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BL
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960283
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b714653
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| Main Entry
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Boyarchenko, Svetlana I.
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| Title & Author
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Non-Gaussian Merton-Black-Scholes theory /\ Svetlana I. Boyarchenko, Sergei Z. Levendorskiĭ.
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| Publication Statement
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Singapore ;River Edge, NJ :: World Scientific,, 2002.
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| Series Statement
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Advanced series on statistical science & applied probability ;; v. 9
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| Page. NO
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1 online resource (xxi, 398 pages) :: illustrations
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| ISBN
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9789810249441
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: 9789812777485
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: 9810249446
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: 9812777482
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9810249446
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| Bibliographies/Indexes
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Includes bibliographical references (pages 385-392) and index.
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| Contents
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Ch. 1. Introduction. 1.1. The Gaussian Merton-Black-Scholes theory. 1.2. Regular Lévy processes of exponential type. 1.3. Pricing of contingent claims. 1.4. The generalized Black-Scholes equation. 1.5. Analytical methods used in the book. 1.6. An overview of the results covered in the book. 1.7. Commentary -- ch. 2. Lévy processes. 2.1. Basic notation and definitions. 2.2. Lévy processes: general definitions. 2.3. Lévy processes as Markov processes. 2.4. Boundary value problems for the Black-Scholes-type equation. 2.5. Commentary -- ch. 3. Regular Lévy Processes of Exponential type in ID. 3.1. Model classes. 3.2. Two definitions of regular Lévy processes of exponential type. 3.3. Properties of the characteristic exponents and probability densities of RLPE. 3.4. Properties of the infinitesimal generators. 3.5. A "naive approach" to the construction of RLPE or why they are natural from the point of view of the theory of PDO. 3.6. The Wiener-Hopf factorization -- ch. 4. Pricing and hedging of contingent claims of European type. 4.1. Equivalent Martingale measures in a Lévy market. 4.2. Pricing of European options and the generalized Black-Scholes formula. 4.3. Generalized Black-Scholes equation and its properties for different RLPE and different choices of EMM, and implications for parameter fitting. 4.4. Other European options. 4.5. Hedging. 4.6. Commentary -- ch. 5. Perpetual American options. 5.1. The reduction to a free boundary problem for the stationary generalized Black-Scholes equation. 5.2. Perpetual American put: the optimal exercise price and the rational put price. 5.3. Perpetual American call. 5.4. Put-like and call-like options: the case of more general payoffs. 5.5. Commentary -- ch. 6. American options: finite time horizon. 6.1. General discussion. 6.2. Approximations of the American put price. 6.3. American put near expiry -- ch. 7. First-touch digitals. 7.1. An overview. 7.2. Exact pricing formulas for first-touch digitals. 7.3. The Wiener-Hopf factorization with a parameter. 7.4. Price near the barrier. 7.5. Asymptotics as [symbol] -- ch. 8. Barrier options. 8.1. Types of barrier options. 8.2. Down-and-out call option without a rebate. 8.3. Asymptotics of the option price near the barrier. 8.4. Commentary -- ch. 9. Multi-asset contracts. 9.1. Multi-dimensional regular Lévy processes of exponential type. 9.2. European-style contracts. 9.3. Locally risk-minimizing hedging with a portfolio of several assets. 9.4. Weighted discretely sampled geometric average -- ch. 10. Investment under uncertainty and capital accumulation. 10.1. Irreversible investment and uncertainty. 10.2. The investment threshold. 10.3. Capital accumulation under RLPE. 10.4. Computational results. 10.5. Approximate formulas and the comparative statics -- ch. 11. Endogenous default and pricing of the corporate debt. 11.1. An overview. 11.2. Endogenous default. 11.3. Equity of a firm near bankruptcy level and the yield spread for junk bonds. 11.4. The case of a solvent firm. 11.5. Endogenous debt level and endogenous leverage. 11.6. Conclusion. 11.7. Auxiliary results.
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Ch. 12. Fast pricing of European options. 12.1. Introduction. 12.2. Transformation of the pricing formula for the European put. 12.3. FFT and IAC. 12.4. Comparison of FFT and IAC -- ch. 13. Discrete time models. 13.1. Bermudan options and discrete time models. 13.2. A perpetual American put in a discrete time model. 13.3. The Wiener-Hopf factorization. 13.4. Optimal exercise boundary and rational price of the option -- ch. 14. Feller processes of normal inverse Gaussian type. 14.1. Introduction. 14.2. Constructions of NIG-like Feller process via pseudodifferential operators. 14.3. Applications for financial mathematics. 14.4. Discussion and conclusions -- ch. 15. Pseudo-differential operators with constant symbols. 15.1. Introduction. 15.2. Classes of functions. 15.3. Space S'[symbol] of generalized functions on R[symbol]. 15.4. Pseudo-differential operators with constant symbols on R[symbol]. 15.5. The action of PDO in the Sobolev spaces on R[symbol]. 15.6. Parabolic equations. 15.7. The Wiener-Hopf equation on a half-line I. 15.8. Parabolic equations on [0,T] x R[symbol]. 15.9. PDO in the Sobolev spaces with exponential weights, in 1D. 15.10. The Sobolev spaces with exponential weights and PDO on a half-line. 15.11. Parabolic equations in spaces with exponential weights. 15.12. The Wiener-Hopf equation on a half-line II. 15.13. Parabolic equations on R x R[symbol] with exponentially growing data -- ch. 16. Elements of calculus of pseudodifferential operators. 16.1. Basics of the theory of PDO with symbols of the class S[symbol]. 16.2. Operators depending on parameters. 16.3. Operators with symbols holomorphic in a tube domain. 16.4. Proofs of auxiliary technical results. 16.5. Change of variables and pricing of multi-asset contracts. 16.6. Pricing of barrier options under Lévy-like Feller processes.
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| Abstract
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This book introduces an analytically tractable and computationally effective class of non-Gaussian models for shocks (regular Lévy processes of the exponential type) and related analytical methods similar to the initial Merton-Black-Scholes approach, which the authors call the Merton-Black-Scholes theory. The authors have chosen applications interesting for financial engineers and specialists in financial economics, real options, and partial differential equations (especially pseudodifferential operators); specialists in stochastic processes will benefit from the use of the pseudodifferential.
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| Subject
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Finance-- Mathematical models.
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Finances-- Modèles mathématiques.
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Finanças (modelos matemáticos)
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Finance-- Mathematical models.
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Finance.
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MATHEMATICS-- Probability Statistics-- Stochastic Processes.
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Modèle mathématique.
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Option (Finances)
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Processos estocasticos.
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Processos gaussianos.
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| Dewey Classification
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519.2/3
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| LC Classification
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HG106.B69 2002eb
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| Added Entry
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Levendorskiĭ, Serge,1951-
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