Optimal Cross Hedging of Insurance Derivatives

Abstract

We consider insurance derivatives depending on an external physical risk process, for example, a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover, we check the optimal investment strategies for standard admissibility criteria. Finally, we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

A dynamic binomial expansion technique for credit risk measurement: a Bayesian filtering approach

Credit risk measurement and management are important and current issues in the modern finance world from both the theoretical and practical perspectives. There are two major schools of thought for credit risk analysis, namely the structural models based on the asset value model originally proposed by Merton and the intensity‐based reduced form models. One of the popular credit risk models used in practice is the Binomial Expansion Technique (BET) introduced by Moody's. However, its one‐period static nature and the independence assumption for credit entities' defaults are two shortcomings for the use of BET in practical situations. Davis and Lo provided elegant ways to ease the two shortcomings of BET with their default infection and dynamic continuous‐time intensity‐based approaches. This paper first proposes a discrete‐time dynamic extension to the BET in order to incorporate the time‐dependent and time‐varying behaviour of default probabilities for measuring the risk of a credit risky portfolio. In reality, the ‘true’ default probabilities are unobservable to credit analysts and traders. Here, the uncertainties of ‘true’ default probabilities are incorporated in the context of a dynamic Bayesian paradigm. Numerical studies of the proposed model are provided.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

Perpetual convertible bonds with credit risk

A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion, analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterised in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially, the continuation region may become a disconnected subset of the state space.     

Weber Correspondence Analysis: The One-Dimensional Case

When designing programs or software for the implementation of Monte Carlo (MC) hypothesis tests, we can save computation time by using sequential stopping boundaries. Such boundaries imply stopping resampling after relatively few replications if the early replications indicate a very large or a very small p value. We study a truncated sequential probability ratio test (SPRT) boundary and provide a tractable algorithm to implement it. We review two properties desired of any MC p value, the validity of the p value and a small resampling risk, where resampling risk is the probability that the accept/reject decision will be different than the decision from complete enumeration. We show how the algorithm can be used to calculate a valid p value and confidence intervals for any truncated SPRT boundary. We show that a class of SPRT boundaries is minimax with respect to resampling risk and recommend a truncated version of boundaries in that class by comparing their resampling risk (RR) to the RR of fixed boundaries with the same maximum resample size. We study the lack of validity of some simple estimators of p values and offer a new, simple valid p value for the recommended truncated SPRT boundary. We explore the use of these methods in a practical example and provide the MChtest R package to perform the methods.     

统计在生命科学中的实践与认识(Ⅲ)二 四格表的“物理”意义和数据处理(下)

上篇涉及两个率的比较、配对计数资料的显著性和对诊断方法的评价。本篇主要涉及重复检查、危险因素分析以及时空聚集性等。     

网络流在清理三角债问题中的应用

本文把清理三角债中两种优化数学模型问题，化成求解相应网络上最小费用流的问题，从而得到（强）多项式算法，并把另外的一种优化数学模型问题。化成线性规划问题.于是解答了文[3]中提出的清理三角债的三个基本问题．     

Copula理论在复合期权定价中的应用

本文使用风险中性评价方法分三部分计算了复合期权的价值,针对需要计算联合分布的第二部分,通过选取边缘分布为GARCH模型的二元正态Copula模型进行推理验证,结果求得的联合分布与使用风险中性评价方法的计算结果一致.进一步计算得到了时间相依的复合期权的价值,并且给出了使用Bayes时序诊断法和Z检验来诊断期权定价时其出现价格大的波动时的局部拐点的方法.     

Dual Representation of the Cost of Designing a Portfolio Satisfying Multiple Risk Constraints

ABSTRACT

We consider, within a Markovian complete financial market, the problem of finding the least expensive portfolio process meeting, at each payment date, three different types of risk criterion. Two of them encompass an expected utility-based measure and a quantile hedging constraint imposed at inception on all the future payment dates, while the other one is a quantile hedging constraint set at each payment date over the next one. The quantile risk measures are defined with respect to a stochastic benchmark and the expected utility-based constraint is applied to random payment dates. We explicit the Legendre-Fenchel transform of the pricing function. We also provide, for each quantile hedging problem, a backward dual algorithm allowing to compute their associated value function by backward recursion. The algorithms are illustrated with a numerical example.