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.
$pdo_t =$ 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
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 $pdi_t$?