An extended CEV model and the Legendre transform–dual–asymptotic solutions for annuity contracts

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Abstract

This paper develops an extended constant elasticity of variance (E-CEV) model to overcome the shortcomings of the general CEV model. Under the E-CEV model, we study the optimal investment strategy before and after retirement in a defined contribution pension plan where benefits are paid by annuity. By applying the Legendre transform, dual theory and an asymptotic expansion approach, we respectively derive two asymptotic strategies for a CRRA and CARA utility functions in two different periods. Furthermore, we find that each asymptotic strategy can be decomposed into an optimal zero-order strategy and a perturbation strategy. The optimal zero-order strategy denotes an investment strategy where the current volatility is just equal to the mean level of the volatility, whereas the perturbation strategy provides an approximation solution to hedge the slow varying nature of the current volatility deviating from mean level. Finally, we find that the optimal zero-order strategy under given conditions will reduce to the results of Devolder et al. (2003), Xiao et al. (2007) and Gao (2009), respectively.

Introduction

In a defined contribution (DC) pension plan, the financial risk is borne by the member and it can be decomposed into two parts: investment risk, during the accumulation phase, and annuity risk, focused at retirement. Recently, due to the demographic evolution and the development of the equity market, DC schemes have become popular in global pension market. So the optimal investment strategy for a pension investor has become an important subject in recent years.

The current actuarial literature about the investment in DC pension schemes is quite rich (e.g., Boulier et al. (2001), Haberman and Vigna (2002), Devolder et al. (2003), Gerrard et al. (2004) and Deelstra et al. (2004)). However, these research generally supposed to be the risky asset price dynamics driven by a geometric Brownian motion (GBM). This means that the volatility of the risky asset price is only a constant, which is contradicted by the empirical evidence of fluctuating historical volatility. In the financial literature, this problem has been extended to many stochastic volatility (SV) models such as mean-reverting models (e.g., Hull and White (1987), Stein and Stein (1991) and Heston (1993)), CEV model (e.g., Cox and Ross (1976) and Cox (1996)), and SV with jump (SVJ) (e.g., Bates (1996), Bakshi et al. (1997), Duffie et al. (2000) and Barndorff-Nielsen and Shephard (2001)). In the related literature devoted to optimal portfolio choice, there have been many papers considering the addition of a mean-reverting SV model (e.g., Jonsson and Sircar (2002) and Munk et al. (2004)), a CEV model (Xiao et al., 2007, Gao, 2009), and a SVJ (cf. Liu et al. (2003), Yang and Zhang (2005) and Branger et al. (2008)). However, Jäckel (2004) reviewed these mean-reverting SV models and pointed that these models cannot fully capture the realistic returns behavior. For instance, the dynamics of the Hull and White (1987) SV model predict that both expectation and most likely value of instantaneous volatility only converge to zero, whereas the dynamics of the Stein and Stein (1991) predict that volatility is very likely to be near zero. The dynamics of Heston (1993) model predict that volatility can reach zero, stay at zero for some time, or stay extremely low or very high for long periods of time. Allowing jumps into the SV models have become apparent in recent studies. These SVJ models, however, have provided two practical problems. First, small sample problems may be severe because jumps that are large enough to have a substantial effect on risky asset price must be infrequent in order to be consistent with the underlying returns data. Second, adding jumps will introduce additional parameters into the risky asset price dynamics problem such as the price of jump risk and the price of intensity risk if the intensity is state dependent. But these risk parameters are hard to interpret, not estimable in the time series alone, and difficult to pin down in the cross section. Moreover, Chernov et al. (2003) found that adding jumps into the SV models has little effect on pricing or hedging longer maturity options and actually worsens hedging performance for short maturities. Thus, it seems reasonable to attempt to preserve the simpler structure of SV models.

The CEV model with stochastic volatility is a natural extension of GBM, and it has the ability of capturing the implied volatility skew. Compared with other stochastic volatility models involving volatility skew (cf. Heston (1993), Bates (1996) and Barndorff-Nielsen and Shephard (2001)), the advantage of the CEV model is that it remains analytical tractability. Thus, under a CEV model, it can be analytically examined the influences of the volatility skew on an investor’s trading strategy. Recently, Xiao et al. (2007) and Gao (2009) began to apply the CEV model to the pension investment and derived the dual solution for the logarithm, power and exponential utilities via a Legendre transform and dual theory.

However, due to the inverse relationship between the volatility and the stock price, the CEV model cannot counteract the inevitable tendency to drive the volatility to zero as the stock price increases over long periods. Considering that DC pension fund investment period is usually long time horizon, generally about 20 to 40 years. Therefore, we develop an extended CEV model (E-CEV) to correct the shortcoming of the CEV model, which will counteract the trend of the volatility to zero over long time horizon. Namely, we introduce a new stochastic parameter to replace the constant k in the general CEV model. In this model, if this new process maintains an upward trending with the stock price increasing over long time and consequently, then it will counteract the inverse relationship between stock price and volatility. As a result, this will (on average) maintain the current level of instantaneous volatility over long time.

Under this E-CEV model, we study the portfolio optimization problem for a DC plan. Following Jonsson and Sircar (2002), Xiao et al. (2007) and Gao (2009) work, we first derive a general framework to the optimization problem by applying Legendre transform, dual theory. Under the given CRRA and CARA utility functions, this general framework can be simplified to several ordinary differential equations and one nonlinear partial differential equation (PDE) via a variable transform technique. However, there is no explicit solution to this nonlinear PDE due to the influences of the stochastic process x. Therefore, we proceed with an asymptotic expansion approach. This approach is effective in accounting for stochastic volatility effect. In the last few years, it has been developed for option pricing problems, see, e.g., Takahashi (1999), Kunitomo and Takahashi (2001) and Widdicks et al. (2005). Recently, Emms et al. (2007) use a perturbation expansion to determine the optimal premium in order to maximize the insurer’s expected total wealth. By applying an asymptotic expansion approach, we respectively obtain two asymptotic strategies for the two utility functions in two periods (before and after retirements).

Furthermore, we find that each asymptotic strategy can be decomposed into two parts: one is the optimal zero-order strategy, and the other is the perturbation strategy. The optimal zero-order strategy denotes an investment strategy where the current volatility is equal to the mean level of the volatility. However, owing to the new stochastic parameter xt cannot completely match the dynamic changing of the risky asset price over the whole time horizons, which results in the current volatility sometimes deviating from mean level. Then, the perturbation strategy provides an approximation solution to hedge the slow varying nature of the current volatility deviating from mean level.

In addition, we compare the difference of the asymptotic strategies (resp. optimal zero-order strategies) for the CRRA utility between before and after retirement periods, and find that in each strategy after retirement we have to use the surplus instead of the total assets, which confirms the empiric idea to decrease the part invested in risky assets. However, there is no such effect in the CARA case.

Finally, we show that under given conditions, the optimal zero-order strategy can degenerate to the corresponding results of Devolder et al. (2003), Xiao et al. (2007) and Gao (2009).

The rest of the paper is organized as follows. In Section 2, we develop an extended CEV model and present the optimization problem. In Section 3, we provide the general framework to solve the optimization problem for the two periods by applying maximum principle, Legendre transform and dual theory. In Sections 4 Asymptotics—period before retirement, 5 Asymptotics—Period after retirement, under the given utility function, we respectively derive an asymptotic solution and optimal zero-order strategies for the two periods via variable transform and asymptotic expansion approach. Conclusions are given in Section 6.

Section snippets

The optimization problem

In this section, we develop an extended CEV model (E-CEV) to describe the risky asset price dynamics and propose the optimization problems for the two periods: before and after retirements.

General framework

This section provides a general framework to the optimization problems (2.4), (2.5) by applying the maximum principle, Legendre transform and dual theory.

Asymptotics—period before retirement

In this section, we try to find an asymptotic solution for the period before retirement via variable transform technique and an asymptotic expansion approach.

Asymptotics—Period after retirement

This section is devoted to finding asymptotic solutions for the CRRA and CARA utility functions under period after retirement.

Conclusions

According to the problem of declining volatility in the CEV model, we developed an extended constant elasticity of variance (E-CEV) model to correct this problem. Under this extended model, we studied the optimal investment strategy before and after retirement in a defined contribution pension plan. Following Jonsson and Sircar (2002), Xiao et al. (2007) and Gao (2009) works and further using variable transform and asymptotic expansion approach, we derived the asymptotic strategies for the CRRA

Acknowledgements

The author is very grateful to the anonymous referees for the insightful comments and suggestions. This research is supported by the Natural Science Foundation of China under Grant No. 70971039 and the Educational Ministry Foundation of China under Grant No. 07JC790025 and the Project Supported by Chinese Universities Scientific Fund No. 09MR46.

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