FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

小孔径超视距目标探测中的OFD-NLFM发射波形设计

针对小孔径超视距目标探测时阵列孔径减小空域滤波性能下降的问题,提出了一种正交频分非线性调频（OFD-NLFM）的发射波形设计方法。首先以正切调频函数为频率函数对发射信号进行建模,详细说明了影响脉压性能的正切函数时间副瓣电平控制因子的选择方法,重点提出一种基于凸优化的脉压信号峰值旁瓣抑制算法,建立了脉压输出噪声功率最小的优化模型并进行求解。仿真表明,提出的正交频分非线性调频信号具有较好的正交性,采用凸优化加权算法后脉压主瓣宽度比传统线性调频信号降低约1/3,峰值旁瓣为-31.22 dB,旁瓣电平平均值小于-100 dB,具备更低的抗噪声和干扰性能,从波形设计和脉压处理角度改善了空域滤波性能的不足。     

EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL‐REPRESENTATION FORMULA

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·) . Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual‐representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex‐duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρ^L(·) which is continuous from below, and if the European option H is essentially bounded.     

HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous-time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the “wealth process” is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model, we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state-price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state-price density for pricing contingent claims in this market. the mathematical tools are those of continuous-time martingales, convex analysis, functional analysis, and duality theory.     

凸函数与不等式

本文由凸函数的定义及图象导出某些重要不等式并利用微分知识给予证明,最后用导出的不等式证明另一些重要不等式。     

A simple measure of skewness

Groeneveld (1986) in discussing the skewness for the Weibull family has pointed out the shortcomings of the classical measures of asymmetry—the standardized third moment and the Pearson measure of skewness. He has shown that a modified form of the Pearson measure b₃= (μ-m)/E|X-m| portrays the skewness of Weibull family quite well. We give another competitive measure of skewness T that is easy to interpret and is based on conditional expectations. The proposed measure satisfies the desirable properties of a skewness measure.     

解析凸函数的两个重要不等式

与凸函数有关的不等式是基础数学理论的重要工具,尤其在不等式的证明中发挥的作用是无可替代的.其中Jensen不等式与Hadamard不等式更是起到重要作用.Jensen不等式通常用来证明有限不等式,它是将无穷项求和与积分联系起来的重要桥梁.利用Hadamard不等式可以对两个正数的几何平均数与算术平均数加细.     

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.     

Update rules for convex risk measures

In the first part of the paper we investigate the properties that describe the intertemporal structure of dynamic convex risk measures. The usual backward approach to dynamic risk assessment leads to strong and weak versions of time consistency. As an alternative, we introduce a forward approach of consecutivity. In the second part we discuss the problem of how to update a convex risk measure when new information arrives. We analyse to what extent the above properties are appropriate update criteria.     

Jensen’s Inequality in Finance

The purpose of this paper is to measure the size and the statistical significance of three inequalities in the field of financial economics. The three are variants of Jensen’s inequality. The first inequality is a comparison of the expected value of a ratio to the ratio of the expected value, a problem that arises in pricing foreign exchange rates. The second is a comparison of the log of the expected value to the expected value of the log, a problem that arises in testing forward market efficiency, money demand, production functions, and trade gravity models. The third is a comparison of the expected utility to the utility of the expected value, and helps in determining the importance of the expected utility paradigm, and the magnitude of the equity risk premium. The methodology used is by simulation of random normal variables, thereby introducing sampling error. Despite this sampling error the conclusion is general: all three inequalities are economically material, and stand statistically as inequalities. The major conclusion is that Jensen’s inequality is not a theoretical and superfluous exercise in finance as some have advocated.