which is highly flexible and can be performed under different model assumptions. Through this pricing method we also quantify the impact of cohort effects and parameter uncertainty on prices of mortality-linked securities. Numerical illustrations based on longevity bonds with different maturities are provided.     

Maxentropic construction of risk neutral measures: discrete market models

The maximum entropy principle provides a variational method to select a measure yielding pre-assigned mean values to a random variable. It can also be invoked to construct measures that render a stochastic process a martingale, thus providing a systematic way of constructing risk-neutral measures and thus closing a market. We carry out this programme for discrete market models. On the one hand these are amenable to numerical implementation and on the other, they provide a stepping stone for more general market models in continuous time.     

A Family of Maximum Entropy Densities Matching Call Option Prices

Abstract

We investigate the position of the Buchen–Kelly density (Peter W. Buchen and Michael Kelly. The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis, 31(1), 143–159, March 1996.) in the family of entropy maximizing densities from Neri and Schneider (Maximum entropy distributions inferred from option portfolios on an asset. Finance and Stochastics, 16(2), 293–318, April 2012.), which all match European call option prices for a given maturity observed in the market. Using the Legendre transform, which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen–Kelly density and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen–Kelly density in the sense of relative entropy. The method presented here can also be used to interpolate between call option prices, and we compare it to a method proposed by Kahalé (An arbitrage-free interpolation of volatilities. Risk, 17(5), 102–106, May 2004). Orozco Rodriguez and Santosa (Estimation of asset distributions from option prices: Analysis and regularization. SIAM Journal on Financial Mathematics, 3(1), 374–401, 2012.) have produced examples in which the Buchen–Kelly algorithm becomes numerically unstable, and we use these as test cases to show that the algorithm given here remains stable and leads to good results.     

Risk measuring under liquidity risk

We present a general framework for measuring the liquidity risk. The theoretical framework defines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a security or a portfolio of securities cannot be sold or bought without causing changes in prices. The risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinate security are increasing monotonic and convex cash sub-additive on long positions. The contrary, in certain situations, holds for the sell positions. By using convex risk measures, we apply our framework to the situation in which large trades are broken into many small ones. Dual representation results are obtained for both positions in securities and portfolios. We give many examples of risk measures and derive for each of them the respective capital requirement. In particular, we discuss the VaR measure.     

On the principle of maximum entropy and the risk analysis of disaster loss

Disasters that occur everywhere in the most disordered way indicate that disaster entropy has reached the maximum value. Under given constraint conditions, when disaster entropy is the maximum value, the disaster loss series should follow P-III distribution. The occurrence interval of disaster loss refers to the average time interval that disaster loss of certain degree happens in the future. We could, according to the field disaster data and using P-III distribution function, calculate the value of future disaster loss with certain recurrence interval. Explicit in concept and easy to use, such a method has significant meaning in practice.     

区间二型模糊集效用函数和熵在风险决策中的应用

针对以区间二型模糊集(IT2FS)为信息环境的多属性决策(MADM)问题,引入IT2FS效用函数,并提出基于IT2FS效用函数,熵和风险因子的风险决策模型。首先基于截集思想提出两种IT2FS效用函数公式,有效提取了IT2FS全部信息,比以往的序值型公式更加科学有效。其次基于已提出的IT2FS三种不确定度量存在的问题提出三种新型不确定度量,并基于此三种不确定度量提出IT2FS熵公式弥补原有熵度量的不足。再次引入风险偏好因子反映决策者不同的风险态度,并改进风险偏好因子范围。构造基于效用函数,熵和风险偏好因子的风险决策模型。最后利用一个实例分析结果表明,该风险决策模型中决策者风险偏好对属性权重以及方案的排序存在影响,该决策思想对风险投资决策和风险管理决策均有一定的参考作用。     

不完全信息下损失风险的研究

在获得损失分布不完全信息情况下,提出用方差和熵共同度量损失风险的方法.在不完全信息条件下,通过最大熵原理在最不确定的情况下得到最大熵损失分布,并获得了损失分布的熵函数值.用熵值度量损失分布对于均匀分布的离散程度,从而度量概率波动带来的风险;用方差度量损失对于均值的离散程度,从而度量状态波动带来的风险.由于熵是与损失变量更高阶矩信息相联系的,所以新方法是从更全面的角度对损失风险的预测.通过算例,进一步看出在获得高阶矩信息下,熵参与风险度量的必要性.     

一类无约束离散Minimax问题的区间调节熵算法

In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C^1. The paper deals with this problem by means of taking the place of maximum-entropy function with adjustable entropy function. By constructing an interval extension of adjustable entropy function and some region deletion test rules, a new interval algorithm is presented. The relevant properties are proven, The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum-entropy algorithm. Both theoretical and numerical results show that the method is reliable and efficient.     

农作物单产的最大熵分布及在费率厘定上的应用

农作物单产分布的确定是农业保险中费率厘定的基础。本文引入最大熵原理,基于最大熵优化模型得出农作物单产的最大熵分布,并以此进行费率厘定。同时以辽宁省主要作物水稻、玉米、大豆和花生为例,确定了该四种农作物的费率,分别为4.45%、6.77%、6.34%、6.43%。结果表明:利用最大熵分布理论进行费率厘定不需要事先假定农作物单产分布的形式,而且考虑了更多作物单产分布的信息,为农业保险费率的合理精算提供一种新的可供选择的方法,有助于农业风险决策的科学化。     

调节熵函数法

１．引言 考虑如下极小极大问题这里ｆｉ（ｘ）是Ｒｎ中连续可微的函数,ｍ≥２是正整数（Ｐ）是一类比较典型的非光滑优化问题,是许多实际问题的数学模型．同时,线性规划的 Ｋａｒｍａｒｋａｒ标准型的对偶也是（Ｐ）的形式,光滑约束优化问题的一类重要罚函数法也是将问题化为类似（Ｐ）的形式．所以,如何有效地求解（Ｐ）,是一个重要问题．近些年发展起来的嫡函数法（或称凝聚函数法）是一种较新颖而实用的方法．它借助信息论中 Ｓｈａｎｎｏｎ熵的概念,推导出一族光滑的极大熵函数Ｆｐ（ｘ）,且Ｆｐ（ｘ）一致逼近要极小化的非光…     

Combining ranked mean value forecasts

A methodology is developed for combining mean value forecasts using not only all the important statistics related to the past performance and the dependence of the individual forecasts, but also a rank ordering of the individual forecasts representing the belief of a decision maker about the future performance of the forecasts. The maximum likelihood combination of the forecasts turns out to be weighted linear combination of the individual forecasts, where the weights are a function of the rank order of the forecasts, correlation coefficients between the forecasts, and relative entropy information measures between the individual forecasts and the actual values. These weights are assessed once in the most general case and once in a special case where the forecasts are normally distributed. The sensitivity of the weights is also investigated. A sample application of this method for predicting U.S. hog prices is also presented.     

非线性l_1问题的极大熵方法

本文给出求解非线性l1问题的极大熵方法．介绍了极大熵函数的性质,极大熵算法及其收敛性,最后给出一个算例。     

Modelling Specific Interest Rate Risk with Estimation of Missing Data

For the treatment of specific interest rate risk, a risk model is suggested, quantifying and combining both market and credit risk components consistently. The market risk model is based on credit spreads derived from traded bond prices. Though traded bond prices reveal a maximum amount of issuer specific information, illiquidity problems do not allow for classical parameter estimation in this context. To overcome this difficulty an efficient multiple imputation method is proposed that also quantifies the amount of risk associated with missing data. The credit risk component is based on event risk caused by correlated rating migrations of individual bonds using a Copula function approach.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

平衡规划问题的熵函数方法及其在混合交通流中的应用

将参变极值问题的极大熵函数方法应用到求解平衡规划问题中,通过先验分布信息和Kullback熵概念,给出了平衡规划问题基于Kullback熵表示的熵函数求解方法,并将平衡规划的极大熵函数方法应用于求解混合交通平衡分配问题．     

Decision risk analysis for an interval TOPSIS method

TOPSIS is a multi-attribute decision making (MADM) technique for ranking and selection of a number of externally determined alternatives through distance measures. When the collected data for each criterion is interval and the risk attitude for a decision maker is unknown, we present a new TOPSIS method for normalizing the collected data and ranking the alternatives. The results show that the decision maker with different risk attitude ranks the different alternatives.     

Option pricing in incomplete discrete markets

Various methods of option pricing in discrete time models are discussed. The classical risk minimization method often results in negative prices and a natural modification is proposed. Another method of risk minimization using an inductive procedure as in the Cox-Ross-Rubinstein model is also proposed. The definition of the risk interpreted as the maximum of possible loss is discussed.     

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.