Malliavin calculus in finance: theory and practice

Table of Contents
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 It^o'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. 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.

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.