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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.
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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.
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Produktdetails
- Produktdetails
- Verlag: Taylor & Francis
- Seitenzahl: 350
- Erscheinungstermin: 13. Juli 2021
- Englisch
- ISBN-13: 9781000403510
- Artikelnr.: 61959844
- Verlag: Taylor & Francis
- Seitenzahl: 350
- Erscheinungstermin: 13. Juli 2021
- Englisch
- ISBN-13: 9781000403510
- Artikelnr.: 61959844
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.
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.
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.
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.
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.