Andrea Pascucci

PDE and martingale methods in option pricing

This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Levy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform.

On a class of degenerate parabolic equations of Kolmogorov type

We adapt the Levis parametrix method to prove existence, estimates and qualitative properties of a global fundamental solution to ultraparabolic partial differential equations of Kolmogorov type. Existence and uniqueness results for the Cauchy problem are also proved.

Free boundary and optimal stopping problems for American Asian options

Abstract We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.

On the complete model with stochastic volatility by Hobson and Rogers

In the complete model with stochastic volatility by Hobson and Rogers, preference independent options prices are solutions to degenerate partial differential equations obtained by including additional state variables describing the dependence on past prices of the underlying. In this paper, we aim to emphasize the mathematical tractability of the model by presenting analytical and numerical results comparable with the known ones in the classical Black–Scholes environment.

The obstacle problem for a class of hypoelliptic ultraparabolic equations

We study the obstacle problem for a class of degenerate parabolic operators with continuous coefficients. This problem arises in the Black–Scholes framework when considering path-dependent American options. We prove the existence of a unique strong solution u to the Cauchy and Cauchy–Dirichlet problems, under rather general assumptions on the obstacle function. We also show that u is a solution in the viscosity sense.

Analytical Approximation of the Transition Density in a Local Volatility Model

We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.

Parametrix Approximation of Diffusion Transition Densities

A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

THE MOSER'S ITERATIVE METHOD FOR A CLASS OF ULTRAPARABOLIC EQUATIONS

We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions.

Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models

We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility,

On the viscosity solutions of a stochastic differential utility problem

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