Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.     

Designing and pricing guarantee options in defined contribution pension plans

The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. However, for DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio and we develop a discrete model that selects the reference portfolio to minimize the cost of a guarantee. While the relation DB–DC is typically viewed as a binary one, the model shows how to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signaling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.     

The pricing of derivatives on assets with quadratic volatility

The basic model of financial economics is the Samuelson model of geometric Brownian motion because of the celebrated Black-Scholes formula for pricing the call option. The asset's volatility is a linear function of the asset value and the model guarantees positive asset prices. In this paper, it is shown that the pricing partial differential equation can be solved for level-dependent volatility which is a quadratic polynomial. If zero is attainable, both absorption and negative asset values are possible. Explicit formulae are derived for the call option: a generalization of the Black-Scholes formula for an asset whose volatiliy is affine, the formula for the Bachelier model with constant volatility, and new formulae in the case of quadratic volatility. The implied Black-Scholes volatilities of the Bachelier and the affine model are frowns, the quadratic specifications imply smiles.     

Option pricing under joint dynamics of interest rates, dividends, and stock prices

This paper proposes a unified framework for option pricing, which integrates the stochastic dynamics of interest rates, dividends, and stock prices under the transversality condition. Using the Vasicek model for the spot rate dynamics, I compare the framework with two existing option pricing models. The main implication is that the stochastic spot rate affects options not only directly but also via an endogenously determined dividend yield and return volatility; consequently, call prices can be decreasing with respect to interest rates.     

依赖时间参数的几种有固定执行价格的亚式期权定价

本文在假定标的资产模型依赖时间参数（即无风险利率,标的资产的期望收益率,波动率及红利率）,利用已建立的亚式期权定价模型,讨论了上限型期权、抵付型期权、双向型期权等,得到相应的期权定价解析公式．     

广义Black-Scholes模型期权定价新方法--保险精算方法

利用公平保费原则和价格过程的实际概率测度推广了Mogens Bladt和Tina Hviid Rydberg的结果．在无中间红利和有中间红利两种情况下，把Black-Scholes模型推广到无风险资产(债券或银行存款)具有时间相依的利率和风险资产(股票)也具有时间相依的连续复利预期收益率和波动率的情况，在此情况下获得了欧式期权的精确定价公式以及买权与卖权之间的平价关系．给出了风险资产(股票)具有随机连续复利预期收益率和随机波动率的广义Black-Scholes模型的期权定价的一般方法．利用保险精算方法给出了股票价格遵循广义Ornstein-Uhlenback过程模型的欧式期权的精确定价公式和买权和卖权之间的平价关系．     

An FFT Approach to Price Guaranteed Minimum Death Benefit in Variable Annuities under a Regime-Switching Model

This paper considers the valuation of guaranteed minimum death benefit in variable annuities under a regime-switching model. More specifically, the risk-free interest rate, the appreciation rate and the volatility of the reference investment fund are modulated by a continuous-time, finite-state, observable Markov chain. A regime-switching Esscher transform is adopted to select an equivalent martingale measure in the incomplete financial market. Inverse Fourier transform is used to derive an analytical pricing formula for the embedded option in variable annuity with guaranteed minimum death benefit. To calculate the fair guarantee charge, fast Fourier transform approach is applied. Numerical examples are provided to illustrate the practical implementation and the relationship between the fair guarantee charges and other parameters.     

认股权证的修正精算定价研究

基于修正Bladt和Rydberg在无市场假设下关于期权定价的保险精算方法的基础上,从评估实际损失和相应概率分布角度,利用公平保费原则建立认股权证的定价模型,并给出定价公式.且当投资者对原生资产期望回报率为无风险利率时,该定价为风险中性价格.     

Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Itô’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel’s principle. Our analysis is relevant to risk management applications since we derive an expression of the delta for the sensitivity analysis of the guarantee fees with respect to changes in the underlying fund value. We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.     

固定执行价格下回望看涨期权保险精算定价

利用保险精算方法,将期权定价问题转化为纯保费确定问题,根据股票价格过程的实际概率测度推导出了无风险利率为常数时,固定执行价格下回望看涨期权定价公式,验证了当标的资产的期望收益率等于无风险利率时,保险精算定价和风险中性定价的一致性.最后通过实例分析了保险精算价格和风险中性价格的差异,并利用Matlab编程得到了保险精算价格与标的资产期望收益率之间的关系.     

Valuing variable annuity guarantees with the multivariate Esscher transform

Variable annuities are usually sold with a range of guarantees that protect annuity holders from some downside market risk. Although it is common to see variable annuity guarantees written on multiple funds, existing pricing methods are, by and large, based on stochastic processes for one single asset only. In this article, we fill this gap by developing a multivariate valuation framework. First, we consider a multivariate regime-switching model for modeling returns on various assets at the same time. We then identify a risk-neutral probability measure for use with the model under consideration. This is accomplished by a multivariate extension of the regime-switching conditional Esscher transform. We further extend our results to the situation when the guarantee being valued is linked to equity indexes measured in foreign currencies. In particular, we derive a probability measure that is risk-neutral from the perspective of domestic investors. Finally, we illustrate our results with a hypothetical variable annuity guarantee.     

Valuing variable annuity guarantees with the multivariate Esscher transform

Variable annuities are usually sold with a range of guarantees that protect annuity holders from some downside market risk. Although it is common to see variable annuity guarantees written on multiple funds, existing pricing methods are, by and large, based on stochastic processes for one single asset only. In this article, we fill this gap by developing a multivariate valuation framework. First, we consider a multivariate regime-switching model for modeling returns on various assets at the same time. We then identify a risk-neutral probability measure for use with the model under consideration. This is accomplished by a multivariate extension of the regime-switching conditional Esscher transform. We further extend our results to the situation when the guarantee being valued is linked to equity indexes measured in foreign currencies. In particular, we derive a probability measure that is risk-neutral from the perspective of domestic investors. Finally, we illustrate our results with a hypothetical variable annuity guarantee.     

随机利率情形下跳-扩散模型的未定权益定价

讨论Vasicek短期利率模型下,风险资产的价格过程服从跳-扩散过程的欧式未定权益定价问题,利用鞅方法得到了欧式看涨期权和看跌期权定价公式及平价关系,最后给出了基于风险资产支付连续红利收益的欧式期权定价公式.     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

Pensions modelling

Valuable insights into the problem of how to fund defined benefitpension schemes can be obtained by analysis using the standardBlack–Scholes/Merton option pricing model, consideringthe pension fund finances jointly with those of the sponsoringcompany. The nature of the fund assets and liabilities is completelydifferent, and this lies behind current controversies aboutthe appropriate discount rate, valuation, financial accountingand preferential status for pension fund claimants in insolvency.     

Equilibrium pricing bounds on option prices

We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kinds of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. Explicit bounds are provided for the call option. Finally, we provide a numerical example.     

Pricing currency options under two-factor Markov-modulated stochastic volatility models

This article investigates the valuation of currency options when the dynamic of the spot Foreign Exchange (FX) rate is governed by a two-factor Markov-modulated stochastic volatility model, with the first stochastic volatility component driven by a lognormal diffusion process and the second independent stochastic volatility component driven by a continuous-time finite-state Markov chain model. The states of the Markov chain can be interpreted as the states of an economy. We employ the regime-switching Esscher transform to determine a martingale pricing measure for valuing currency options under the incomplete market setting. We consider the valuation of the European-style and American-style currency options. In the case of American options, we provide a decomposition result for the American option price into the sum of its European counterpart and the early exercise premium. Numerical results are included.     

Equity-linked pension schemes with guarantees

This paper analyses the relationship between the level of a return guarantee in an equity-linked pension scheme and the proportion of an investor’s contribution needed to finance this guarantee. Three types of schemes are considered: investment guarantee, contribution guarantee and surplus participation. The evaluation of each scheme involves pricing an Asian option, for which relatively tight upper and lower bounds can be calculated in a numerically efficient manner.We find a negative (and for two contract specifications also concave) relationship between the participation in the surplus return of the investment strategy and the guarantee level in terms of a minimum rate of return. Furthermore, the introduction of the possibility of early termination of the contract (e.g. due to the death of the investor) has no qualitative and very little quantitative impact on this relationship.     

国内外利率为随机的双币种重置型期权定价

双币种重置期权的特征是指在终端期T时的收益依赖于预先设定的t<,0>时刻标的资产的价格与执行价K>0(事先给定)的大小关系重新设置期权的执行价从而给出其定价,这种期权是投资于外国资产的一种合约,其风险不仅依赖外国资产价格的变化,还受外国货币的汇率以及国内外两种利率波动的影响,所以在实际应用方面十分广泛.本文首先就标的资...