Analysis, geometry, and modeling in finance : advanced methods in option pricing /

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Bibliographic Details
Author / Creator:	Henry-Labordère, Pierre.
Imprint:	Boca Raton : CRC Press, c2009.
Description:	383 p. : ill. ; 25 cm.
Language:	English
Series:	Chapman & Hall/CRC financial mathematics series
Subject:	Options (Finance) -- Mathematical models. Options (Finance) -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/7643776

Hidden Bibliographic Details
ISBN:	9781420086997 (alk. paper) 1420086995 (alk. paper)
Notes:	"A Chapman & Hall book." Includes bibliographical references (p. 369-378) and index.

Table of Contents:

1. Introduction
2. A Brief Course in Financial Mathematics
2.1. Derivative products
2.2. Back to basics
2.2.1. Sigma-algebra
2.2.2. Probability measure
2.2.3. Random variables
2.2.4. Conditional probability
2.2.5. Radon-Nikodym derivative
2.3. Stochastic processes
2.4. Ito process
2.4.1. Stochastic integral
2.4.2. Ito's lemma
2.4.3. Stochastic differential equations
2.5. Market models
2.6. Pricing and no-arbitrage
2.6.1. Arbitrage
2.6.2. Self-financing portfolio
2.7. Feynman-Kac's theorem
2.8. Change of numeraire
2.9. Hedging portfolio
2.10. Building market models in practice
2.10.1. Equity asset case
2.10.2. Foreign exchange rate case
2.10.3. Fixed income rate case
2.10.4. Commodity asset case
2.11. Problems
3. Smile Dynamics and Pricing of Exotic Options
3.1. Implied volatility
3.2. Static replication and pricing of European option
3.3. Forward starting options and dynamics of the implied volatility
3.3.1. Sticky rules
3.3.2. Forward-start options
3.3.3. Cliquet options
3.3.4. Napoleon options
3.4. Interest rate instruments
3.4.1. Bond
3.4.2. Swap
3.4.3. Swaption
3.4.4. Convexity adjustment and CMS option
3.5. Problems
4. Differential Geometry and Heat Kernel Expansion
4.1. Multi-dimensional Kolmogorov equation
4.1.1. Forward Kolmogorov equation
4.1.2. Backward Kolmogorov's equation
4.2. Notions in differential geometry
4.2.1. Manifold
4.2.2. Maps between manifolds
4.2.3. Tangent space
4.2.4. Metric
4.2.5. Cotangent space
4.2.6. Tensors
4.2.7. Vector bundles
4.2.8. Connection on a vector bundle
4.2.9. Parallel gauge transport
4.2.10. Geodesics
4.2.11. Curvature of a connection
4.2.12. Integration on a Riemannian manifold
4.3. Heat kernel on a Riemannian manifold
4.4. Abelian connection and Stratonovich's calculus
4.5. Gauge transformation
4.6. Heat kernel expansion
4.7. Hypo-elliptic operator and Hormander's theorem
4.7.1. Hypo-elliptic operator
4.7.2. Hormander's theorem
4.8. Problems
5. Local Volatility Models and Geometry of Real Curves
5.1. Separable local volatility model
5.1.1. Weak solution
5.1.2. Non-explosion and martingality
5.1.3. Real curve
5.2. Local volatility model
5.2.1. Dupire's formula
5.2.2. Local volatility and asymptotic implied volatility
5.3. Implied volatility from local volatility
6. Stochastic Volatility Models and Geometry of Complex Curves
6.1. Stochastic volatility models and Riemann surfaces
6.1.1. Stochastic volatility models
6.1.2. Riemann surfaces
6.1.3. Associated local volatility model
6.1.4. First-order asymptotics of implied volatility
6.2. Put-Call duality
6.3. [lambda]-SABR model and hyperbolic geometry
6.3.1. [lambda]-SABR model
6.3.2. Asymptotic implied volatility for the [lambda]-SABR
6.3.3. Derivation
6.4. Analytical solution for the normal and log-normal SABR model
6.4.1. Normal SABR model and Laplacian on H[superscript 2]
6.4.2. Log-normal SABR model and Laplacian on H[superscript 3]
6.5. Heston model: a toy black hole
6.5.1. Analytical call option
6.5.2. Asymptotic implied volatility
6.6. Problems
7. Multi-Asset European Option and Flat Geometry
7.1. Local volatility models and flat geometry
7.2. Basket option
7.2.1. Basket local volatility
7.2.2. Second moment matching approximation
7.3. Collaterized Commodity Obligation
7.3.1. Zero correlation
7.3.2. Non-zero correlation
7.3.3. Implementation
8. Stochastic Volatility Libor Market Models and Hyperbolic Geometry
8.1. Introduction
8.2. Libor market models
8.2.1. Calibration
8.2.2. Pricing with a Libor market model
8.3. Markovian realization and Frobenius theorem
8.4. A generic SABR-LMM model
8.5. Asymptotic swaption smile
8.5.1. First step: deriving the ELV
8.5.2. Connection
8.5.3. Second step: deriving an implied volatility smile
8.5.4. Numerical tests and comments
8.6. Extensions
8.7. Problems
9. Solvable Local and Stochastic Volatility Models
9.1. Introduction
9.2. Reduction method
9.3. Crash course in functional analysis
9.3.1. Linear operator on Hilbert space
9.3.2. Spectrum
9.3.3. Spectral decomposition
9.4. 1D time-homogeneous diffusion models
9.4.1. Reduction method
9.4.2. Solvable (super)potentials
9.4.3. Hierarchy of solvable diffusion processes
9.4.4. Natanzon (super)potentials
9.5. Gauge-free stochastic volatility models
9.6. Laplacian heat kernel and Schrodinger equations
9.7. Problems
10. Schrodinger Semigroups Estimates and Implied Volatility Wings
10.1. Introduction
10.2. Wings asymptotics
10.3. Local volatility model and Schrodinger equation
10.3.1. Separable local volatility model
10.3.2. General local volatility model
10.4. Gaussian estimates of Schrodinger semigroups
10.4.1. Time-homogeneous scalar potential
10.4.2. Time-dependent scalar potential
10.5. Implied volatility at extreme strikes
10.5.1. Separable local volatility model
10.5.2. Local volatility model
10.6. Gauge-free stochastic volatility models
10.7. Problems
11. Analysis on Wiener Space with Applications
11.1. Introduction
11.2. Functional integration
11.2.1. Functional space
11.2.2. Cylindrical functions
11.2.3. Feynman path integral
11.3. Functional-Malliavin derivative
11.4. Skorohod integral and Wick product
11.4.1. Skorohod integral
11.4.2. Wick product
11.5. Fock space and Wiener chaos expansion
11.5.1. Ornstein-Uhlenbeck operator
11.6. Applications
11.6.1. Convexity adjustment
11.6.2. Sensitivities
11.6.3. Local volatility of stochastic volatility models
11.7. Problems
12. Portfolio Optimization and Bellman-Hamilton-Jacobi Equation
12.1. Introduction
12.2. Hedging in an incomplete market
12.3. The feedback effect of hedging on price
12.4. Non-linear Black-Scholes PDE
12.5. Optimized portfolio of a large trader
A. Saddle-Point Method
B. Monte-Carlo Methods and Hopf Algebra
B.1. Introduction
B.1.1. Monte Carlo and Quasi Monte Carlo
B.1.2. Discretization schemes
B.1.3. Taylor-Stratonovich expansion
B.2. Algebraic Setting
B.2.1. Hopf algebra
B.2.2. Chen series
B.3. Yamato's theorem
References
Index

Analysis, geometry, and modeling in finance : advanced methods in option pricing /

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