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Pricing equity-linked guaranteed minimum death benefits with surrender risk by complex Fourier series expansion method
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In this paper, we consider an efficient method to price the equity index annuity with return guarantees are two class cliquet-styles, which are popular in the insurance market. In order to describe persistent changes of the economic state in the financial market, we assume that the underlying index return process is a regime-switching Lévy model and the model parameters rely on a continuous-time Markov chain with finite economy states. The complex Fourier series approach was used to give the explicit formulae for the price of equity index annuity. The error analysis proves that the complex Fourier series approach can achieve an exponential convergence rate and the experiment results demonstrate the efficiency and accuracy of this method.
In this paper, we analyze a form of equity-linked Guaranteed Minimum Death Benefit (GMDB), whose payoff depends on a dollar cost averaging (DCA) style periodic investment in the risky index, with rider premiums paid at regular intervals. This rider is a very natural insurance vehicle for equity-linked variable annuities, and the DCA feature has a tendency to reduce the uncertainty associated with the final payoff, beyond the minimum benefit guaranteed to the beneficiary upon death of the insured. This makes the insurer risk easier to manage as the contract ages. From the policyholder's perspective , the protection is cheaper than a standard GMDB whose payoff depends solely on the risky index value upon death, but still offers the upside potential from an investment made at regular intervals. We derive closed-form valuation formulas under the fairly broad class of exponential Lévy models for the risky index, which includes Black-Scholes as a special case. Closed-form valuation is provided by a fundamental link between this contract and a series of Asian options, for which valuation is well established. We provide several several valuation strategies, and demonstrate the soundness of the framework.
Since 2016, Solvency II introduced in the insurance industry sector new capital requirements rules for the evaluation of life insurance liabilities based on VaR risk measure. This paper aims to analyze the effect of the dynamic policyholder behavior on the evaluation of lapse risk of a portfolio of participating life insurance policies.
Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unifiedapproach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull-White, Stein-Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.
The actuarial pricing of mortality insurance contracts including the withdrawal cause of decrement is revisited. Based on the discrete-time Markov chain model of life insurance, we decompose the stochastic underwriting gain in investment, insurance and expense risk components. In this framework, a premium tariff is called actuarial fair provided the expected value of the gross insurance risk component, defined as the sum of its insurance and expense risk components, vanishes at each discrete time of valuation. We show that the actuarial fair pricing conditions can be achieved in two different ways according to whether the original tariff uses a single or double decrement mortality table. The method allows to recover on the one side the withdrawal benefits and on the other side it provides a new discrete time justification of the Zillmer reserve as withdrawal benefit.
This article presents evidence in favour of time-varying Markov regime-Switching (MS) properties in all shares stock returns in the USA. The model specifications include the US Institute for Supply Management's (ISM) manufacturing and Nonmanufacturing Business Activity Index (NMBAI) in the transition equations. We find that the developments in the ISM manufacturing index affect the regime-switching probabilities in both bull and bear stock market periods. The business activity in nonmanufacturing sectors, on the other hand, has a bearing only on bull market periods. We also test for the possibility of a common factor influencing both stock returns and business confidence in the manufacturing sector by estimating a time-varying MS model with the US industrial production in the transition equation. We find that the null hypothesis of a fixed transition probability MS model cannot be rejected when the US industrial production index is included in the transition equation of a time-varying MS model. We conclude that the information content in the ISM manufacturing confidence index, such as expectational shifts, has a separate influence on the stock market regimes over and above that of actual developments in industrial production.
In this paper, we consider equity-linked life insurance con- tracts that give their holder the possibility to surrender their policy before maturity. Such contracts can be val- ued using simulation methods proposed for the pricing of American options, but the mortality risk must also be taken into account when pricing such contracts. Here, we use the least-squares Monte Carlo approach of Longstaff and Schwartz coupled with quasi-Monte Carlo sampling and a control variate in order to construct efficient estimators for the value of such contracts. We also show how to incorpo- rate the mortality risk into these pricing algorithms without explicitly simulating it.
Typescript. Thesis (Ph. D.)--University of Iowa, 2003. Includes bibliographical references (leaves 156-166).
This work studies the valuation and optimal surrender of variable (equity-linked) annuities under a Lévy-driven equity market with mortality risk. We consider a practical periodic fee structure which can vary over time and is assessed as a proportion of the fund value. At maturity, the fund value is returned to the policyholder according to a guaranteed minimum accumulation benefit (GMAB). Mortality risk is also modeled discretely, and the contract offers a guaranteed minimum death benefit (GMBD) prior to maturity. The benefits accommodate caps on the growth of funds (in addition to the rising floor) to reduce the fee level and as a disincentive to early surrender. Interest rates are modeled via a deterministic discounting term structure, which can be calibrated (bootstrapped) to the rates market, according to market convention. An efficient and accurate valuation framework is developed, along with closed form pricing formulas in the case where policy surrender is not permitted. Numerous experiments are conducted to illustrate the interplay between contract parameters and the decision to surrender, and we provide an extensive analysis that investigates how to structure contracts to disincentivize early surrender.
This paper presents a flexible valuation approach for variable annuity (VA) contracts embedded with guaranteed minimum maturity benefit (GMMB) riders written on an underlying fund that evolves according to a general regime-switching framework. Unlike the classical regime-switching models which only allow model parameters to change upon regime switches, our framework allows, more importantly, model structures to vary. With mild assumptions on the characteristic function of the log-stock price, our model settings enable the study of fundamental features of the market dynamics, such as stochastic volatility and jumps, on the underlying fund value of GMMB in a unified framework. This novel idea is illustrated by a three-regime model whose environments can be characterised by either the geometric Brownian motion process, the Heston (1993) stochastic volatility process, or double exponential process. Three versions of the GMMB riders are considered; a fixed or roll-up guarantee, a geometric average guarantee and a ratchet guarantee. With the Fourier Cosine (COS) method which utilises characteristic functions, explicit valuation expressions for various contracts are derived, and numerical illustrations are performed to analyse the efficiency of the approach in terms of computational speed and accuracy. The paper makes a unique contribution by presenting regime-dependent bounds and an algorithm for determining the optimal grid points required for the COS method to achieve a specific level of accuracy. Numerical experiments for the valuation framework reveal that i) as the likelihood of regime shifts increases, the price difference of VA contracts with different initial regimes diminishes, which is consistent with financial intuition; ii) the generalised regime-switching models are more suitable for VA pricing when the time-to-maturity spans over several decades, during which structural changes, although infrequent, are inevitable.
In this paper, we consider the valuation problem of equity-linked annuity with guaranteed minimum death benefit (GMDB) by complex Fourier series method under regime-switching jump diffusion models. We show that the price formulas can be expressed by some discounted density functions associated with the residual lifetime random variable. Fourier transforms for discounted density functions are derived, and complex Fourier series expansion method is applied to recover the discounted density functions. Some explicit formulas for computing the GMDB price are given. Finally, we give some numerical results to show effectiveness of our method.
Highly accurate approximation pricing formulae and option Greeks are obtained for European-type options using a complex Fourier series. We assume that risky assets are driven by exponential Lévy processes and affine stochastic volatility models. We provide a succinct error analysis to demonstrate that we can achieve an exponential convergence rate in the pricing method in many cases as long as we choose the correct truncated computational interval. As a novel pricing method, we also numerically demonstrate that the complex Fourier series performs either favourably or comparably with existing techniques in numerical experiments.
We introduce a new numerical method called the complex Fourier series (CFS)method proposed by Chan (2017)to price options with an early-exercise feature—American, Bermudan and discretely monitored barrier options—under exponential Lévy asset dynamics. This new method allows us to quickly and accurately compute the values of early-exercise options and their Greeks. We also provide an error analysis to demonstrate that, in many cases, we can achieve an exponential convergence rate in the pricing method as long as we choose the correct truncated computational interval. Our numerical analysis indicates that the CFS method is computationally more comparable or favourable than the methods currently available. Finally, the superiority of the CFS method is illustrated with real financial data by considering Standard & Poor's depositary receipts (SPDR)exchange-traded fund (ETF)on the S&P 500® index options, which are American options traded from November 2017 to February 2018 and from 30 January 2019 to 21 June 2019.
We study the risk-neutral valuation of participating life insurance policies with surrender guarantees when an early default mechanism, forcing an insurance company to be liquidated once a solvency threshold is reached, is imposed by a regulator. The early default regulation affects the policies’ value not only directly via changing the policies’ payment stream but also indirectly via influencing policyholder's surrender. In this paper, we endogenize surrender risk by assuming a representative policyholder's surrender intensity bounded from below and from above and uncover the impact of the regulation on the policyholder's surrender decision making. A partial differential equation is derived to characterize the price of a participating policy and solved with the finite difference method. We discuss the impacts of the early default regulation and insurance company's reaction to the regulation in terms of its investment strategy on the policyholder's surrender as well as on the contract value, which depend on the policyholder's rationality level.
This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.
This article presents a numerical method of pricing the surrender risk in Ratchet equity-index annuities (EIAs). We assume that log-returns of the underlying fund belong to a class of regime-switching models where the parameters are allowed to change randomly according to a hidden Markov chain. The defining feature of these models is the fact that in each regime the characteristic function of log-returns is assumed to have an analytical form. The presented method provides an unified pricing framework within this class and includes the recently developed COS method as a particular case. This aspect of the method is particularly useful when pricing Ratchet options embedded in EIAs, for which the COS method exhibits a low rate of convergence. Our numerical results confirm that for models considered in this article the proposed approach improves convergence of the COS method without increasing the computational burden.
The paper analyzes one of the most common life insurance products — the so-called participating (or with profits) policy. This type of contract stands in contrast to unit-linked (UL) products in that interest is credited to the policy periodically according to some mechanism which smoothes past returns on the life insurance company’s (LIC) assets. As is the case for UL products, the participating policies are typically equipped with an interest rate guarantee and possibly also an option to surrender (sell-back) the policy to the LIC before maturity. The paper shows that the typical participating policy can be decomposed into a risk free bond element, a bonus option, and a surrender option. A dynamic model is constructed in which these elements can be valued separately using contingent claims analysis. The impact of various bonus policies and various levels of the guaranteed interest rate is analyzed numerically. We find that values of participating policies are highly sensitive to the bonus policy, that surrender options can be quite valuable, and that LIC solvency can be quickly jeopardized if earning opportunities deteriorate in a situation where bonus reserves are low and promised returns are high. ©2000 Elsevier Science B.V. All rights reserved.
After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull-White, Stein-Stein, α-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.
In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox-Ingersoll-Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.
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 for the sensitivity of the guarantee fees with respect to changes in the underlying fund value (called the " delta "). 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.
This paper is concerned with the valuation of equity-linked annuities with mortality risk under a double regime-switching model, which provides a way to endogenously determine the regime-switching risk. The model parameters and the reference investment fund price level are modulated by a continuous-time, finite-time, observable Markov chain. In particular, the risk-free interest rate, the appreciation rate, the volatility and the martingale describing the jump component of the reference investment fund are related to the modulating Markov chain. Two approaches, namely, the regime-switching Esscher transform and the minimal martingale measure, are used to select pricing kernels for the fair valuation. Analytical pricing formulae for the embedded options underlying these products are derived using the inverse Fourier transform. The fast Fourier transform approach is then used to numerically evaluate the embedded options. Numerical examples are provided to illustrate our approach.
We investigate equity-linked investment products with a threshold expense strategy, under which an insurance company will collect expenses continuously from the policyholder’s account only when the account value is lower than a pre-specified level. The logarithmic value of the policyholder’s account, before deducting any fees, is described by a jump diffusion process which is independent of the time-to-death random variable. The distribution of the time-to-death random variable is approximated by a combination of exponential distributions, which are dense in the space of density functions on . We characterize the Laplace transform of the distribution of a general refracted jump diffusion process through some integro-differential equations. Besides, the distribution of a refracted double exponential jump diffusion process at an independent exponential random variable is derived, from which closed-form formulas to evaluate the total expenses and the fair fee rates are obtained. Finally, we illustrate our results by some numerical examples.
In this article, we consider the problem of computing the expected discounted value of a death benefit, e.g. in Gerber
et al.
(2012, 2013), in a regime-switching economy. Contrary to their proposed discounted density approach, we adopt the Laplace transform to value the contingent options. By this alternative approach, closed-form expressions for the Laplace transforms of the values of various contingent options, such as call/put options, lookback options, barrier options, dynamic fund protection and the dynamic withdrawal benefits, have been obtained. The value of each contingent option can then be recovered by the numerical Laplace inversion algorithm, and this efficient approach is documented by several numerical illustrations. The strength of our methodology becomes apparent when we tackle the valuations of exotic contingent options in the cases when (1) the contracts have a finite expiry date; (2) when the time-until-death variable is uniformly distributed in accordance with De Moivre's law.
This paper tests for nonlinear effects of asset prices on the US fiscal policy. By modeling government spending and taxes as time-varying transition probability Markovian processes (TVPMS), we find that taxes significantly adjust in a nonlinear fashion to asset prices. In particular, taxes respond to housing and (to a smaller extent) to stock price changes during normal times. However, at periods characterized by high financial volatility, government taxation only counteracts stock market developments (and not the dynamics of the housing sector). As for government spending, it is neutral vis-a-vis the asset market cycles. We conclude that, correcting the fiscal balance and, notably, the revenue side for time-varying effects of asset prices provides a more accurate assessment of the fiscal stance and its sustainability.
In this article we propose a lattice algorithm for pricing simple Ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR++ stochastic interest rates. In addition we present an asymptotic expansion technique that permits us to obtain a first-order approximation formula for the price of simple Ratchet EIAs without early surrender risk and without a global minimum contract value. Numerical comparisons show the reliability of the proposed methods.
This study extends the endogenous Markov-switching model of Kim, Piger, and Startz’ (2008) to a general state-space model. It also complements Kim's (1994) regime switching dynamic linear model by allowing the discrete regime to be jointly determined with observed or unobserved continuous state variables. The estima- tion framework involves a Bayesian Markov chain Monte Carlo (MCMC) scheme to simulate the latent state variable that controls the regime shifts. A simula- tion exercise shows that neglecting endogeneity leads to biased inference. This method is then applied to the dynamic Nelson-Siegel yield curve model where the unobserved time-varying level, slope and curvature factors are contemporaneously correlated with the Markov switching volatility regimes. The estimation results indicate that the high volatility tends to be associated with positive innovations in the level and slope factors. More importantly, we find that the endogenous regime-switching dynamic Nelson-Siegel model outperforms the model with and without exogenous regime-switching in terms of out-of-sample prediction accuracy. (JEL classification: C11; C51; C58) This article is protected by copyright. All rights reserved.
I review the burgeoning literature on applications of Markov regime switching models in empirical finance. In particular, distinct attention is devoted to the ability of Markov Switching models to fit the data, filter unknown regimes and states on the basis of the data, to allow a powerful tool to test hypothesesformulated in the light of financial theories, and to their forecasting performance with reference to both point and density predictions. The review covers papers concerning a multiplicity of sub-fields in financial economics, ranging from empirical analyses of stock returns, the term structure of default-free interest rates, the dynamics of exchange rates, as well as the joint process of stock and bond returns.
In this paper we consider an equity-indexed annuity (EIA) investor who wants to determine when he should surrender the EIA in order to maximize his logarithmic utility of the wealth at surrender time. We model the dynamics of the index using a geometric Brownian motion with regime switching. To be more realistic, we consider a finite time horizon and assume that the Markov chain is unobservable. This leads to the optimal stopping problem with partial information. We give a representation of the value function and an integral equation satisfied by the boundary. In the Bayesian case which is a special case of our model, we obtain analytical results for the value function and the boundary.
In this paper I analyze two American-type options related to life and pension insurance contract. I use Monte Carlo simulations combined with the Longstaff and Schwartz approach for the valuation of American options to find the value of a typical surrender option. I find that the values may be much lower than previously indicated. This reduction of value is due to a different treatment of bonuses, limiting the customers’ ability to forecast the return of their policies. The numerical results show that the value may be higher than the corresponding surrender option.
We tackle the problem of computing fair periodical premiums of an equity-linked policy with a maturity guarantee and an embedded surrender option. We consider the policy as a Bermudan-style contingent claim that can be exercised at the premium payment dates. The evaluation framework is based on a discretization of a bivariate model that considers the joint evolution of the equity value with stochastic interest rates. To deeply reduce the computational complexity of the pricing problem we use the singular points framework that allows us to compute accurate upper and lower estimates of the policy premiums.
We consider the fair valuation of a participating life insurance policy with surrender options when the market values of the asset are modelled by Markov-modulated Geometric Brownian Motion (GBM). We reduce the dimension of the optimal stopping problem for the policy by changing probability measures. We also provide a decomposition result for the value of the policy. The Barone–Adesi–Whaley approximation has been employed to approximate the solution of the free boundary problem for the policy by second-order piecewise linear ordinary differential equations (ODEs). The fair valuation of participating perpetual American contracts are also considered.
Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including L\'evy processes and Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.
This work develops numerical approximation methods for quantile hedging involving mortality components for contingent claims in incomplete markets, in which guaranteed minimum death benefits (GMDBs) could not be perfectly hedged. A regime-switching jump-diffusion model is used to delineate the dynamic system and the hedging function for GMDBs, where the switching is represented by a continuous-time Markov chain. Using Markov chain approximation techniques, a discrete-time controlled Markov chain with two component is constructed. Under simple conditions, the convergence of the approximation to the value function is established. Examples of quantile hedging model for guaranteed minimum death benefits under linear jumps and general jumps are also presented.
We develop an efficient Fourier-based numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The two-dimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826-848, F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27-62], and high-order quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence.
The effect of secondary markets on equity-linked life insurance with surrender guarantees
- Hilpert
The effect of policyholders' rationality on unit-linked life insurance contracts with surrender guarantees
- Li
Business confidence and stock returns in the USA: a time-varying Markov regime-switching model
- Evik