The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.     

Valuation of cash flows under random rates of interest: A linear algebraic approach

This paper reformulates the classical problem of cash flow valuation under stochastic discount factors into a system of linear equations with random perturbations. Using convergence results, a sequence of uniform approximations is developed. The new formulation leads to a general framework for deriving approximate statistics of cash flows for a broad class of models of stochastic interest rate process. We show applications of the proposed method by pricing default-free and defaultable cash flows. The methodology developed in this paper is applicable to a variety of uncertain cash flow analysis problems.     

A stochastic programming model for funding single premium deferred annuities

Single Premium Deferred Annuities (SPDAs) are investment vehicles, offered to investors by insurance companies as a means of providing income past their retirement age. They are mirror images of insurance policies. However, the propensity of individuals to shift part, or all, of their investment into different annuities creates substantial uncertainties for the insurance company. In this paper we develop amultiperiod, dynamic stochastic program that deals with the problem of funding SPDA liabilities. The model recognizes explicitly the uncertainties inherent in this problem due to both interest rate volatility and the behavior of individual investors. Empirical results are presented with the use of the model for the funding of an SPDA liability stream using government bonds, mortgage-backed securities and derivative products. Research partially supported by NSF grants CCR-9104042 and SES-91-00216, and AFOSR grant 91-0168. Computing resources were made available by AHPCRC at the University of Minnesota, by NPAC at Syracuse University, New York, and by the GRASP Laboratory at Computer Science Department at University of Pennsylvania.     

Analysis of survivorship life insurance portfolios with stochastic rates of return

A general portfolio of survivorship life insurance contracts is studied in a stochastic rate of return environment with a dependent mortality model. Two methods are used to derive the first two moments of the prospective loss random variable. The first one is based on the individual loss random variables while the second one studies annual stochastic cash flows. The distribution function of the present value of future losses at a given valuation time is derived. For illustrative purposes, an AR(1) process is used to model the stochastic rates of return, and the future lifetimes of a couple are assumed to follow a copula model. The effects of the mortality dependence, the portfolio size and the policy type, as well as the impact of investment strategies on the riskiness of portfolios of survivorship life insurance policies are analyzed by means of moments and probability distributions.     

On pricing and reserving with-profits life insurance contracts

As a first approximation, asset and liability management issues faced by life insurance companies originate from the sale of with-profits contracts. These contracts are bond-type products with several rate guarantees and other interestsensitive embedded options. Benefits paid out to policyholders mostly depend on the investment performance of a given asset portfolio in which premiums are invested. Thus, guarantees and options granted to policyholders may become effective when the investment performance of the asset portfolio is poor. Issuing a with-profits contract is therefore not equivalent to issuing plain-vanilla debt. The purpose of this paper is to value with-profits liabilities in a consistent option-pricing framework and to develop efficient asset or liability strategies to manage profitability and variability of shareholder value.     

Valuation of equity-indexed annuity under stochastic mortality and interest rate

An equity-indexed annuity (EIA) contract offers a proportional participation in the return on a specified equity index, in addition to a guaranteed return on the single premium. In this paper, we discuss the valuation of equity-indexed annuities under stochastic mortality and interest rate which are assumed to be dependent on each other. Employing the method of change of measure, we present the pricing formulas in closed form for the most common product designs: the point-to-point and the annual reset. Finally, we conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.     

Polynomial diffusion models for life insurance liabilities

In this paper we study the pricing and hedging problem of a portfolio of life insurance products under the benchmark approach, where the reference market is modelled as driven by a state variable following a polynomial diffusion on a compact state space. Such a model can be used to guarantee not only the positivity of the OIS short rate and the mortality intensity, but also the possibility of approximating both pricing formula and hedging strategy of a large class of life insurance products by explicit formulas.     

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

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.     

A comparison of inventory policies for perishable goods

Past research on inventory management of perishables introduced models in which demand is sensitive to the age of the product. For such models, we prove that a fixed-order quantity policy is optimal under certain conditions and show that its expected cost is closer to optimal than that of the base-stock level policy when there is demand for units of all ages. We also show numerically when substituting older products to fulfill the demand for new (or vice versa) is beneficial.     

A systematic approach to pricing and hedging international derivatives with interest rate risk: analysis of international derivatives under stochastic interest rates

This paper deals with the valuation and the hedging of non-path-dependent European options on one or several underlying assets in a model of an international economy allowing for both, interest rate risk and exchange rate risk. Using martingale theory and, in particular, the change of numeraire technique we provide a unified and easily applicable approach to pricing and hedging exchange options on stocks, bonds, futures, interest rates and exchange rates. We also cover the pricing and hedging of compound exchange options.     

一类具有随机利率的跳扩散模型的期权定价

假定股票价格的跳过程为比Po isson过程更一般的跳过程一类特殊的更新过程,在风险中性的假设下,推导出了具有随机利率的跳扩散模型的欧式期权定价公式.从而推广了文[3]的结果.     

MIN-MAD life: A multi-period optimization model for life insurance company investment decisions

This paper presents a model, called the MIN-MAD Life Model, for managing the investments of a life insurance company over a multiperiod planning horizon. The MIN-MAD Life Model is a linear programming under uncertainty model based on Markowitz portfolio theory. Given the insurance company's current position and its forecasts of possible future developments with their associated probabilities, the model helps determine the set of efficient investment decisions over the planning horizon subject to market constraints and to the insurance company's legal and policy constraints. The senior executives of the life insurance company need examine only the set of efficient investment decisions to determine their optimal investment decisions.     

A survey of personalized treatment models for pricing strategies in insurance

We consider a model for price calculations based on three components: a fair premium; price loadings reflecting general expenses and solvency requirements; and profit. The first two components are typically evaluated on a yearly basis, while the third is viewed from a longer perspective. When considering the value of customers over a period of several years, and examining policy renewals and cross-selling in relation to price adjustments, many insurers may prefer to reduce their short-term benefits so as to focus on their most profitable customers and the long-term value. We show how models of personalized treatment learning can be used to select the policy holders that should be targeted in a company’s marketing strategies. An empirical application of the causal conditional inference tree method illustrates how best to implement a personalized cross-sell marketing campaign in this framework.     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

A competitive hub location and pricing problem

We formulate and solve a new hub location and pricing problem, describing a situation in which an existing transportation company operates a hub and spoke network, and a new company wants to enter into the same market, using an incomplete hub and spoke network. The entrant maximizes its profit by choosing the best hub locations and network topology and applying optimal pricing, considering that the existing company applies mill pricing. Customers’ behavior is modeled using a logit discrete choice model. We solve instances derived from the CAB dataset using a genetic algorithm and a closed expression for the optimal pricing. Our model confirms that, in competitive settings, seeking the largest market share is dominated by profit maximization. We also describe some conditions under which it is not convenient for the entrant to enter the market.     

Mslip: A computer code for the multistage stochastic linear programming problem

This paper describes an efficient implementation of a nested decomposition algorithm for the multistage stochastic linear programming problem. Many of the computational tricks developed for deterministic staircase problems are adapted to the stochastic setting and their effect on computation times is investigated. The computer code supports an arbitrary number of time periods and various types of random structures for the input data. Numerical results compare the performance of the algorithm to MINOS 5.0.     

A generalized pricing framework addressing correlated mortality and interest risks: a change of probability measure approach

Annuity-contingent derivatives involve both mortality and interest risks, which could have a correlation. In this article, we propose a generalized pricing framework in which the dependence between the two risks can be explicitly modelled. We also utilize the change of measure technique to simplify the valuation expressions. We illustrate our methodology in the valuation of a guaranteed annuity option (GAO). Using both forward measure associated with the bond price as numéraire and the newly introduced concept of endowment-risk-adjusted measure, we derive a simplified formula for the GAO price under the generalized framework. Numerical results show that the methodology proposed in this article is highly efficient and accurate.     

时间分数阶期权定价模型的一类有效差分方法

时间分数阶期权定价模型(时间分数阶Black-Scholes方程)数值解法的研究具有重要的理论意义和实际应用价值.对时间分数阶Black-Scholes方程构造了显-隐格式和隐-显差分格式,讨论了两类格式解的存在唯一性,稳定性和收敛性.理论分析证实,显-隐格式和隐-显格式均为无条件稳定和收敛的,两种格式具有相同的计算量.数值试验表明:显-隐和隐-显格式的计算精度与经典Crank-Nicolson(C-N)格式的计算精度相当,其计算效率(计算时间)比C-N格式提高30%.数值试验验证了理论分析,表明本文的显-隐和隐-显差分方法对求解时间分数阶期权定价模型是高效的,证实了时间分数阶Black-Scholes方程更符合实际金融市场.     

Reverse mortgage pricing and risk analysis allowing for idiosyncratic house price risk and longevity risk

Reverse mortgages provide an alternative source of funding for retirement income and health care costs. The two main risks that reverse mortgage providers face are house price risk and longevity risk. Recent real estate literature has shown that the idiosyncratic component of house price risk is large. We analyse the combined impact of house price risk and longevity risk on the pricing and risk profile of reverse mortgage loans in a stochastic multi-period model. The model incorporates a new hybrid hedonic–repeat-sales pricing model for houses with specific characteristics, as well as a stochastic mortality model for mortality improvements along the cohort direction (the Wills–Sherris model). Our results show that pricing based on an aggregate house price index does not accurately assess the risks underwritten by reverse mortgage lenders, and that failing to take into account cohort trends in mortality improvements substantially underestimates the longevity risk involved in reverse mortgage loans.