# On which model is based the Finite Differences method for implied volatility computations?

I am very new to finance, so I don't know if my question makes sense but I have seen that there are different methods to estimate the implied volatility of an American Option.

One of them is the finite differences method (used in the RQuantlib package in R), but since it is a mathematical method, what financial theory can it be used with to get the implied volatility ? Is it based on the Black and Scholes model ? Also is there any article or book I could read to better understand ?

Implied volatility is obtained by taking the observed market price of an option and solving for the necessary volatility in the Black Scholes formula to give that price. The finite difference method is just a numerical method to solve PDEs like the Black Scholes equation on a computer.

• Thanks for the reply. Can the finite difference method be used with other models, like the Heston model? Feb 13 '20 at 8:49
• Yes. You can use finite difference methods with any options model or PDE. It is just a form of numerical analysis to solve a PDE on a computer.
– roz
Feb 13 '20 at 20:20

The BS implied vol is the vol parameter in the BS formula that makes it hit the observed price of an European option.

To price an American option you need an assumption on the underlying dynamics, say geometric Brownian motion with constant diffusion coefficient (which happens to be also named BS dynamics). Then you need to get the derivative pricing PDE (free-boundary problem) which, in turn, can be solved using the FD method.

The constant diffusion coefficient that allows the FD PDE pricer to hit an observed American option price is NOT a BS implied volatility (as defined above for European options).