Editing Implied volatility (section)

==Solving the inverse pricing model function==
In general, a pricing model function, ''f'', does not have a closed-form solution for its inverse, ''g''. Instead, a [[root finding]] technique is often used to solve the equation:

:<math>f(\sigma_\bar{C}, \cdot) - \bar{C} = 0 \,</math>

While there are many techniques for finding roots, two of the most commonly used are [[Newton's method]] and [[Brent's method]]. Because options prices can move very quickly, it is often important to use the most efficient method when calculating implied volatilities.

Newton's method provides rapid convergence; however, it requires the first partial derivative of the option's theoretical value with respect to volatility; i.e., <math>\frac{\partial C}{\partial \sigma} \,</math>, which is also known as ''vega'' (see [[Greeks (finance)|The Greeks]]). If the pricing model function yields a closed-form solution for ''vega'', which is the case for [[Black–Scholes model]], then Newton's method can be more efficient. However, for most practical pricing models, such as a [[Binomial options pricing model|binomial model]], this is not the case and ''vega'' must be derived numerically. When forced to solve for ''vega'' numerically, one can use the Christopher and Salkin method or, for more accurate calculation of out-of-the-money implied volatilities, one can use the Corrado-Miller model.<ref>{{cite web|last1=Akke|first1=Ronald|title=Implied Volatility Numerical Methods|url=http://www.ronakke.com/BSIV-Numerical-Methods.html|website=RonAkke.com|access-date=9 June 2014}}</ref>

Specifically in the case of the Black[-Scholes-Merton] model, Jaeckel's "Let's Be Rational"<ref>{{citation|title=Let's be rational|url=http://www.jaeckel.org| author = Jaeckel, P. | journal = Wilmott Magazine |date=January 2015 |volume=2015|issue=75| pages = 40–53|doi=10.1002/wilm.10395|url-access=subscription}}</ref> method computes the implied volatility to full attainable (standard 64 bit floating point) machine precision for all possible input values in sub-microsecond time. The algorithm comprises an initial guess based on matched asymptotic expansions, plus (always exactly) two Householder improvement steps (of convergence order 4), making this a three-step (i.e., non-iterative) procedure. A reference implementation<ref>{{cite web|url=http://www.jaeckel.org|title=Reference Implementation of 'Let's Be Rational'|website=www.jaeckel.org|year=2013| author = Jaeckel, P.}}</ref> in C++ is freely available.
Besides the above mentioned [[root finding]] techniques, there are also methods that approximate the multivariate [[inverse function]] directly. Often they are based on [[polynomials]] or [[rational functions]].<ref name="rat">{{cite journal  | title = A parametrized barycentric approximation for inverse problems with application to the Black–Scholes formula
 | author = Salazar Celis, O. | journal = [[IMA Journal of Numerical Analysis]] | volume = 38| issue = 2| pages = 976–997| year = 2018| doi = 10.1093/imanum/drx020| hdl = 10067/1504500151162165141| hdl-access = free}}</ref>

For the Bachelier ("normal", as opposed to "lognormal") model, Jaeckel<ref>{{cite journal|title=Implied Normal Volatility|url=http://www.jaeckel.org| author = Jaeckel, P. | journal = Wilmott Magazine |date=March 2017 | pages = 52–54}} '''Note''' The print version contains typesetting errors in the formulae which have been correct on www.jaeckel.org.</ref> published a fully analytic and comparatively simple two-stage formula that gives full attainable (standard 64 bit floating point) machine precision for all possible input values.