Answer to Question #187907 in Financial Math for Umairec

Question

The price S(t) of a share follows the geometric Brownian motion S(t) = Seμt+σW(t). The parameters S and μ are given, but the volatility sigma is not known.

You are asked to compute the implied volatility as an estimate for the parameter σ using the data provided by the market.

What data you may want to use? What is the volatility smile?

Answer 1

Given, Geometric Brownian motion $S(t) = S_eμt+σW(t)$

and $S_e$ and $\mu$ is known.

Implied volatility is calculated by taking the observed option price in the market and a pricing formula such as the Black–Scholes formula that will be introduced below and backing out the volatility that is consistent with the option price given other input parameters such as the strike price of the option.

The data i.e. genrally used for calculating the implied volatility is the trading pronlems. Generally Traders sell their goods and services. So Trading data is used to calculate implied volatility.

Volatility smile: A volatility smile is a common graph shape that results from plotting the strike price and implied volatility of a group of options with the same underlying asset and expiration date.

The graph of volatility smile is-