## Applications to finance

### On pricing and hedging models for energy prices

The standard approach to model electricity futures prices is the so-called Gaussian factor model which is closely related to the Heath-Jarrow-Morton model developped for interest rates. In this approach, the dynamics of spot prices is then derived as the exponential of a Gaussian Ornstein-Uhlenbeck process. This model has the advantage to lead to closed pricing formulae, while the Gaussian property is unable to take into account spot price spikes. In this presentation, we propose instead a Lévy factor model for futures prices where spot prices are then represented by the exponential of a Lévy Ornstein-Uhlenbeck process. We consider the variance Optimal criteria for this incomplete market model and analyse in simulations and on real market data the performances of the resulting hedging strategy and compare it to the standard Black-Scholes strategy, for some simple European vanilla options.

### Minimal variance hedging in incomplete markets: stochastic differentiation and the Clark-Ocone formula

Giulia Di Nunno (Universitetet i Oslo)

In a market driven by a Lévy martingale, we consider a square integrable claim. We study the problem of minimal variance hedging and we give an overview of various approaches to an explicit representation of the minimal variance hedging portfolio. We discuss the Clark-Ocone formula and its extensions to fit this application to hedging. We also present different representations of the hedging portfolio by means of a non-anticipating differentiation and we discuss its relations with the Clark-Ocone formula and the Malliavin derivative.

### Malliavin Fractional Smothness for Lévy Processes and Discrete Time Hedging

Christel Geiss (Jyväskylä Yliopisto), joint work with S. Geiss and E. Laukkarinen

Let $(X_t)$ be a Lévy process such that $(S_t)$ given by $S_t =e^{X_t}$ (standing for the discounted price process) is a square integrable process. For a pay-off $f(S_1) \in L_2$ where $f$ is any Borel function we consider {\bf variance optimal hedging}, i.e. we choose the constant $c$ and the predictable process $(\phi_t)$ (satisfying an integrability condition) such that it holds \inf \mathbb{E} (f(S_1) -c -G_1(\phi))^2 = \text{ minimal}, where $G_1(\phi) = \int_{(0,1]} \phi_t dS_t$ denotes the cumulative gains.

We investigate the relation between Malliavin fractional smoothness of $G_1(\phi$ and the rate of convergence $r$ of the discretization problem \bigg \|\int_{(0,1]} \phi_t dS_t - \sum_{k=1}^n \phi_{t_{k-1}} (S_{t_k}-S_{t_{k-1}}) \bigg \|_{L_2} \le cn^{-r}, \,\,\, n \to \infty. and the it pattern of time nets $0=t_0< \ldots < t_n=1$.

### Malliavin Greeks for complex Asian options in a jump diffusion setting

Lucia Caramellino (Università di Roma)

We deal with a unifying Malliavin calculus in a jump diffusion context and develop a integration by parts formulas giving the starting point of our work. The results are then applied to study representation formulas for the delta of complex Asian options. Several examples are detailed and equipped with numerical studies.

### Applications of Malliavin Calculus in Statistical Inference

José Manual Corcuera (Universitat de Barcelona)

In this talk the author reviews some recent work where Malliavin Calculus is applied to study problems in statistical inference. Problems like the LAMN property of the model or the asymptotic behaviour of the multipower variation of certain processes.

#### References:

Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M. (2009) Power variation for Gaussian processes with stationary increments. Stochastic Processes and their Applications, 119, 1845-1865.
Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M. (2009) Multipower variation for Brownian Semistationary Processes. Bernoulli (In press).
Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M. (2009) Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. Mathematics Preprint Series. No. 413. IMUB. Universitat de Barcelona. Submitted.
Corcuera, J.M. (2010) New central limit theorems for functionals of Gaussian processes and their applications. Preprint. Submitted.
Kohatsu-Higa, A. (2010) Statistical Inference and Malliavin Calculus. Progress in Probabiity, Vol 63, 59-82.