Finite Differences, American Option Pricing, and Implied Volatility
Summary
The answers distinguish implied volatility from the finite-difference method. For a European option, Black–Scholes implied volatility is the volatility input that makes the model price match the observed market price. Finite differences are a numerical technique for approximating solutions to differential equations, including the Black–Scholes pricing equation; they are not themselves a volatility model or an implied-volatility definition.
For an American option, pricing requires assumptions about the underlying’s dynamics. Under geometric Brownian motion with constant diffusion, the American-option pricing equation is a free-boundary problem that can be solved numerically with finite differences. The volatility parameter adjusted until that American model price matches the market quote is a model-calibrated parameter, not the standard Black–Scholes implied volatility defined for European options. The discussion is introductory and does not cover numerical implementation, exercise-boundary treatment, or alternative dynamics. It also notes that models such as Heston may use finite differences for pricing and can be calibrated directly to option prices without first computing implied volatilities.
Key ideas
- European Black–Scholes implied volatility is found by matching the model price to the observed option price.
- Finite differences numerically solve pricing equations and do not define implied volatility.
- American-option valuation requires an underlying dynamics assumption and an early-exercise pricing treatment.
- A volatility calibrated through an American-option finite-difference pricer is not the standard European Black–Scholes implied volatility.
- A stochastic-volatility model can be calibrated directly to option prices.
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# On which model is based the Finite Differences method for implied volatility computations? # 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 ## Answer by roz (score 0, accepted) https://quant.stackexchange.com/a/51129 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. ## Answer by ir7 (score 2) https://quant.stackexchange.com/a/51165 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). ## Answer by Valometrics.com (score 0) https://quant.stackexchange.com/a/51143 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.
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