Malliavin Calculus in Finance: Theory and Practice aims to introduce the study of stochastic volatility (SV) models via Malliavin Calculus.

Malliavin calculus has had a profound impact on stochastic analysis. Originally motivated by the study of the existence of smooth densities of certain random variables, it has proved to be a useful tool in many other problems. In particular, it has found applications in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks.

The objective of this book is to offer a bridge between theory and practice. It shows that Malliavin calculus is an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results.


  • Intermediate-advanced level text on quantitative finance, oriented to practitioners with a basic background in stochastic analysis, which could also be useful for researchers and students in quantitative finance
  • Includes examples on concrete models such as the Heston, the SABR and rough volatilities, as well as several numerical experiments and the corresponding Python scripts
  • Covers applications on vanillas, forward start options, and options on the VIX.
  • The book also has a Github repository with the Python library corresponding to the numerical examples in the text. The library has been implemented so that the users can re-use the numerical code for building their examples. The repository can be accessed here: https://bit.ly/2KNex2Y.


section I|54 pages

A primer on option pricing and volatility modelling

chapter 2Chapter 1|22 pages

The option pricing problem

chapter Chapter 2|30 pages

The volatility process

section II|76 pages

Mathematical tools

chapter 56Chapter 3|16 pages

A primer on Malliavin Calculus

chapter Chapter 4|38 pages

Key tools in Malliavin Calculus

chapter Chapter 5|20 pages

Fractional Brownian motion and rough volatilities

section III|116 pages

Applications of Malliavin Calculus to the study of the implied volatility surface

chapter 132Chapter 6|52 pages

The ATM short-time level of the implied volatility

chapter Chapter 7|30 pages

The ATM short-time skew

chapter Chapter 8|32 pages

The ATM short-time curvature

section IV|56 pages

The implied volatility of non-vanilla options

chapter 248Chapter 9|32 pages

Options with random strikes and the forward smile

chapter Chapter 10|22 pages

Options on the VIX