# Valuation Down-And-Out Put Option via Rubinstein Closed-Form Solution

I am trying to understand the closed form solution for evaluating a down-and-out put option of Rubinstein and Reiner (1991) as stated in Baule and Tallau (2011) for the valuation of bonus certificates.

Used notation:

$$pdo_t =$$ price of a down-and-out put in time t

$$p =$$ price of a plain vanilla put in time t

$$pdi_t =$$ price of a down-and-in put in time t

$$\Phi=$$ cumulative function of the normal distribution

$$T=$$ Maturity date

$$K=$$ Strike

$$H=$$ Barrier

I am struggeling with understanding equation (8):

Why does this formula model a put-payoff (see yellow parts) and the payoff of a call (see blue-marked parts) to receive the value of a down-and-in put $$pdi_t$$?

Thank you.