A non-option financial instrument that has embedded optionality, such as an interest rate cap, can also have an implied volatility.
However, in general, the value of an option depends on an estimate of the future realized price volatility, σ, of the underlying.
Or, mathematically: where C is the theoretical value of an option, and f is a pricing model that depends on σ, along with other inputs.
In general, it is not possible to give a closed form formula for implied volatility in terms of call price (for a review see [1]).
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: 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.,
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.
[5] Specifically in the case of the Black[-Scholes-Merton] model, Jaeckel's "Let's Be Rational"[6] 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.
Besides the above mentioned root finding techniques, there are also methods that approximate the multivariate inverse function directly.
[8] For the Bachelier ("normal", as opposed to "lognormal") model, Jaeckel[9] 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.
The reason is that the underlying needed to hedge the call option can be sold for a higher price.
It is a mistake to confuse a price, which implies a transaction, with the result of a statistical estimation, which is merely what comes out of a calculation.
However, the above view ignores the fact that the values of implied volatilities depend on the model used to calculate them: different models applied to the same market option prices will produce different implied volatilities.
There exist few known parametrisation of the volatility surface (Schonbusher, SVI, and gSVI) as well as their de-arbitraging methodologies.