Volatility models

The assumption of constant volatility usually made in class is inconsistent with evidence that implied
volatilities computed for a class of options with the same expiry date can vary across multiple strike prices,
often displaying a convex “volatility smile”. Statistically, it has also been observed that volatilities display
clustering in time, and appear to be negatively correlated to asset prices. Together these observations
have led to the development of local volatility models such as the CEV model and two-factor volatility
models such as the Heston and SABR models.
In this project we will look at the construction and properties of such models in continuous time, as well
as their numerical solution. We can then investigate their use in pricing, examining for example the
emergence of volatility smiles from simulated option prices, option sensitivity analysis under stochastic or
local volatility, and model calibration.
We will draw comparison with examples from the Open Source Risk project [1], a platform for pricing and
risk analysis supported by Acadia, and the associated software project Open Source Risk Engine (ORE)
which is based on QuantLib and written in Python.