CONTINUOUS-TIME MEAN–VARIANCE OPTIMIZATION FOR DEFINED CONTRIBUTION PENSION FUNDS WITH REGIME-SWITCHING

Deals with a continuous-time version of the Markowitz mean-variance portfolio selection problem. The proposed model is concerned with a basket of securities consisting of one bank account and multiple stocks. One distinct feature of the model is that the market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches between a finite number of states. It is assumed that the random regime switching is independent of the random sources that drive the stock prices. Although the model is more realistic and takes into account possible random volatility, it essentially renders the underlying market incomplete. We use a Markov-chain modulated diffusion formulation to model the problem. By using techniques of stochastic linear-quadratic control, we obtain mean-variance efficient portfolios and efficient frontiers in explicitly closed forms, based on solutions of two systems of linear ordinary differential equations. We also address related issues such as minimum variance portfolio and mutual fund theorem. The results are notably different from those for the case when there is no regime switching. Nonetheless, if the interest rate is independent of the Markov chain and is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stocks appreciation and volatility rates are Markov-modulated.

ABSTRACTIn this paper, we consider the ruin probability minimization of an insurance company that buys proportional reinsurance and invests in markets where borrowing is constrained. We use a diffusion approximation model for the surplus process of this company and assume that the company invests its surplus into a riskless and multiple risk assets that are modeled as geometric Brownian motions. To find the results, we introduce an auxiliary market parametrized with fictitious processes to relax the borrowing constraint and apply the techniques of stochastic optimal control. In this way, we find the optimal proportional reinsurance and investment strategy of an insurance company investing into multiple risky assets to minimize its ruin probability under the borrowing constraint. Furthermore, from our solutions, we show how our results connect to economic survival analysis and how investment and reinsurance strategies are related.

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal with the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

This paper considers a continuous‐time mean‐variance asset‐liability management problem with incompletely observable information. An investor can only observe the prices of the asset and liability and the dynamics of the unobservable states of the underlying financial market is described by a hidden Markovian chain. The price of the risky asset is assumed to be governed by a hidden Markovian regime switching geometric Brownian motion and the liability is assumed to follow a hidden Markovian regime switching Brownian motion with drift, respectively. The appreciation rates of the risky asset and the liability are modulated by the hidden Markovian chain. By using the separation principle, the filtering‐estimation problem and the mean‐variance asset‐liability management problem are discussed. The explicit expressions for the optimal asset‐liability management strategy and the mean‐variance efficient frontier are determined by using the stochastic maximum principle.

This paper considers the optimal investment strategy for an insurer under the criterion of mean-variance. The risk process is a compound Poisson process and the insurer can invest in a risk-free asset and multiple risky assets. We obtain the optimal investment policy using the stochastic liner-quadrant (LQ) control theory. Then the efficient strategy (optimal investment strategy) and efficient frontier are derived explicitly by a verification theorem with the classical solution of Hamilton-Jacobi-Bellman (HJB) equation.

The Bachelier model for pricing options on futures spreads (OFS) assumes changes in the underlying .futures prices and spread follow unrestricted arithmetic Brownian motion (UABM). The assumption of UABM allows for a convenient analytic solution for the price of an OFS. The same is not possible under the more traditional assumption of geometric Brownian motion (GBM). Given the additional complexity of methods for pricing and hedging OFS using GBM such as Monte Carlo simulation and binomial trees, it is worth investigating how results from the Bachelier model compare to these other methods. The Bachelier model is presented and then extended to price an OFS with three underlying commodities. Hedge parameters for both models are provided. Results indicate that for OFS with sufficiently low volatility, differences between the Bachelier model and methods assuming GBM are quite small.

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures.

We study a discrete-time version of Markowitz's mean-variance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finite-state Markov chain, based on which a discrete-time Markov modulated portfolio selection model is presented. Such models either arise from multiperiod portfolio selections or result from numerical solution of continuous-time problems. The natural connections between discrete-time models and their continuous-time counterpart are revealed. Since the Markov chain frequently has a large state space, to reduce the complexity, an aggregated process with smaller state-space is introduced and the underlying portfolio selection is formulated as a two-time-scale problem. We prove that the process of interest yields a switching diffusion limit using weak convergence methods. Next, based on the optimal control of the limit process obtained from our recent work, we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality.

A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.

This paper proposes a way to forecast the future closing price of small size companies in Bursa Malaysia by using geometric Brownian motion (GBM). Forecasting is restricted to short term investment because most of the investors aim to gain profit in short period of time. The reasons of choosing small size companies are because the asset prices are lower, hence the asset are affordable for all level of investors. In this paper, we suggest that GBM which involves randomness, volatility, and drift can help investor in making their investment decision wisely. This research shows the model is highly accurate model in forecasting stock prices and it is proven by the lower value of mean absolute percentage error (MAPE). Although it is highly accurate, we try to find the suitable volatility measurements that give the forecast value closer to the actual movement of stock prices. The result shows that by using highs-lows-close volatility, the forecast stock prices are closest to the actual prices. This volatility measurement and GBM model are suggested to the investor to forecast future prices for a maximum of two week investment.

The calendar spread options (CSOs) on agricultural commodities, most notably corn, soybeans and wheat, allow market participants to hedge the roll-over risk of futures contracts. Despite the interest from agricultural businesses, there is lack of both theoretical and empirical research on pricing and hedging performances of CSOs. We propose to price and hedge CSOs under geometric Brownian motion (GBM) and stochastic volatility (SV) models. We estimate the model parameters by using implied state-generalized method of moments (IS-GMM) and evaluate the in-sample and out- of-sample pricing and hedging performances. We find that the average pricing errors of the SV model are 0.79% for corn, 0.75% for soybeans and 1.2% for wheat; the pricing and hedging performance of the SV model are mostly superior to the benchmark GBM model, both in and out of sample, with only one exception where the out-of-sample hedging error for the GBM model for market makers is slightly better than the SV model.

This paper aim at studying a mean-variance portfolio selection problem with stochastic salary, proportional administrative costs and taxation in the accumulation phase of a defined contribution (DC) pension scheme. The fund process is subjected to taxation while the contribution of the pension plan member (PPM) is tax exempt. It is assumed that the flow of contributions of a PPM are invested into a market that is characterized by a cash account and a stock. The optimal portfolio processes and expected wealth for the PPM are established. The efficient and parabolic frontiers of a PPM portfolios in mean-variance are obtained. It was found that capital market line can be attained when initial fund and the contribution rate are zero. It was also found that the optimal portfolio process involved an inter-temporal hedging term that will offset any shocks to the stochastic salary of the PPM.<strong> </strong>

We use a white noise approach to study the problem of optimal insider control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system. These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the logarithmic utility optimal insider portfolio in a generalized Black-Scholes type financial market. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable, i.e. the maximal utility is infinite. We consider two extensions to delay of this result and prove the following:•If the risky asset price is given by a stochastic delay equation, the resulting insider market is still not viable.•If, however, there is delay in the information flow to the insider, the market becomes viable.

Under heterogeneous expectations, the mean–variance model of capital market equilibrium is employed to determine the effect restricting short sales has on equilibrium asset prices. Two equivalent markets differing only with respect to short sale restrictions are compared. It is shown that, in general, risky asset prices can either rise or fall due to short sale constraints. However, under a homogeneity of beliefs for the covariance matrix of future prices, short sale constraints will only increase risky asset prices.

Sentiment asset pricing model with consumption