Valuation of Continuous-installment Shout Option

In order to study the pricing problem of continuous-installment options, a parabolic variational inequality that the option price satisfies is derived, and the existence of an explicit expression for its barrier function in a subset of the solution domain is proved. The variational inequality has two free boundaries, one is the optimal shouting boundary, and the other is the optimal stopping boundary. The location and properties of these two free boundaries are discussed through qualitative analysis, and the penalty method is used to solve the variational inequalities so as to give numerical examples with different parameters. The results show that if the risk-free interest rate is greater than the dividend yield, the optimal shouting boundary tends to infinity when the time is far from the maturity date; but as the maturity date approaches, both free boundaries tend to strike prices. In addition, both the shouting right and the installment rate have a significant impact on the free boundaries.