# Implied Volatility - Motivation

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:

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, will result in a particular value for C.

Put in other terms, assume that there is some inverse function g = f−1, such that

where is the market price for an option. The value is the volatility implied by the market price, or the implied volatility.

In general, it is not possible to give a closed form formula for implied volatility in terms of call price. 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.