This master’s thesis deals with implied volatility and its usage in financial mathematics. First, in introduction we give some basic information about models used in this paper, e.g. the Black and Scholes model and Cox-Ross-Rubinstein model of binomial tree. First section gives introduction to implied volatility, estimated volatility of a security’s price, most commonly used when pricing options. To derive implied volatilities the Black and Scholes formula is solved for the constant volatility parameter \(\sigma\) using observed option prices. It appears to be non flat across strikes, a stylized fact which has been called ”smile” effect. In the end of this chapter we use two variants of principal components, standard principal component analysis to identify the dominant factor components driving term structure movements of options, and the common principal components approach that enables modeling not only term structure movements of implied volatilities but the dynamics of the ”smile” as well. Then we introduce the implied binomial tree (IBT) method which constructs a numerical procedure consistent with the volatility smile. The empirical fact that the market implied volatilities decrease with the strike level, and increase with the time to maturity of options is better reflected by this construction than standard CRR trees. The basic purpose of the IBT is its use in hedging and calculations of implied probability distributions (or state price density (SPD)), and volatility surfaces. Two algorithms for construction of IBT for a liquid European option are given. Firstly, we follow the Derman and Kani algorithm, discuss its possible shortcomings, and then present the Barle and Cakici construction. In the last chapter we talk about state price density and ways of obtaining it. The central idea is that the price of a financial security is equal to the expected net present value of its future payoffs under the risk-neutral probability density function. This method constitutes a no-arbitrage approach to recover the SPD. No assumption on the underlying asset dynamics are required. First, we propose ways for semiparametric estimation of the call pricing function and the necessary steps to recover the SPD. Then, in the end, an numerical example is given.