# 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 ?

Thank you in advance

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? Commented Feb 13, 2020 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
Commented Feb 13, 2020 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).

• Thanks for your reply Commented Feb 17, 2020 at 14:28

Finite difference method is used to compute derivatives of functions as it is the case for greeks estimation. Regarding the IV computation, one can use an algorithm to get volatility by inverting the theoritical price formula to match quoted prices.

As for Heston model, we can use finite difference method to approximate the solution but its calibration consists on all parameters estimation from quoted options prices. It is more complicated to get parameters that injected on the heston prices formulas, allows you to match quoted options prices but as you can see Heston model calibration deosn't need IV computation.