- First the distribution of stock returns exhibits heavier tails than the Gaussian distribution.
- Implied volatilities of options depend on strike prices and maturities
When we take a look at commonly used equity models we observe that:
- The classical Black Scholes model cannot reproduce market volatilities as these exhibit smiles and/or skewnesses.
- Dupire’s local volatility can – in principle – fit market volatilities but still does not contain stochasticity of volatility
- Advanced volatility models (e.g. Heston) cover additional features and are popular in the community.
- Adding jumps to diffusion models (Bates) or modelling the evolution of the stock prices using jumps only (NIG,VG) may increase the coverage of additional features or provides an alternative way to stochastic volatility for closing the gap between models and reality.
In the next blog posts we will pick one model after the other and highlight their advantageous and disadvantageous from a modelling as well as from a numerical point of view.