Can "Turbo warrants" be priced using the Black & Scholes model?

I am trying to model the pricing of an asset called a "Turbo warrant", which to me looks a lot like a Down-and-Out Barrier option with leverage. When the price of the underlying asset hits a certain barrier (B), the contract becomes worthless. The issuer of these Turbo warrants indicates that their price is calculated as follows:
$$P = frac{S – F}{ratio} $$

(Note: the ratio is used in case the price of the underlying
asset is high, like in the case of Amazon stock which
is around $3,000. The ratio is often 10 or 100)

But I wonder if this is the correct way to model the price of these options. I do not fully understand why the Black and Scholes model or a variant is not used (like is sometimes used with Barrier options), so that Greeks can also be calculated for these Turbos. Could someone explain?

Edit

To be entirely complete, the issuer of the Turbo charges an interest of around 2% on the financing level $F$, which is paid daily by increasing the level of $F$ and consequently $B$ everyday. So that:

$$ F(t) = F(0) (1+r)^t$$