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E-grāmata: Nonlinear Option Pricing

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(Bloomberg LP, New York, New York, USA), (Société Générale, Paris, France)
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New Tools to Solve Your Option Pricing Problems

For nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research—including Risk magazine’s 2013 Quant of the Year—Nonlinear Option Pricing compares various numerical methods for solving high-dimensional nonlinear problems arising in option pricing. Designed for practitioners, it is the first authored book to discuss nonlinear Black-Scholes PDEs and compare the efficiency of many different methods.

Real-World Solutions for Quantitative Analysts

The book helps quants develop both their analytical and numerical expertise. It focuses on general mathematical tools rather than specific financial questions so that readers can easily use the tools to solve their own nonlinear problems. The authors build intuition through numerous real-world examples of numerical implementation. Although the focus is on ideas and numerical examples, the authors introduce relevant mathematical notions and important results and proofs. The book also covers several original approaches, including regression methods and dual methods for pricing chooser options, Monte Carlo approaches for pricing in the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection method and the particle method for calibrating local stochastic volatility models to market prices of vanilla options with/without stochastic interest rates, the a + b? technique for building local correlation models that calibrate to market prices of vanilla options on a basket, and a new stochastic representation of nonlinear PDE solutions based on marked branching diffusions.

Recenzijas

"... provides a wide overview of the advanced modern techniques applied in financial modeling. It gives an optimal combination of analytical and numerical tools in quantitative finance. It could provide guidance on the development of nonlinear methods of option pricing for practitioners as well as for analysts." Nikita Y. Ratanov, from Mathematical Reviews Clippings, January 2015

" anyone with interest in quantitative finance and partial differential equations/continuous time stochastic analysis will not only greatly enjoy this book, but he or she will find both many numerical ideas of real practical interest as well as material for academic research, perhaps for years to come." Peter Friz, The Bachelier Finance Society

"This textbook provides a comprehensive treatment of numerical methods for nonlinear option pricing problems." Zentralblatt MATH 1285

"It is the only book of its kind. The contribution of this book is threefold: (a) a practical, intuitive, and self-contained derivation of various of the latest derivative pricing models driven by diffusion processes; (b) an exposition of various advanced Monte Carlo simulation schemes for solving challenging nonlinear problems arising in financial engineering; (c) a clear and accessible survey of the theory of nonlinear PDEs. The authors have done a brilliant job providing just the right amount of rigorous theory required to understand the advanced methodologies they present. Julien Guyon and Pierre Henry-Labordčre, as befitting their reputations as star quants, have done an excellent job presenting the latest theory of nonlinear PDEs and their applications to finance. Much of the material in the book consists of the authors own original results. I highly recommend this book to seasoned mathematicians and experienced quants in the industry Mathematicians will be able to see how practitioners argue heuristically to arrive at solutions of the toughest problems in financial engineering; practitioners of quantitative finance will find the book perfectly balanced between mathematical theory, financial modelling, and schemes for numerical implementation." Quantitative Finance, 2014

"Ever since Black and Scholes solved their eponymous linear PDE in 1969, the complexity of problems plaguing financial practitioners has exploded (non-linearly!). How fitting it is that nonlinear PDEs are now routinely used to extend the original framework. Written by two leading quants at two leading financial houses, this book is a tour de force on the use of nonlinear PDEs in financial valuation." Peter Carr, PhD, Global Head of Market Modeling, Morgan Stanley, New York, and Executive Director of Masters in Mathematical Finance, Courant Institute of Mathematical Sciences, New York University

"Finance used to be simple; you could go a long way with just linearity and positivity but this is not the case anymore. This superb book gives a wide array of modern methods for modern problems." Bruno Dupire, Head of Quantitative Research, Bloomberg L.P.

"In this unique and impressive book, the authors apply sophisticated modern tools of pure and applied mathematics, such as BSDEs and particle methods, to solve challenging nonlinear problems of real practical interest, such as the valuation of guaranteed equity-linked annuity contracts and the calibration of local stochastic volatility models. Not only that, but sketches of proofs and implementation details are included. No serious student of mathematical finance, whether practitioner or academic, can afford to be without it." Jim Gatheral, Presidential Professor, Baruch College, CUNY, and author of The Volatility Surface

"Guyon and Henry-Labordčre have produced an impressive textbook, which covers options and derivatives pricing from the point of view of nonlinear PDEs. This book is a comprehensive survey of nonlinear techniques, ranging from American options, uncertain volatility, and uncertain correlation models. It is aimed at graduate students or quantitative analysts with a strong mathematical background. They will find the book reasonably self-contained, i.e., discussing both the mathematical theory and the applications, in a very balanced approach. A must-read for the serious quantitative analyst." Marco Avellaneda, Courant Institute of Mathematical Sciences, New York University

Preface xvii
1 Option Pricing in a Nutshell
1(26)
1.1 The super-replication paradigm
1(13)
1.1.1 Models of financial markets
1(1)
1.1.2 Self-financing portfolios
2(1)
1.1.3 Arbitrage and arbitrage-free models
3(2)
1.1.4 Super-replication
5(6)
1.1.5 Complete models versus incomplete models
11(2)
1.1.6 Pricing in practice
13(1)
1.2 Stochastic representation of solutions of linear PDEs
14(10)
1.2.1 The Cauchy problem
14(2)
1.2.2 The Dirichlet problem
16(3)
1.2.3 The Neumann problem
19(5)
1.3 Exercises
24(3)
1.3.1 Local martingales and supermartingales
24(1)
1.3.2 Timer options on two assets
24(1)
1.3.3 Azema-Yor martingales
24(1)
1.3.4 Azema-Yor martingales and the Skorokhod embedding problem
25(2)
2 Monte Carlo
27(22)
2.1 The Monte Carlo method
27(5)
2.1.1 Principle of the method
27(1)
2.1.2 Sampling error, reduction of variance
28(2)
2.1.3 Discretization error, reduction of bias
30(2)
2.2 Euler discretization error
32(6)
2.2.1 Weak error
33(1)
2.2.2 Results concerning smooth or bounded payoffs
33(1)
2.2.3 Results for general payoffs
34(2)
2.2.4 Application to option pricing and hedging
36(2)
2.3 Romberg extrapolation
38(11)
2.3.1 Standard Romberg extrapolation
38(2)
2.3.2 Iterated Romberg extrapolation
40(1)
2.3.3 Numerical experiments with vanilla options
41(3)
2.3.4 Numerical experiments with path-dependent options
44(5)
3 Some Excursions in Option Pricing
49(16)
3.1 Complete market models
49(11)
3.1.1 Modeling the dynamics of implied volatility: Schonbucher's market model
50(1)
3.1.2 Modeling the dynamics of log-contract prices: Bergomi's variance swap model
51(2)
3.1.3 Modeling the dynamics of power-payoff prices: an HJM-like model
53(2)
3.1.4 The case of a family of power-payoffs
55(2)
3.1.5 Modeling the dynamics of local volatility: Carmona and Nadtochiy's market model
57(3)
3.2 Beyond replication and super-replication
60(3)
3.2.1 The utility indifference price
60(1)
3.2.2 Quantile hedging
61(1)
3.2.3 Minimum variance hedging
62(1)
3.3 Exercises
63(2)
3.3.1 Super-replication and quantile hedging
63(1)
3.3.2 Duality for the utility indifference price
63(1)
3.3.3 Utility indifference price for exponential utility
63(1)
3.3.4 Minimum variance hedging for the double lognormal stochastic volatility model
63(2)
4 Nonlinear PDEs: A Bit of Theory
65(28)
4.1 Nonlinear second order parabolic PDEs: Some generalities
65(6)
4.1.1 Parabolic PDEs
66(1)
4.1.2 What do we mean by a well-posed PDE?
67(1)
4.1.3 What do we mean by a solution?
67(1)
4.1.4 Comparison principle and uniqueness
67(4)
4.2 Why is a pricing equation a parabolic PDE?
71(2)
4.3 Finite difference schemes
73(4)
4.3.1 Introduction
73(1)
4.3.2 Consistency
74(1)
4.3.3 Stability
75(1)
4.3.4 Convergence
75(2)
4.4 Stochastic control and the Hamilton-Jacobi-Bellman PDE
77(6)
4.4.1 Introduction
77(1)
4.4.2 Standard form
78(1)
4.4.3 Bellman's principle
79(1)
4.4.4 Formal derivation of the HJB PDE
80(3)
4.5 Optimal stopping problems
83(1)
4.6 Viscosity solutions
84(6)
4.6.1 Motivation
84(2)
4.6.2 Definition
86(2)
4.6.3 Viscosity solutions to HJB equations
88(1)
4.6.4 Numerical scheme: The Barles-Souganadis framework
89(1)
4.7 Exercises
90(3)
4.7.1 The American timer put
90(1)
4.7.2 Super-replication price in a stochastic volatility model
91(1)
4.7.3 The robust utility indifference price
92(1)
5 Examples of Nonlinear Problems in Finance
93(32)
5.1 American options
93(1)
5.2 The uncertain volatility model
94(3)
5.2.1 The model
94(1)
5.2.2 Pricing vanilla options
95(1)
5.2.3 Robust super-replication
95(2)
5.2.4 A finite difference scheme
97(1)
5.3 Transaction costs: Leland's model
97(2)
5.4 Illiquid markets
99(11)
5.4.1 Feedback effects
99(3)
5.4.2 A simple model of a limit order book
102(2)
5.4.3 Continuous-time limit: Some intuition
104(1)
5.4.4 Admissible trading strategies
105(1)
5.4.5 Perfect replication paradigm
106(1)
5.4.6 The parabolic envelope of the Hamiltonian
107(3)
5.5 Super-replication under delta and gamma constraints
110(6)
5.5.1 Super-replication under delta constraints
110(2)
5.5.2 HJB and variational inequality
112(1)
5.5.3 Adding gamma constraints
112(1)
5.5.4 Numerical algorithm
113(3)
5.5.5 Numerical experiments
116(1)
5.6 The uncertain mortality model for reinsurance deals
116(3)
5.7 Different rates for borrowing and lending
119(1)
5.8 Credit valuation adjustment
120(2)
5.9 The passport option
122(2)
5.10 Exercises
124(1)
5.10.1 Transaction costs
124(1)
5.10.2 Super-replication under delta constraints
124(1)
6 Early Exercise Problems
125(48)
6.1 Super-replication of American options
125(4)
6.2 American options and semilinear PDEs
129(2)
6.3 The dual method for American options
131(2)
6.4 On the ownership of the exercise right
133(1)
6.5 On the finiteness of exercise dates
133(2)
6.6 On the accounting of multiple coupons
135(2)
6.7 Finite difference methods for American options
137(1)
6.8 Monte Carlo methods for American options
138(10)
6.8.1 Why is it difficult?
138(1)
6.8.2 Estimating Vti: The Tsitsiklis-Van Roy algorithm
139(1)
6.8.3 Estimating Ti*: The Longstaff-Schwartz algorithm
140(2)
6.8.4 Estimating M*: The Andersen-Broadie algorithm
142(2)
6.8.5 Conditional expectation approximation
144(1)
6.8.6 Parametric methods
145(1)
6.8.7 Quantization methods
146(1)
6.8.8 Mesh methods
147(1)
6.8.9 The case of continuous exercise dates
147(1)
6.9 Case study: Pricing and hedging of a multi-asset convertible bond
148(9)
6.9.1 Introduction
148(1)
6.9.2 Modeling assumptions
149(1)
6.9.3 Pricing parameters and regressors
149(1)
6.9.4 Price and exercise strategy
150(1)
6.9.5 Hedge ratios
151(2)
6.9.6 First order ratios need no reoptimization of exercise strategy
153(1)
6.9.7 Why exercise the convertible bond?
154(3)
6.10 Introduction to chooser options
157(1)
6.11 Regression methods for chooser options
158(5)
6.11.1 The "naive" primal algorithm
158(2)
6.11.2 Efficient Monte Carlo valuation of chooser options
160(3)
6.12 The dual algorithm for chooser options
163(3)
6.12.1 The case of two tokens
163(2)
6.12.2 The general case
165(1)
6.13 Numerical examples of pricing of chooser options
166(5)
6.13.1 The multi-times restrikable put
166(4)
6.13.2 The passport option
170(1)
6.14 Exercise: Bounds on prices of American call options
171(2)
7 Backward Stochastic Differential Equations
173(14)
7.1 First order BSDEs
173(7)
7.1.1 Introduction
173(4)
7.1.2 First order BSDEs provide an extension of Feynman-Kac's formula for semilinear PDEs
177(1)
7.1.3 Numerical simulation
178(2)
7.2 Reflected first order BSDEs
180(1)
7.3 Second order BSDEs
181(4)
7.3.1 Introduction
181(1)
7.3.2 Second order BSDEs provide an extension of Feynman-Kac's formula for fully nonlinear PDEs
182(1)
7.3.3 Numerical simulation
183(2)
7.4 Exercise: An example of semilinear PDE
185(2)
8 The Uncertain Lapse and Mortality Model
187(38)
8.1 Reinsurance deals
187(1)
8.2 The deterministic lapse and mortality model
188(3)
8.3 The uncertain lapse and mortality model
191(3)
8.4 Path-dependent payoffs
194(2)
8.4.1 Payoffs depending on realized volatility
194(1)
8.4.2 Payoffs depending on the running maximum
195(1)
8.5 Pricing the option on the up-and-out barrier
196(1)
8.6 An example of PDE implementation
196(2)
8.7 Monte Carlo pricing
198(3)
8.8 Monte Carlo pricing of the option on the up-and-out barrier
201(1)
8.9 Link with first order BSDEs
202(3)
8.10 Numerical results using PDEs
205(6)
8.10.1 Vanilla GMxB deal
205(4)
8.10.2 GMxB with variable fees
209(1)
8.10.3 GMxB with variable put nominal
210(1)
8.10.4 GMxB with cliqueting strikes
211(1)
8.11 Numerical results using Monte Carlo
211(14)
8.11.1 Vanilla GMxB deal
213(3)
8.11.2 GMxB with variable fee and variable put nominal
216(9)
9 The Uncertain Volatility Model
225(24)
9.1 Introduction
225(2)
9.2 The model
227(4)
9.3 The parametric approach
231(4)
9.3.1 The ideas behind the algorithm
231(1)
9.3.2 The algorithm
232(1)
9.3.3 Choice of the parameterization
233(1)
9.3.4 The single-asset case
234(1)
9.3.5 The two-asset case
234(1)
9.4 Solving the UVM with BSDEs
235(5)
9.4.1 A new numerical scheme
236(1)
9.4.2 First example: At-the-money call option
237(1)
9.4.3 The algorithm
238(1)
9.4.4 About the generation of the first N1 paths
239(1)
9.5 Numerical experiments
240(8)
9.5.1 Options with one underlying
240(3)
9.5.2 Options with two underlyings and no uncertainty on correlation
243(3)
9.5.3 Options with two underlyings and uncertainty on correlation
246(2)
9.6 Exercise: UVM with penalty term
248(1)
10 McKean Nonlinear Stochastic Differential Equations
249(22)
10.1 Definition
249(4)
10.2 The particle method in a nutshell
253(1)
10.3 Propagation of chaos and convergence of the particle method
254(11)
10.3.1 Building intuition: The BBGKY hierarchy
254(5)
10.3.2 Propagation of chaos and convergence of the particle method
259(2)
10.3.3 The McKean-Vlasov SDE propagates the chaos
261(4)
10.4 Exercise: Random matrices, Dyson Brownian motion, and McKean SDEs
265(2)
10.A The Monge-Kantorovich distance and its financial interpretation
267(4)
11 Calibration of Local Stochastic Volatility Models to Market Smiles
271(44)
11.1 Introduction
271(2)
11.2 The calibration condition
273(1)
11.3 Existence of the calibrated local stochastic volatility model
274(1)
11.4 The PDE method
274(2)
11.5 The Markovian projection method
276(4)
11.6 The particle method
280(2)
11.6.1 The particle method for the calibration of LSVMs to market smiles
280(1)
11.6.2 Regularizing kernel
281(1)
11.6.3 Acceleration techniques
281(1)
11.6.4 The algorithm
281(1)
11.7 Adding stochastic interest rates
282(14)
11.7.1 The calibration condition
283(2)
11.7.2 The PDE method
285(1)
11.7.3 The particle method
285(1)
11.7.4 First example: The particle method for the hybrid Ho-Lee/Dupire model
286(1)
11.7.5 Second example: The particle method for the hybrid Hull-White/Dupire model
287(3)
11.7.6 Malliavin representation of the local volatility
290(4)
11.7.7 Local stochastic volatility combined with Libor Market Models
294(2)
11.8 The particle method: Numerical tests
296(8)
11.8.1 Hybrid Ho-Lee/Dupire model
296(3)
11.8.2 Bergomi's local stochastic volatility model
299(2)
11.8.3 Existence under question
301(1)
11.8.4 Hybrid Ho-Lee/Local Bergomi model
302(2)
11.9 Exercise: Dynamics of forward variance swaps for the double lognormal SVM
304(1)
11.A Proof of Proposition 11.2
304(4)
11.B Proof of Formula (11.57)
308(1)
11.C Including (discrete) dividends
309(6)
11.C.1 Calibration of the Dupire local volatility
310(1)
11.C.2 Calibration of the SIR-LSVM
311(1)
11.C.3 Approximate formula for the call option in the Black-Scholes model with discrete dividends
311(4)
12 Calibration of Local Correlation Models to Market Smiles
315(64)
12.1 Introduction
315(5)
12.2 The FX triangle smile calibration problem
320(2)
12.3 A new representation of admissible correlations
322(3)
12.4 The particle method for local correlation
325(1)
12.5 Some examples of pairs of functions (a, b)
326(2)
12.6 Some links between local correlations
328(2)
12.7 Joint extrapolation of local volatilities
330(1)
12.8 Price impact of correlation
331(7)
12.8.1 The price impact formula
331(2)
12.8.2 Equivalent local correlation
333(1)
12.8.3 Implied correlation
334(2)
12.8.4 Impact of correlation on price
336(1)
12.8.5 Uncertain correlation model
337(1)
12.9 The equity index smile calibration problem
338(4)
12.10 Numerical experiments on the FX triangle problem
342(22)
12.10.1 Calibration
342(3)
12.10.2 Pricing
345(19)
12.11 Generalization to stochastic volatility, stochastic interest rates, and stochastic dividend yield
364(6)
12.11.1 The FX triangle smile calibration problem
364(4)
12.11.2 The equity index smile calibration problem
368(2)
12.12 Path-dependent volatility
370(5)
12.13 Exercises
375(2)
12.13.1 Arbitrage-freeness condition on cross-currency smiles: The case of one maturity
375(1)
12.13.2 Arbitrage-freeness condition on cross-currency smiles: The case of finitely many maturities
376(1)
12.A Calibration of an LSVM with stochastic interest rates and stochastic dividend yield
377(2)
13 Marked Branching Diffusions
379(48)
13.1 Nonlinear Monte Carlo algorithms for some semilinear PDEs
379(7)
13.1.1 A brute force algorithm
380(1)
13.1.2 Backward stochastic differential equations
380(1)
13.1.3 Gradient representation
381(5)
13.2 Branching diffusions
386(4)
13.2.1 Branching diffusions provide a stochastic representation of solutions of some semilinear PDEs
386(2)
13.2.2 Superdiffusions
388(2)
13.3 Marked branching diffusions
390(15)
13.3.1 Definition and main result
390(3)
13.3.2 Diagrammatic interpretation
393(1)
13.3.3 When is u bounded?
394(4)
13.3.4 Optimal probabilities pk
398(3)
13.3.5 Marked superdiffusions
401(1)
13.3.6 Numerical experiments
402(3)
13.4 Application: Credit valuation adjustment
405(10)
13.4.1 Introduction
405(3)
13.4.2 Algorithm: Final recipe
408(1)
13.4.3 Complexity
409(1)
13.4.4 Numerical examples
409(1)
13.4.5 CVA formulas
410(3)
13.4.6 Building polynomial approximations of the positive part
413(2)
13.5 System of semilinear PDEs
415(3)
13.5.1 Introduction
415(2)
13.5.2 Stochastic representation using multi-species marked branching diffusions
417(1)
13.6 Fully nonlinear PDEs
418(4)
13.6.1 A toy example: The Kardar-Parisi-Zhang PDE
418(1)
13.6.2 Bootstrapping
418(1)
13.6.3 The uncertain volatility model
419(1)
13.6.4 Exact simulation of one-dimensional SDEs
420(2)
13.7 Exercises
422(5)
13.7.1 The law of the number of branchings
422(2)
13.7.2 A high order PDE
424(1)
13.7.3 Hyperbolic (nonlinear) PDEs
425(2)
References 427(14)
Index 441
Julien Guyon, Pierre Henry-Labordere
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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