Non-Gaussian Merton-Black-Scholes Theory

The following sections are included:

• Preface

• Contents

The following sections are included:

• The Gaussian Merton-Black-Scholes theory
  • Strong points
  • Drawbacks
    • Fat tails and anomalous skewness and kurtosis
    • The Black-Scholes market and formula
    • Volatility smile and volatility surface
  • Remedies for the MBS-theory
    • General remarks
    • Jump-diffusion models and stochastic volatility models
    • Lévy processes

• Regular Lévy Processes of Exponential type
  • Characteristic function and exponent, Lévy measure and Lévy-Khintchine formula
  • Definition of RLPE in 1D
  • Infinitesimal generators of RLPE as PDO

• Pricing of contingent claims
  • Discrete time models with a discrete space of states: No-arbitrage and equivalent martingale measures
  • Discrete time models with a discrete space of states: Completeness of the market, and pricing of derivative securities
  • Absence of arbitrage, EMM and completeness in the Gaussian Black-Scholes model market
  • Sufficient condition for no-arbitrage in a Lévy market and incompleteness of a Lévy market. The pricing formula for contingent claims of European type and the problem of a choice of EMM
  • On pricing based on the utility maximization

• The Generalized Black-Scholes equation
  • The informal derivation
  • An outline of the reduction of the pricing of contingent claims to boundary value problems for the generalized Black-Scholes equation: barrier options
  • The case of interest bearing securities
  • The generalized Merton-Black-Scholes theory
  • Optimal stopping problems and the smooth pasting condition
  • The case of a dividend-paying stock

• Analytical methods used in the book
  • The Fourier transform, and Complex Analysis
  • The Wiener-Hopf factorization and the Wiener-Hopf equation
  • The case of the non-stationary Black-Scholes equation and the constant barrier
  • The case of the non-constant barrier and multi-asset contracts
  • Pseudodifferential operators

• An overview of the results covered in the book
  • Elements of the theory of Lévy processes
  • Option pricing
  • Investment under uncertainty and capital accumulation
  • Endogenous default and pricing of the corporate debt
  • Numerical methods
  • Extensions
  • Basics of PDO theory

• Commentary

The following sections are included:

• Basic notation and definitions
  • Random variables
  • Conditional expectation
  • The Fourier transform and the Laplace transform. Characteristic functions

• Lévy processes: general definitions
  • Lévy processes and infinitely divisible distributions
  • The Lévy-Khintchine formula and generating triplet
  • Constructions of Lévy processes
    • Explicit constructions
    • Subordinators and subordinated processes
    • Linear transformations of Lévy processes
  • The Wiener-Hopf factorization

• Lévy processes as Markov processes
  • Markov property and Feller property
  • Stopping times and the strong Markov property
  • Resolvent operator and infinitesimal generator
  • Dynkin's formula
  • Duality
  • Absolutely continuous resolvents
  • Operators

• Boundary value problems for the Black-Scholes-type equation
  • Boundary problems for the stationary Black-Scholes type equation
  • Boundary problems for the non-stationary Black-Scholes type equation

• Commentary

The following sections are included:

• Model Classes
  • An overview
  • KoBoL family: direct construction via the Lévy-Khintchine formula
    • Definition in terms of the Lévy measure
    • The characteristic exponent
    • Moments of a KoBoL process
    • Graphs of characteristic exponents and probability densities of different KoBoL processes
    • Proofs of technical lemmas
  • Normal Inverse Gaussian processes and Normal Tempered Stable Lévy Processes: construction via subordination
  • Variance Gamma Processes
  • Hyperbolic Processes and Generalized Hyperbolic Processes
  • Comparison of the tail behaviour of probability densities for different model classes of processes

• Two definitions of Regular Lévy Processes of Exponential type
  • Definition in terms of the Lévy measure
  • Definition in terms of the characteristic exponent

• Properties of the characteristic exponents and probability densities of RLPE

• Properties of the infinitesimal generators

• A “naive approach” to the construction of RLPE or why they are natural from the point of view of the theory of PDO
  • Modified Stable Processes and RLPE

• The Wiener-Hopf factorization
  • The case of Lévy processes of exponential type
  • The Wiener-Hopf factorization for RLPE
  • Approximate formulas for the factors in the case of NIG, HP, KoBoL and NTS Lévy processes

The following sections are included:

• Equivalent Martingale Measures in a Lévy market
  • Esscher transform
  • General case

• Pricing of European options and the generalized Black-Scholes formula
  • Pricing formulas: convolution with the pricing kernel and the Fourier-inversion formula
  • Call and put options
  • Numerical examples: the comparison with the Black-Scholes formula, and the shapes of the smile
  • The problem of the model fitting and evaluation

• Generalized Black-Scholes equation and its properties for different RLPE and different choices of EMM, and implications for parameter fitting

• Other European options
  • Power options
  • Digital options
  • Combinations
  • General case: the condition on the rate of growth of the payoff at the infinity and the origin

• Hedging
  • General discussion
  • Locally risk-minimizing hedging ratio
  • Continuity of the locally risk-minimizing hedging ratio till the expiry
  • Numerical Examples

• Commentary

The following sections are included:

• The reduction to a free boundary problem for the stationary generalized Black-Scholes equation
  • General discussion
  • Free boundary value problem for the price of the perpetual American option
  • Main Lemma

• Perpetual American put: the optimal exercise price and the rational put price
  • Main Theorem
  • Proof of optimality in the class
  • Proof of optimality in the class
  • Failure of the smooth pasting principle for some RLPE's and its substitute
  • Approximate formulas for the case of model RLPE
  • Proof of Theorem 5.2

• Perpetual American call
  • Main results
  • Proof of optimality in the case of unbounded payoffs

• Put-like and call-like options: the case of more general payoffs
  • Put-like options
  • Call-like options

• Commentary

The following sections are included:

• General discussion
  • The free boundary problem
  • The role of dividends and non-zero interest rate

• Approximations of the American put price
  • Geske-Johnson approximation
  • Analytic method of lines
  • Carr's randomization

• American put near expiry
  • Behaviour of the early exercise boundary near the maturity date
  • general equation for h(Δ) in the case of NTS processes
  • The case of processes of order ν ∈ [1,2)

The following sections are included:

• An overview

• Exact pricing formulas for first-touch digitals

• The Wiener-Hopf factorization with a parameter
  • Formulas for the factors revisited
  • Analytic continuation of the factors w.r.t. λ

• Price near the barrier
  • Asymptotics as x → +0, and τ > 0 is fixed
    • Main Theorem
    • Numerical example
    • Proof of Theorem 7.4
  • Asymptotics as x → +0 and τ → +∞

• Asymptotics as τ → +∞

The following sections are included:

• Types of barrier options
  • Standard barrier options
  • Options with a rebate

• Down-and-out call option without a rebate
  • Reduction to the boundary problem for the Generalized Black-Scholes equation
  • Reduction to the Cauchy problem for an ordinary differential operator with the operator-valued coefficient
  • Construction of the resolvent, and the Wiener-Hopf factorization with a parameter
  • Proof of Eq. (8.17) and continuity of the RHS in Eq. (8.19)

• Asymptotics of the option price near the barrier

• Commentary

The following sections are included:

• Multi-dimensional Regular Lévy Processes of Exponential type
  • Multi-dimensional KoBoL family
  • Normal Inverse Gaussian processes and Normal Tempered Stable Lévy processes
  • Hyperbolic Processes, Generalized Hyperbolic Processes and Variance Gamma Processes
  • Definition of multi-dimensional RLPE

• European-style contracts
  • The Esscher transform
  • Pricing of power forwards
  • Pricing of European options on one asset
  • Pricing of exchange options and basket options

• Locally risk–minimizing hedging with a portfolio of several assets
  • The set-up
  • Weights of the hedging portfolio
  • Hedging of European claims
  • Locally risk-minimizing hedging of “power forwards”
  • Approximate locally risk-minimizing hedging: general case

• Weighted discretely sampled geometric average

The following sections are included:

• Irreversible investment and uncertainty

• The investment threshold

• Capital accumulation under RLPE

• Computational results

• Approximate formulas and the comparative statics

The following sections are included:

• An overview

• Endogenous default

• Equity of a firm near bankruptcy level and the yield spread for junk bonds

• The case of a solvent firm

• Endogenous debt level and endogenous leverage

• Conclusion

• Auxiliary results
  • Proof of Eq. (11.36)–Eq. (11.38)
  • Proof of Eq. (11.45)–Eq. (11.46)

The following sections are included:

• Introduction

• Transformation of the pricing formula for the European put

• FFT and IAC
  • Fast Fourier Transform
  • Integration-along-cut method
    • Case a)
    • Case b)

• Comparison of FFT and IAC

The following sections are included:

• Bermudan options and discrete time models

• A perpetual American put in a discrete time model
  • Process specification
  • The pricing problem for a perpetual American option as an optimal stopping problem

• The Wiener-Hopf factorization
  • General formulas
  • Some useful properties of the factors a_±
  • The case of exponential polynomials

• Optimal exercise boundary and rational price of the option

The following sections are included:

• Introduction

• Constructions of NIG-like Feller process via pseudodifferential operators
  • Naive construction: NIG-like generators with state-dependent parameters
  • Constructions via infinitesimal generators: subordination of semigroups of operators

• Applications for financial mathematics
  • Generalised Black-Scholes equation for contingent claims under NIG-like Feller processes
  • Pricing of European options under NIG-like Feller processes obtained via subordination
  • Pricing of European options under NIG-like Feller processes obtained by the naive approach

• Discussion and conclusions

The following sections are included:

• Introduction

• Classes of functions
  • General notation and remarks
  • The space
  • The Fourier transform

• Space of generalized functions on Rⁿ
  • Generalized functions (distributions)
  • Operations in
  • The Fourier transform of generalized functions
  • The regularization of oscillatory integrals
  • Tensor product of generalized functions
  • Generalized functions on an open set. The support of a generalized function

• Pseudo-differential operators with constant symbols on Rⁿ
  • Definition of PDO with constant symbols
  • The Sobolev spaces H^s(Rⁿ)
  • Action of PDO in the scale H^s(Rⁿ)
  • Elliptic equations
  • Restriction to a hyperplane
  • The Sobolev embedding theorem
  • Action in the Hölder spaces

• The action of PDO in the Sobolev spaces on Rⁿ_±
  • Spaces and PDO in a half-space
  • Spaces H^s(Rⁿ_±)
  • The Cauchy-type integral and its decomposition

• Parabolic equations
  • The Cauchy problem for the parabolic equation
  • Reduction to a family of ODE on R₊
  • Parabolic equation as ODE with the operator-valued coefficient
  • Parabolic equations and the Hölder-Zygmund spaces
  • The inhomogeneous parabolic equation and the asymptotics of the solution as t → +∞

• The Wiener-Hopf equation on a half-line I
  • Statement of the problem
  • The Wiener-Hopf factorization
  • Main Theorems
  • The asymptotics of the solution near the boundary

• Parabolic equations on [0,T] × R₊
  • The statement of the boundary problem
  • The Wiener-Hopf factorization with a parameter
  • Reduction to ODE with the operator-valued coefficient and the solution

• PDO in the Sobolev spaces with exponential weights, in 1D
  • Generalized functions
  • The Sobolev spaces H^s,γ(R)
  • PDO with the symbols holomorphic in a strip, and their action in the scale H^s,γ(R)
  • Action of PDO in the Hölder-Zygmund spaces with exponential weights

• The Sobolev spaces with exponential weights and PDO on a half-line
  • Spaces
  • Spaces H^s,γ(R_±)
  • The Cauchy-type integral and its decomposition

• Parabolic equations in spaces with exponential weights

• The Wiener-Hopf equation on a half-line II
  • Statement of the problem
  • The Wiener-Hopf factorization of symbols holomorphic in a strip
  • Main Theorems
  • The asymptotics of solutions near the boundary and the free boundary problems
  • The asymptotics of solutions at the infinity

• Parabolic equations on R × R₊ with exponentially growing data

The following sections are included:

• Basics of the theory of PDO with symbols of the class
  • Asymptotic summation and the composition theorem
  • Parametrix and approximate square root
  • Boundedness theorem
  • The inverse operator and the square root

• Operators depending on parameters

• Operators with symbols holomorphic in a tube domain

• Proofs of auxiliary technical results

• Change of variables and pricing of multi-asset contracts

• Pricing of barrier options under Lévy-like Feller processes

The following sections are included:

• Bibliography

• Index