Realization Utility With Path-dependent Reference Point

Under the assumption of rational people,traditional finance considers a problem of maximizing expected utility.This article considers the problem of maximizing realization utility under the assumption of irrational people.The realization utility is a burst of utility brought about by investors selling stocks relative to the reference level to gain or loss.We use the S-type utility function in behavioral finance to measure realization utility to formulate the model.This article assumes that the asset price process is subject to a Geometric Brownian Movement with dividends,and investors can reinvest after trading,taking into account the impact of factors such as proportional transaction fees and external shocks that investors may be exposed to.We found that investors sometimes will also realize losses voluntarily and use the related concepts of disposition effect and loss aversion in behavioral finance to explain.It is proved that the value function is the only continuous viscosity solution to the HJB equation of the corresponding optimal stopping problem.And we reduce the problem to one-dimension problem through the variable transformation.Finally,we use the penalty method to numerically solve the HJB equation and visualize the effect of different parameters on the best investment strategy and give explanations accordingly.