# How to use the parity parameter when pricing third-party warrants with BS?

I attempt a second basic question. Let me know if https://money.stackexchange.com/ would have been more suitable for that.

Third-party warrants are very similar to call options. One of their main characteristics is the parity as defined here:

Parity: this represents the number of warrants needed to exercise the right on a given underlying. A parity of 10 on a call warrant on a share means that 10 call warrants need to be exercised at expiration in order to buy 1 share at the exercise price.

(from https://wholesale.banking.societegenerale.com/en/news-insights/glossary/warrants/)

If I use the Black-Scholes formula to price a warrant, where is the parity supposed to appear?

Please note in "Options, Futures, and Other Derivatives, 5th edition", John C. Hull has a section p249 about "warrants issued by a company on its own stock". I think it is a different topic, as there is no dilution in third-party issued warrants.

## 1 Answer

My intuition is we can price a third-party warrant with price $$c_w$$ and parity $${parity}$$ by pricing the equivalent option with price $$c_o = c_w\times{parity}$$.

$$c_w\times{parity}=N(d_1)S-N(d_2)Ke^{-r(T-t)},$$

$$d_1$$ and $$d_2$$ calculated as usual.