Giulia Di Nunno

White noise analysis for Lévy processes

We construct a white noise theory for Levy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark–Haussmann–Ocone formula for Levy processes F(ω)=E[F]+∑m⩾1∫0TE[Dt(m)F|Ft]♢Y•t(m)dt. Here E[F] is the generalized expectation, the operators Dt(m)F,m⩾1 are (generalized) Malliavin derivatives, ♢ is the Wick product and for all m⩾1Y•t(m) is the white noise of power jump processes Yt(m). In particular, Y•t(1) is the white noise of the Levy process. The formula holds for all F∈G∗⊃L2(μ), where G∗ is a space of stochastic distributions and μ is a white noise probability measure. Finally, we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by Levy processes.

MALLIAVIN CALCULUS AND ANTICIPATIVE ITÔ FORMULAE FOR LÉVY PROCESSES

We introduce the forward integral with respect to a pure jump Levy process and prove an Ito formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Ito formula for the Skorohod integral.

Optimal portfolio for an insider in a market driven by Levy processes

We consider a financial market driven by a Lévy process with filtration . An insider in this market is an agent who has access to more information than an honest trader. Mathematically, this is modelled by allowing a strategy of an insider to be adapted to a bigger filtration . The corresponding anticipating stochastic differential equation of the wealth is interpreted in the sense of forward integrals. In this framework, we study the optimal portfolio problem of an insider with logarithmic utility function. Explicit results are given in the case where the jumps are generated by a Poisson process. §Dedicated to the memory of Axel Grorud.

Advanced mathematical methods for finance

Dynamic risk measures.- Ambit processes and stochastic partial differential equations.- Fractional processes as models in stochastic finance.- Credit contagion in a long range dependent macroeconomic factor model.- Modeling information flows in financial markets.- An overview of comonotonicity and its applications in finance and insurance.- A general maximum principle for anticipative stochastic control and applications to insider trading.- Analyticity of the Wiener-Hopf factors and valuation of exotic options in Levy models.- Optimal liquidation of a pairs trade.- A PDE-based approach or pricing mortgage-backed securities.- Nonparametric methods for volatility density estimation.- Fractional smoothness and applications in finance.- Liquidity models in continuous and discrete times.- Some new BSDE results for an infinite-horizon stochastic control problem.- Functionals associated with gradient stochastic flows and nonlinear SPDEs.- Fractional smoothness and applications in Finance modeled by F-doubly stochastic Markov chains.- Exotic derivatives under stochastic volatility models with jumps.- Asymptotics of HARA utility from terminal wealth under proportional transaction costs with decision lag or execution delay and obligatory diversification.

Stochastic Integrals and Adjoint Derivatives

1. Stochastic measures and functions. Space-time products. Stochastic measures with independent values. The events generated. The deFinetti-Kolmogorov law. Non-anticipating and predictable stochastic functions. 2. The Ito non-anticipating integral. A general definition and related properties. The stochastic Poisson integral. The jumping stochastic processes. Gaussian-Poisson stochastic measures. 3. The non-anticipating integral representation. Multilinear polynomials and Ito multiple integrals. Integral representations with Gaussian-Poisson integrators. Homogeneous integrators. 4. The non-anticipating derivative. A general definition and related properties. Differentiation formulae. 5. Anticipating derivative and integral. Definitions and related properties. The closed anticipating extension of the Ito nonanticipating integral.

Levy Models Robustness and Sensitivity

We study the robustness of the sensitivity with respect to parameters in expec- tation functionals with respect to various approximations of a Levy process. As sensitivity parameter, we focus on the delta of an European option as the deriva- tive of the option price with respect to the current value of the underlying asset. We prove that the delta is stable with respect to natural approximations of a Levy process, including approximating the small jumps by a Brownian motion. Our methods are based on the density method, and we propose a new conditional den- sity method appropriate for our purposes. Several examples are given, including numerical examples demonstrating our results in practical situations.

Minimal-Variance Hedging in Large Financial Markets: Random Fields Approach

We study a large financial market where the discounted asset prices are modeled by martingale random fields. This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount. We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information. An explicit representation of the minimal variance hedging portfolio is suggested. Techniques of stochastic differentiation are applied to achieve the main results. Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market.

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.

Optimal Portfolio, Partial Information and Malliavin Calculus.

In a market driven by Lévy processes, we consider an optimal portfolio problem for a dealer who has access to some information in general smaller than the one generated by the market events. In this sense, we refer to this dealer as having partial information. For this generally incomplete market and within a non-Markovian setting, we give a characterization for a portfolio maximizing the expected utility of the final wealth. Techniques of Malliavin calculus are used for the analysis.

RANDOM FIELDS: NON-ANTICIPATING DERIVATIVE AND DIFFERENTIATION FORMULAS

The non-anticipating stochastic derivative represents the integrand in the best L2-approximation for random variables by Ito non-anticipating integrals with respect to a general stochastic measure with independent values on a space–time product. In this paper some explicit formulas for this derivative are obtained.