Options, Pricing, and Risk Management Part III

This third part of our quantitative finance training course “Options, Pricing and Risk Management” will be on the modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

The course is composed of many videos, quizzes, applications and tutorials in Python.

A certificate of achievement will be delivered once the course has been completed with success.

In the first week, we will discuss the limits of the Black-Scholes model, particularly the fact that the volatility surface is not flat in practice. We will give an introduction on the modelling of the volatility surface implied by option prices with a quick overview of some of the main methods used to build the volatility surface. We will present the Newton-Raphson method for extracting the implied volatility from option prices.

We will measure the impact of the skewness and kurtosis of the return distribution on the volatility smile using the Cornish Fisher expansion, and we will introduce the Breeden-Litzenberger formula which links the second order derivative of vanilla option prices with the risk-neutral density function and can be very useful to price exotic option payoffs.

In the second week, we will focus on parametric methods and particularly on the Stochastic Volatility Inspired (SVI) model, popular on equities and some of its extensions. We will discuss on the calibration and the limits of the model.

The SVI Jump-Wings (SVI-JW) parameterisation is an interesting alternative method where the different parameters have a concrete interpretation, each of them controlling a specific aspect of the smile, making model calibration easier and more robust. We will see how to price exotic payoffs with this method. We will finally present the Surface SVI (SSVI) an extension of the model for the whole volatility surface, free of arbitrage under certain conditions.

The second part of the course will be on stochastic volatility models.

We will focus on the Heston model during the third week,  which is one of the most used stochastic volatility model. The model assumes that the variance of the asset price is stochastic, it is correlated with the asset price and follows a mean-reverting process. We will present the model, how its parameters impact the return distributions and the volatility surface. One strong advantage of the model is that despite its relative complexity, there is a semi-analytic solution for the price of European vanilla options using the characteristic function instead of the density function of the log price, which is very useful for the calibration to market prices. We will discuss on the limits of the model and see how it can be used to price path-dependent and path-independent exotic options.

The fourth week will be on the SABR model, another popular stochastic volatility model, particularly on interest rates. The model assumes that the forward price and its variance are both stochastic and correlated. We will introduce the model with its four parameters. There is no closed-form solution for the pricing of vanilla options under the SABR model except when the parameter beta is equal to zero or one but there is a good asymptotic estimation. We will see how the different parameters impact the moments of the return distribution and the implied volatility curve. We will discuss on how to calibrate parameters to market prices, on the limits of the asymptotic estimate and we will see how to price and risk manage exotic option payoff with the SABR model. We will finally discuss the limits of the model.