Editing Implied volatility (section)

==Motivation==
An option pricing model, such as Black–Scholes, uses a variety of inputs to derive a theoretical value for an option. Inputs to pricing models vary depending on the type of option being priced and the pricing model used. However, in general, the value of an option depends on an estimate of the future realized price volatility, σ, of the underlying. Or, mathematically:

:<math>C = f(\sigma, \cdot) \,</math>

where ''C'' is the theoretical value of an option, and ''f'' is a pricing model that depends on σ, along with other inputs.

The function ''f'' is [[monotonically increasing]] in σ, meaning that a higher value for volatility results in a higher theoretical value of the option. Conversely, by the [[inverse function theorem]], there can be at most one value for σ that, when applied as an input to <math>f(\sigma, \cdot) \,</math>, will result in a particular value for ''C''.

Put in other terms, assume that there is some inverse function ''g'' = ''f''<sup>&minus;1</sup>, such that

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

where <math>\scriptstyle \bar{C} \,</math> is the market price for an option. The value <math>\sigma_\bar{C} \,</math> is the volatility '''implied''' by the market price <math>\scriptstyle \bar{C} \,</math>, or the '''implied volatility'''.

In general, it is not possible to give a closed form formula for implied volatility in terms of call price (for a review see <ref>{{Cite journal |last1=Orlando |first1=Giuseppe |last2=Taglialatela |first2=Giovanni |date=2017-08-15 |title=A review on implied volatility calculation |journal=Journal of Computational and Applied Mathematics |language=en |volume=320 |pages=202–220 |doi=10.1016/j.cam.2017.02.002 |issn=0377-0427|doi-access=free }}</ref>). However, in some cases (large strike, low strike, short expiry, large expiry) it is possible to give an [[asymptotic expansion]] of implied volatility in terms of call price.<ref>[https://ssrn.com/abstract=1965977 Asymptotic Expansions of the Lognormal Implied Volatility], Grunspan, C. (2011)</ref> A different approach based on closed form approximations has been also investigated.<ref>{{Cite journal |last1=Mininni |first1=Michele |last2=Orlando |first2=Giuseppe |last3=Taglialatela |first3=Giovanni |date=2021-06-01 |title=Challenges in approximating the Black and Scholes call formula with hyperbolic tangents |journal=Decisions in Economics and Finance |language=en |volume=44 |issue=1 |pages=73–100 |doi=10.1007/s10203-020-00305-8 |s2cid=224879802 |issn=1129-6569|arxiv=1810.04623 }}</ref><ref>{{Citation |last1=Mininni |first1=Michele |title=A generalized derivation of the Black-Scholes implied volatility through hyperbolic tangents |date=2022 |url=https://www.dbc.wroc.pl/dlibra/publication/156539 |publisher=Argumenta Oeconomica, 2022, Nr 2 (49) |access-date=2022-12-11 |last2=Orlando |first2=Giuseppe |last3=Taglialatela |first3=Giovanni}}</ref> 

===Example===
A [[European call option]], <math>C_{XYZ}</math>, on one share of non-dividend-paying XYZ Corp with a strike price of $50 expires in 32 days. The [[risk-free interest rate]] is 5%. XYZ stock is currently trading at $51.25 and the current market price of <math>C_{XYZ}</math> is $2.00. Using a standard Black–Scholes pricing model, the volatility implied by the market price <math>C_{XYZ}</math> is 18.7%, or:

:<math>\sigma_\bar{C} = g(\bar{C}, \cdot) = 18.7\%</math>

To verify, we apply implied volatility to the pricing model, ''f ,'' and generate a theoretical value of $2.0004:

:<math>C_{theo} = f(\sigma_\bar{C}, \cdot) = \$2.0004</math>

which confirms our computation of the market implied volatility.