2nd Edition

# Malliavin Calculus in Finance Theory and Practice

** Malliavin Calculus in Finance: Theory and Practice, Second Edition** introduces the study of stochastic volatility (SV) models via Malliavin Calculus. Originally motivated by the study of the existence of smooth densities of certain random variables, Malliavin calculus has had a profound impact on stochastic analysis. In particular, it has been found to be an effective tool in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks.

This book aims to bridge the gap between theory and practice and demonstrate the practical value of Malliavin calculus. It offers readers the chance to discover 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.

**New to the Second Edition**

- Includes a new chapter to study implied volatility within the Bachelier framework.
- Chapters 7 and 8 have been thoroughly updated to introduce a more detailed discussion on the relationship between implied and local volatilities, according to the new results in the literature.

**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. Chapter’s 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 Ito's formula. 4.4. Chapter’s 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. 6.3. 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. Section V Non log-normal models. 11. The Bachelier implied volatility. 11.1. Bachelier-type Models. 11.2. A Decomposition formula for option prices. 11.3. A Decomposition formula for implied volitality. 11.4. The Bachelier ATM skew. 11.5. Chapter's digest. **Bibliography. Index.**

### Biography

**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.