# 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. – LocalVolatility 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. – twhale 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. – LocalVolatility 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. – twhale 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? – twhale Jul 24 '20 at 19:39

## 1 Answer

This looks similar to a cliquet or "ratchet" option: an option with a strike price which resets occasionally. The Wikipedia definition of a cliquet is a bit too restrictive since one of the most common uses of such options was by Japanese firms which issued warrants and convertible bonds in the 1990s after the implosion of the Japanese real estate bubble. Also, beware since many analyses look at cliquet options with caps and floors.

The Japanese warrants and convertible bonds had strikes which could only reset downward (often to ATM or 10%-15% ITM). This increased the probability of expiring ITM. Here, your strikes reset upward. Both can be handled with the same perspective: treat the option as a series of forward-start options where early exercise extinguishes the later options.

• A historical note: After the Japanese real estate bubble burst, many firms were undercapitalized and holding bad debt. What they needed was to issue equity to recapitalize. The Ministry of Finance prohibited equity issuance since that would dilute shareholders and (supposedly) sour them on the Japanese market. However, the MoF allowed warrants and convertible bonds to be issued because <magical thinking> exercising those creates new shares but that will not lead to dilution or scare off equity investors</magical thinking>. – kurtosis Jul 24 '20 at 16:14
• Thanks a lot, the historical context is really interesting and I more or less get your answer: "treat the option as a series of forward-start options where early exercise extinguishes the later options." But would you be able to guide me towards a mathematical interpretation of it that could then be operationalised in code for example? – twhale Jul 24 '20 at 19:48
• Sure. Basically, you want to simulate vectors of prices at expirations of these, say, one-year options. In each simulated path, start at the earliest option and exercise early if the value of exercising is above the expected value of later options. (Those later values will change, but this should all converge.) Average over many simulated paths, get option values, and then re-simulate to use those. Repeat until convergence. That's not code nor equations, but hopefully that helps! – kurtosis Jul 24 '20 at 19:55
• Cliquet or ratchet? Perhaps, but I am skeptical. According to Wikipedia turbo warrants are a "barrier option of the down and out type". So who is right? en.wikipedia.org/wiki/Turbo_warrant – noob2 Jul 25 '20 at 18:42
• The cliquet setup lets you handle the changing strike. The individual options you price as barrier options. – kurtosis Jul 25 '20 at 19:34