0.1 Motivation
The implied volatility curve is symmetric before 1987, i.e. the slope of the cure \(K\mapsto I(T,K)\) is close to zero at \(K=S_{0}\) (at-the-money) but it became asymmetric after 1987, i.e. the slope of the curve is negative at \(K=S_0\).
Empirical observation from the stock market implies that the implied volatility based on options with time to maturity \(T\) and strike \(K\) denoted \(I(T,K)\) is not flat in both \(T\) and \(K\). Naturally, the Geometric Brownian Motion of the Black-Scholes-Merton Model is not a good model for stock price since it assumes constant volatility \(\sigma\).
As a result, changes are made to the Black-Scholes Model in primarily two directions, introducing jumps in the stock price or to change the volatility setting.
0.2 Areas
- Constant Volatility (Black-Scholes-Merton Model)
- Time-Dependent Volatility Model
- Local Volatility Model
- Stochastic Volatility Model
- Uncertain Volatility
- Fractional Brownian Motion Volatility Models