However, many phenomena, such as the volatility clustering, the volatility smile, the heavy-tailed nature of return distributions, etc., cannot be explained within the framework of deterministic volatility models. In most of the existing literature, it is standard to assume that the price of the risky asset (stock) follows a geometric Brownian motion, which implies that the volatility of risky asset price is a constant or a deterministic function. Pan and Xiao study an optimal M–V ALM problem with stochastic interest rates and inflation risks. Chiu and Wong investigate a M–V ALM problem with asset correlation risks, which are modeled by a multivariate Wishart process. , respectively, consider a continuous-time M–V ALM problem and a multi-period M–V ALM problem with uncertain time horizon. Chiu and Wong apply the backward stochastic differential equation (BSDE) method to study the M–V ALM problems with cointegrated risky assets. to the cases with a Markovian regime switching market. and Chen and Yang extend the work of Chiu and Li and Leippold et al. also study a continuous-time ALM problem while the liability process is governed by a Brownian motion with drift. By using the stochastic linear-quadratic control theory, Chiu and Li study a continuous-time ALM problem where the risky assets’ prices and the liability value are both governed by geometric Brownian motions. investigate a multi-period ALM problem and derive explicit expressions for the efficient investment strategy and the efficient frontier. Based on the multi-period M–V framework, Leippold et al. Keel and Müller study a portfolio choice with liabilities and show that liabilities affect the efficient frontier. Sharpe and Tint first consider an ALM problem under the static M–V framework. These studies consider optimization problems of selecting optimal portfolios that can yield sufficient returns in compensating the corresponding liability. In recent years, dynamic allocation strategies for mean–variance (M–V) ALM problems have been studied widely. Asset-liability management (ALM) is essential for financial security systems such as banks, life insurance companies, property insurance companies and pension funds.
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