Asymptotic analysis of option pricing in a Markov modulated market

We address asymptotic analysis of option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of the stocks depend on a finite state Markov chain. We study two variations of the chain namely, when the chain is moving very fast compared to the underlying asset price and when it is moving very slow. Using quadratic hedging and asymptotic expansion, we derive corrections on the locally risk minimizing option price.     

Importance sampling for integrated market and credit portfolio models

A sophisticated approach for computing the total economic capital needed for various stochastically dependent risk types is the bottom-up approach. In this approach, usually, market and credit risks of financial instruments are modeled simultaneously. As integrating market risk factors into standard credit portfolio models increases the computational burden of calculating risk measures, it is analyzed to which extent importance sampling techniques previously developed either for pure market portfolio models or for pure credit portfolio models can be successfully applied to integrated market and credit portfolio models. Specific problems which arise in this context are discussed. The effectiveness of these techniques is tested by numerical experiments for linear and non-linear portfolios.     

Dynamic pricing and warranty policies for products with fixed lifetime

We consider a repairable product with known market entry and departure times. A warranty policy is offered with product purchase, under which a customer can have a failed item repaired free of charge in the warranty period. It is assumed that customers are heterogeneous in their risk attitudes toward uncertain repair costs incurred after the warranty expires. The objective is to determine a joint dynamic pricing and warranty policy for the lifetime of the product, which maximizes the manufacturer’s expected profit. In the first part of the analysis, we consider a linearly decreasing price function and a constant warranty length. We first study customers’ purchase patterns under several different pricing strategies by the manufacturer and then discuss the optimal pricing and warranty strategy. In the second part, we assume that the warranty length can be altered once during the product lifetime in developing a joint pricing and warranty policy. Numerical studies show that a dynamic warranty policy can significantly outperform a fixed-length warranty policy.     

Affordable and adequate annuities with stable payouts: Fantasy or reality?

This paper introduces a class of unit-linked annuities that extends existing annuities by allowing portfolio shocks to be gradually absorbed into the annuity payouts. Consequently, our new class enables insurers to offer an affordable and adequate annuity with a stable payout stream. We show how to price and adequately hedge the annuity payouts in a general financial environment. In particular, our model accounts for various stylized facts of stock returns such as asymmetry and heavy-tailedness. Furthermore, the generality of our framework makes it possible to explore the impact of a parameter misspecification on the annuity price and the hedging performance.     

Risk externalities and too big to fail

This paper establishes the case for a fallacy of economies of scale in large aggregate institutions and the effects of scale risks. The problem of rogue trading and excessive risk taking is taken as a case example. Assuming (conservatively) that a firm exposure and losses are limited to its capital while external losses are unbounded, we establish a condition for a firm not to be allowed to be too big to fail. In such a case, the expected external losses second derivative with respect to the firm capital at risk is positive. Examples and analytical results are obtained based on simplifying assumptions and focusing exclusively on the risk externalities that firms too big to fail can have.     

Data breaches: Goodness of fit,pricing, and risk measurement

Some research on cyber risk has been conducted in the field of information technology, but virtually no research exists in the actuarial domain. As a first step toward a more profound actuarial discussion, we use multidimensional scaling and goodness-of-fit tests to analyze the distribution of data breach information. Our results show that different types of data breaches need to be modeled as distinct risk categories. For severity modeling, the log-skew-normal distribution provides promising results. The findings add to the recent discussion on the use of skewed distributions in actuarial modeling (Vernic, 2006; Bolancé et al., 2008; Eling, 2012). Moreover, they provide useful insights for actuaries working on the implementation of cyber insurance policies. We illustrate the usefulness of our results in two applications on risk measurement and pricing.     

The credit risk+ model with general sector correlations

We consider an enhancement of the credit risk⁺ model to incorporate correlations between sectors. We model the sector default rates as linear combinations of a common set of independent variables that represent macro-economic variables or risk factors. We also derive the formula for exact VaR contributions at the obligor level.     

Fourier Inversion Formulas in Option Pricing and Insurance

Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max (S – K, 0) also arises in excess-of-loss or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for payoffs which occur in insurance. Numerical examples are provided.      

永泰县城峰镇旗山小区后山滑坡现场应急调查及处置分析

2010年5月20日,受连续降雨影响,福建省永泰县城峰镇旗山小区后山发现有滑坡地质灾害隐患,后山中下部产生多条35~80m长拉裂缝,坡脚崩塌不断,潜在滑坡规模约8×10⁴m³,存在继续下滑的危险,情况十分危急,威胁坡脚居民2565人的生命和约5.38亿元财产。本文在对灾害现场进行详细地质调查的基础上,结合现场测绘、监测等手段,对该滑坡体的基本特征进行了较深入的调查研究,对滑坡发生及成灾原因进行了初步分析。结果表明,滑坡区地形条件及岩土体特征是滑坡发生的基本条件,连续降雨的饱水加载作用以及雨水沿节理裂隙结构面的下渗软化作用是诱发滑坡发生的直接原因。同时,根据滑坡险情特征,提出了应处置建议及应急卸载措施。最后,根据监测结果信息化指导应急卸载工作,确保了施工安全及受灾群众的生命财产安全。本次成功排险,对指导福建海沿地区地质灾害应急处置工作提供了科学依据。     

Arbitrage valuation and bounds for sinking-fund bonds with multiple sinking-fund dates

The paper tackles the problem of pricing, under interest-rate risk, a default-free sinking-fund bond which allows its issuer to recurrently retire part of the issue by (a) a lottery call at par, or (b) an open market repurchase. By directly modelling zero-coupon bonds as diffusions driven by a single-dimensional Brownian motion, a pricing formula is supplied for the sinking-fund bond based on a backward induction procedure which exploits, at each step, the martingale approach to the valuation of contingent-claims. With more than one sinking-fund date, however, the pricing formula is not in closed form, not even for simple parametrizations of the process for zerocoupon bonds, so that a numerical approach is needed. Since the computational complexity increases exponentially with the number of sinking-fund dates, arbitrage-based lower and upper bounds are provided for the sinking-fund bond price. The computation of these bounds is almost effortless when zero-coupon bonds are as described by Cox, Ingersoll and Ross. Numerical comparisons between the price of the sinking-fund bond obtained via Monte Carlo simulation and these lower and upper bounds are illustrated for different choices of parameters.