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

Question

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.

Answer 1

The premium formula of down and in put has nothing to do with the call payoff.

Actually, it is a correction term to take into account the barrier feature in the payoff.

I used the same formula to price barrier options in my website Valometrics.com in case you want to test Rubinstein and Reiner barrier options formulas.