# 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:

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

• See e.g. papers.ssrn.com/sol3/papers.cfm?abstract_id=2388734. Disclaimer: I am one of the authors. Jul 24 '20 at 14:15
• Thanks! Is there a way to download the paper? Also, can you answer the question, so that others can also read and perhaps add. Jul 24 '20 at 14:59
• There is a "download" button when you follow the link..? Sorry - don't have the time to provide an actual answer to your question though the first sections of the paper show you how to price such an option in a Black-Scholes world. Jul 24 '20 at 15:21
• Have downloaded it. Will have to study it a few times, it is quite technical and I must admit I do not fully understand all the notation. So not easy to understand your pricing model with enough depth. Jul 24 '20 at 16:26
• @LocalVolatility: I have studied your paper. If I understand it correctly, your valuation formula takes the Brownian motion which determines the spot price S as exogenous. But shouldn't a proper pricing formula treat S as endogenous - i.e. calculate the likely value of S as part of the pricing function? Jul 24 '20 at 19:39