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E-book: Malliavin Calculus in Finance: Theory and Practice

(Universitat Pompeu Frabra, Spain),
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"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. Features 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 suchas 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"--

This book introduces the study of stochastic volatility (SV) models via Malliavin Calculus. Malliavin calculus has had a profound impact on stochastic analysis. 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 SV modeling.



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.

Features

  • 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.

 

Reviews

"Malliavin calculus, alongside Ito calculus, is emerging as a vital tool for researchers in the area of financial engineering. This book provides an unprecedented and balanced account, taking the reader from theoretical foundations to practical applications, including state-of-the-art research topics like rough volatility and VIX option skew." Colin Turfus

"This book is a very valuable addition to the existing literature, demonstrating that the cutting-edge research in Mathematical Finance doesn't have to be far from commonly accepted quant practice." Vladimir Lucic, Visiting Professor, Dept. of Mathematics, Imperial College London

"The book is an excellent guide to the applications of the Malliavin calculus to finance. Starting with classical questions of non-arbitrage pricing and the Black-Scholes formula, the authors smoothly continue with volatility processes, studying, in particular, implied, spot and local volatilities. Various models with stochastic volatilities are considered, including models based on fractional Brownian motion and rough volatilities. Variance swaps and the VIX, volatility, and other types of swaps are studied. Then the main tools of Malliavin calculus are presented, together with applications of Malliavin calculus to the implied volatility surface and the implied volatility of non-vanilla options. So, the book is very promising for both mathematicians and practitioners and both mathematicians and practitioners will enjoy the beauty of the mathematical description of the world of real finance." Yuliya Mishura, Taras Shevchenko National University of Kyiv

I. A primer on option pricing and volatility modeling.
1. The option
pricing problem. 1.1. Derivatives. 1.2. Non-arbitrage prices and the
Black-Scholes formula. 1.3. The Black-Scholes model. 1.4. The Black-Scholes
implied volatility and the non-constant volatility case. 1.5.
Chapter's
digest.
2. The volatility process. 2.1. The estimation of the integrated and
the spot volatility. 2.2. Local volatilities. 2.3. Stochastic volatilities.
2.4. Stochastic-local volatilities 2.5. Models based on the fractional
Brownian motion and rough volatilities. 2.6. Volatility derivatives. 2.7.
Chapters Digest. II. Mathematical tools.
3. A primer on Malliavin Calculus.
3.1. Definitions and basic properties. 3.2. Computation of Malliavin
Derivatives. 3.3. Malliavin derivatives for general SV models. 3.4.
Chapter's
digest.
4. Key tools in Malliavin Calculus. 4.1. The Clark-Ocone-Haussman
formula. 4.2. The integration by parts formula. 4.3. The anticipating It^o's
formula. 4.4.
Chapters Digest.
5. Fractional Brownian motion and rough
volatilities. 5.1. The fractional Brownian motion. 5.2. The Riemann-Liouville
fractional Brownian motion. 5.3. Stochastic integration with respect to the
fBm. 5.4. Simulation methods for the fBm and the RLfBm. 5.5. The fractional
Brownian motion in finance. 5.6. The Malliavin derivative of fractional
volatilities. 5.7.
Chapter's digest. III. Applications of Malliavin Calculus
to the study of the implied volatility surface.
6. The ATM short time level
of the implied volatility. 6.1. Basic definitions and notation. 6.2. The
classical Hull and White formula. An extension of the Hull and White formula
from the anticipating Itô's formula. 6.4. Decomposition formulas for implied
volatilities. 6.5. The ATM short-time level of the implied volatility. 6.6.
Chapter's digest.
7. The ATM short-time skew. 7.1. The term structure of the
empirical implied volatility surface. 7.2. The main problem and notations.
7.3. The uncorrelated case. 7.4. The correlated case. 7.5. The short-time
limit of implied volatility skew. 7.6. Applications. 7.7. Is the volatility
long-memory, short memory, or both?. 7.8. A comparison with jump-diffusion
models: the Bates model. 7.9.
Chapter's digest. 8.0. The ATM short-time
curvature. 8.1. Some empirical facts. 8.2. The uncorrelated case. 8.3. The
correlated case. 8.4. Examples. 8.5.
Chapter's digest. IV. The implied
volatility of non-vanilla options.
9. Options with random strikes and the
forward smile. 9.1. A decomposition formula for random strike options. 9.2.
Forward start options as random strike options. 9.3. Forward-Start options
and the decomposition formula. 9.4. The ATM short-time limit of the implied
volatility. 9.5. At-the-money skew. 9.6. At-the-money curvature. 9.7.
Chapter's digest.
10. Options on the VIX. 10.1. The ATM short time level and
skew of the implied volatility. 10.2. VIX options. 10.3.
Chapter's digest.
Bibliography. Index.
Elisa Alòs holds a Ph.D. in Mathematics from the University of Barcelona. She is an Associate Professor in the Department of Economics and Business at Universitat Pompeu Fabra (UPF) and a Barcelona GSE Affiliated Professor. In the last fourteen years, her research focuses on the applications of the Malliavin calculus and the fractional Brownian motion in mathematical finance and volatility modeling.

David Garcia Lorite currently works in Caixabank as XVA quantitative analyst and he is doing a Ph.D. at Universidad de Barcelona under the guidance of Elisa Alòs with a focus in Malliavin calculus with application to finance. For the last fourteen years, he has worked in the financial industry in several companies but always working with hybrid derivatives. He has also strong computational skills and he has implemented several quantitative and not quantitative libraries in different languages throughout his career.
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