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One of the inherent hazards of investing in financial market is the risk incurred by the sudden large shock in security prices and volatilities [1]-[8]. So far Value at Risk (VaR) has gained increasingly popularity in recent years as a risk measure to understand the maximum loss of a portfolio. In the investment horizon, the problem of intertemporal optimization problem under VaR constraints is resolved [1]-[8]. Conventional works showed several VaR regulation scheme, however, they are restricted to cases having asset with ordinary Brownian motions without risk incurred by sudden large shocks (jumps). Also in previous works, we demonstrated VaR regulation method with asset having jump diffusion processes, but the solution is restricted to two-asset case [8]. It is also necessary to estimate diffusion processes from time series. In this paper, we show the extension of VaR regulation. This paper deals with the dynamic asset allocation with Value-at-Risk regulation described by variables having jump diffusion processes for multiple assets. At first, we assume in the model the security price follows jump-diffusion processes which are triggered by a Poisson event [9]-[14]. Because of the tractability provided by the affine structure of the model, we can reduce the Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) which are allowing us to obtain the optimal solution for investment [9]-[14]. In the model, it is assumed that VaR is bounded at time t by an exogenous limit proportional to the current wealth directly for a given time horizon, then the problem becomes to be tractable enough. By using the first-order approximation of the wealth process, we find the optimal dynamic portfolio in which we switch the weight for the risky asset depending on the boundaries of weight[5][7][8]. As a result, the suppression of loss in investments and increase of profit are realized by VaR regulation compared to cases without regulation. Since the estimation of parameters defining diffusion processes may affect the VaR regulation results, we also show the fuzzy based (multistage fuzzy) inference for estimating the jump diffusion processes [14]-[17]. In the followings, in Section 2, we treat the basic model of asset price dynamics and the optimal investment. Section 3 gives the first order approximation of evaluation function after a elapse of time and the impact of VaR regulation. In Section 4, we describe applications for the dynamic asset allocation having risky assess with jump diffusion processes under VaR regulation. |
