Malliavin Calculus in Finance: Theory and Practice

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Foreword
Preface
Section I: A primer on option pricing and volatility modelling
	Chapter 1: The option pricing problem
		1.1. DERIVATIVES
			1.1.1. Forwards and futures
			1.1.2. Options
		1.2. NON-ARBITRAGE PRICES AND THE BLACK-SCHOLES FORMULA
			1.2.1. The forward contract
			1.2.2. The price of a European option as a risk-neutral expectation
			1.2.3. The price of a vanilla option and the Black-Scholes formula
		1.3. THE BLACK-SCHOLES MODEL
			1.3.1. From the Black-Scholes formula to the Black-Scholes model
			1.3.2. Option replication and delta hedging in the Black-Scholes model
		1.4. THE BLACK-SCHOLES IMPLIED VOLATILITY AND THE NON-CONSTANT VOLATILITY CASE
			1.4.1. The implied volatility surface
			1.4.2. The implied and spot volatilities
		1.5. CHAPTER’S DIGEST
	Chapter 2: The volatility process
		2.1. THE ESTIMATION OF THE INTEGRATED AND THE SPOT VOLATILITY
			2.1.1. Methods based on the realised variance
			2.1.2. Fourier estimation of volatility
			2.1.3. Properties of the spot volatility
		2.2. LOCAL VOLATILITIES
			2.2.1. Mimicking processes
			2.2.2. Forward equation and Dupire formula
		2.3. STOCHASTIC VOLATILITIES
			2.3.1. The Heston model
			2.3.2. The SABR model
		2.4. STOCHASTIC-LOCAL VOLATILITIES
		2.5. MODELS BASED ON THE FRACTIONAL BROWNIAN MOTION AND ROUGH VOLATILITIES
		2.6. VOLATILITY DERIVATIVES
			2.6.1. Variance swaps and the VIX
			2.6.2. Volatility swaps
			2.6.3. Weighted variance swaps and gamma swaps
		2.7. CHAPTER’S DIGEST
Section II: Mathematical tools
	Chapter 3: A primer on Malliavin Calculus
		3.1. DEFINITIONS AND BASIC PROPERTIES
			3.1.1. The Malliavin derivative operator
				3.1.1.1. Basic properties
			3.1.2. The divergence operator
		3.2. COMPUTATION OF MALLIAVIN DERIVATIVES
			3.2.1. The Malliavin derivative of an It ˆo process
			3.2.2. The Malliavin derivative of a diffusion process
				3.2.2.1. The Malliavin derivative of a diffusion process as a solution of a linear SDE
				3.2.2.2. Representation formulas for the Malliavin derivative of a diffusion process
		3.3. MALLIAVIN DERIVATIVES FOR GENERAL SV MODELS
			3.3.1. The SABR volatility
			3.3.2. The Heston volatility
			3.3.3. The 3/2. Heston volatility
		3.4. CHAPTER’S DIGEST
	Chapter 4: Key tools in Malliavin Calculus
		4.1. THE CLARK-OCONE-HAUSSMAN FORMULA
			4.1.1. The Clark-Ocone-Haussman formula and the martingale representation theorem
			4.1.2. Hedging in the Black-Scholes model
			4.1.3. A martingale representation for spot and integrated volatilities
				4.1.3.1. The SABR volatility
				4.1.3.2. The Heston volatility
			4.1.4. A martingale representation for non-log-normal assets
		4.2. THE INTEGRATION BY PARTS FORMULA
			4.2.1. The integration-by-parts formula for the Malliavin derivative and the Skorohod integral operators
			4.2.2. Delta, Vega, and Gamma in the Black-Scholes model
				4.2.2.1. The delta
				4.2.2.2. The vega
				4.2.2.3. The gamma
			4.2.3. The Delta of an Asian option in the Black-Scholes model
			4.2.4. The Stochastic volatility case
				4.2.4.1. The delta in stochastic volatility models
				4.2.4.2. The gamma in stochastic volatility models
		4.3. THE ANTICIPATING ITOˆ ’S FORMULA
			4.3.1. The anticipating It ˆ o’s formula as an extension of It ˆ o’s formula
			4.3.2. The law of an asset price as a perturbation of a mixed log-normal
			4.3.3. The moments of log-prices in stochastic volatility models
			4.3.4. Some applications to volatility derivatives
				4.3.4.1. Leverage swaps and gamma swaps
				4.3.4.2. Arithmetic variance swaps
		4.4. CHAPTER’S DIGEST
	Chapter 5: Fractional Brownian motion and rough volatilities
		5.1. THE FRACTIONAL BROWNIAN MOTION
			5.1.1. Correlated increments
			5.1.2. Long and short memory
			5.1.3. Stationary increments and self-similarity
			5.1.4. Hölder continuity
			5.1.5. The p-variation and the semimartingale property
			5.1.6. Representations of the fBm
		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
			5.6.1. Fractional Ornstein-Uhlenbeck volatilities
			5.6.2. The rough Bergomi model
			5.6.3. A fractional Heston model
		5.7. CHAPTER’S DIGEST
Section III: Applications of Malliavin Calculus to the study of the implied volatility surface
	Chapter 6: The ATM short-time level of the implied volatility
		6.1. BASIC DEFINITIONS AND NOTATION
		6.2. THE CLASSICAL HULL AND WHITE FORMULA
			6.2.1. Two proofs of the Hull and White formula
				6.2.1.1. Conditional expectations
				6.2.1.2. The Hull and White formula from classical It ˆ o’s formula
		6.3. AN EXTENSION OF THE HULL AND WHITE FORMULA FROM THE ANTICIPATING ITOˆ ’S FORMULA
		6.4. DECOMPOSITION FORMULAS FOR IMPLIED VOLATILITIES
		6.5. THE ATM SHORT-TIME LEVEL OF THE IMPLIED VOLATILITY
			6.5.1. The uncorrelated case
			6.5.2. The correlated case
			6.5.3. Approximation formulas for the ATMI
			6.5.4. Examples
				6.5.4.1. Diffusion models
				6.5.4.2. Local volatility models
				6.5.4.3. Fractional volatilities
			6.5.5. Numerical experiments
		6.6. CHAPTER’S DIGEST
	Chapter 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
			7.6.1. Diffusion stochastic volatilities: finite limit of the ATM skew slope
				7.6.1.1. Models based on the Ornstein-Uhlenbeck process
				7.6.1.2. The SABR model
				7.6.1.3. The Heston model
				7.6.1.4. The two-factor Bergomi model
			7.6.2. Local volatility models: the one-half rule and dynamic inconsisten
			7.6.3. Stochastic-local volatility models
			7.6.4. Fractional stochastic volatility models
				7.6.4.1. Fractional Ornstein-Uhlenbeck volatilities
				7.6.4.2. The rough Bergomi model
				7.6.4.3. The approximation of fractional volatilities by Markov processes
			7.6.5. Time-varying coefficients
		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
	Chapter 8: The ATM short-time curvature
		8.1. SOME EMPIRICAL FACTS
		8.2. THE UNCORRELATED CASE
			8.2.1. A representation for the ATM curvature
			8.2.2. Limit results
			8.2.3. Examples
				8.2.3.1. Diffusion stochastic volatilities
				8.2.3.2. Fractional volatility models
		8.3. THE CORRELATED CASE
			8.3.1. A representation for the ATM curvature
			8.3.2. Limit results
			8.3.3. The convexity of the short-time implied volatility
		8.4. EXAMPLES
			8.4.1. Local volatility models
			8.4.2. Diffusion volatility models
			8.4.3. Fractional volatilities
				8.4.3.1. Models based on fractional Ornstein-Uhlenbeck processes
		8.5. CHAPTER’S DIGEST
Section IV: The implied volatility of non-vanilla options
	Chapter 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
			9.5.1. Local volatility models
			9.5.2. Stochastic volatility models
			9.5.3. Fractional volatility models
			9.5.4. Time-depending coefficients
		9.6. AT-THE-MONEY CURVATURE
			9.6.1. The uncorrelated case
			9.6.2. The correlated case
		9.7. CHAPTER’S DIGEST
	Chapter 10: Options on the VIX
		10.1. THE ATM SHORT-TIME LEVEL AND SKEW OF THE IMPLIED VOLATILITY
			10.1.1. The ATMI short-time limit
			10.1.2. The short-time skew of the ATMI volatility
		10.2. VIX OPTIONS
			10.2.1. The short-end level of the ATMI of VIX options
			10.2.2. The ATM skew of VIX options
		10.3. CHAPTER’S DIGEST
Bibliography
Index