Stochastic Analysis Seminar, 2012

The seminars aim to share knowledge of the results achieved and open discussion on the ongoing research. The style is informal and the speakers include both students and researchers at CMA and visiting fellows. Student at master level are welcome to take active part in the series.

Next seminar:

May 16: Paul Kruehner, University of Kiel, Germany

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: On a term structure approach for stock options

Abstract: A usual practice for modeling stock markets is to specify a model class for the underlying and calibrate the class to the market data, i.e. one chooses the parameters in such a way that the real market option prices coincide with the theoretic option prices. However, the calibrated parameters for the different time points usually don’t coincide. That means that the chosen model class cannot explain the joint behavior of real market securities. In the spirit of [Carmona 09] we try to specify the dynamics for the options. However, special care has to be taken since arbitrage-free option prices observed at one time point have awkward restrains, see [Davis Hobson 07]. To avoid this constraints we map the possible option price configurations into a more suitable space, namely a convex cone, and specify their dynamics in this cone. Finally, we characterize absence of arbitrage and existence of an underlying dynamic generating the option price processes.

References:

[Carmona 09], Carmona, R., HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets, Lecture Notes in Mathematics, vol. 1919, 2009.

[Davis Hobson 07], Davis, M. and Hobson, D., The Range of Traded Option Prices, Mathematical Finance, vol. 17, 2007.

May 9: Nils Framstad, UiO

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Generalizations of elliptical distributions and their portfolio separation properties

Abstract: The elliptical distributions, originating with Schoenberg ('Metric spaces and completely monotone functions', Ann. of Math 1938), can with minor modifications replace the Gaussian in linear regression and in models like CAPM and the mutual fund theorem.  The latter, also known as the portfolio separation property, states conditions under which a financial market can be replaced by a few (two in the prototypical model) indices ('funds') without welfare loss to the investors. Such conditions can be formulated in terms of the investors' preferences (the Cass--Stiglitz type theorems) or in terms of the returns distributions (Ross-type theorem).

In the talk I will show how the (Ross) portfolio separation property fares under various generalizations of ellipticity in a single period model.  Distribution classes to be discussed, are (i) generalizations of the skew-elliptical (in the Azzalini sense), composed of conditioning some coordinates of an intracorrelated elliptical vector; (ii) the pseudo-isotropic distributions, where the ellipticity is formulated in a quasi-norm, rather than the usual weighted Euclidean norm; (iii) the class where the defining property of the pseudo-isotropic class is weakened to hold only for convex combinations; (iv) the alpha-stable class.

A trick employed by Khanna and Kulldorff ('A generalization of the mutual fund theorem', Finance Stoch. 1999) will be used to extend the portfolio separation properties of the single-period case to a the corresponding geometric Lévy process model if the distribution is infinitely divisible.  The construction itself can be simplified to a level where there is hardly any stochastic analysis left, but both infinite divisibility itself and applicability of the models, will raise unsolved problems.  The talk will be based on my 'Portfolio separation properties of the skew-elliptical distributions, with generalizations' (Statistics and Probability Letters, 2011, http://dx.doi.org/10.1016/j.spl.2011.07.006 ), my preprint 'Portfolio Separation with α-symmetric and Psuedo-isotropic Distributions' ( http://www.sv.uio.no/econ/forskning/memorandum/pdf-filer/2011/Memo-12-2011.pdf ), and work in (and out of) progress.

May 2: Imran Taib, CMA

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Pricing of temperature index insurance

Abstract: The aim of this talk is to introduce pricing of weather insurance contracts based on temperature indices. Three different pricing methods are analysed: the classical burn approach, index modelling and temperature modelling. We take the data from Malaysia as our empirical case. Our results show that there is a significant difference between the burn and index pricing approaches on one hand, and the temperature modelling method on the other. The latter approach is pricing the insurance contract using a seasonal autoregressive time series model for daily temperature variations, and thus provides a precise probabilistic model for the fine structure of temperature evolution. We complement our pricing analysis by an investigation of the profit/loss distribution from the contract, in the perspective of both the insured and the insurer.

Joint work with Professor Fred Espen Benth.

Apr 25. Cancelled.

Apr 11: André Suess, Univ. Barcelona, Spain

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: The Martingale-Measure Approach to SPDEs

Abstract: In this talk we will focus on the martingale-measure approach to stochastic partial differential equations introduced by John B. Walsh in his 1984 Saint Flour lecture notes. The main difference to other approaches such as the "SDEs in Hilbert spaces" approach (see the book of DaPrato and Zabczyk) is that the solutions are random fields in time and space. The talk consists of a thorough introduction to martingale measures as well as stochastic integration with respect to them. Afterwards, depending on time and the previous knowledge of the participants, we will treat more advanced issues related to the random-field solutions derived by the martingale-measure approach, such as p-moments (p>2), path regularity, existence of densities, large deviation principles, Varadhan-Léandre estimates ...

Mar 28: Krzystzof Paczka, CMA

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Malliavin calculus for G-Brownian motion

Abstract: In the talk I'll introduce the Malliavin derivative and the Skorohod integral for G-Brownian motion defined in a sublinear expectation space. The standard results like integration by parts, chain rule and Clark-Ocone formula will be given in this new framework.

Mar 20 (NB Tuesday): John Hosking, CMA

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: On the realization of jump-diffusion processes

Abstract: We will report work in progress on the problem of constructing a solution to the martingale problem with respect to a Lévy-type operator in which the Lévy measure term is x-dependent, with x being the argument of the function on which the operator acts. The x-dependence in the Lévy measure term is not assumed to be in the form of an image measure of a standard Lévy measure by an x-dependent jump-size function.

Results on the existence and uniqueness for this form of martingale problem are available in the literature; however, we are interested in the possible construction of solutions. We will present an approach to construct such a solution in the case where, for each x, the corresponding Lévy measure is of finite mass. However, it is known in literature that there exists a simple approach to construct solutions under certain conditions which allow, in particular, for cases where the Lévy measures are not necessarily of finite mass. We are in the process of trying make use of our approach to treat the problem in the case of more general Lévy measures than those with finite mass. Some issues in this direction will be discussed is this talk.

This is joint work with Mark H. A. Davis.

Mar 14: Salah Mohammed, Uni. Southern Illinois (USA)

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Linear Stochastic Partial Differential Equations

Abstract: In this talk we will answer the following question: Does a linear stochastic partial differential equation with a random initial condition admit a solution

Mar 7: Matthijs Pronk, TU Delft (NL)

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Malliavin calculus in UMD Banach spaces

Abstract: I will discuss some results on extending the Malliavin calculus to a Banach-valued setting.

Feb 22: Olivier Menokeu Pamen (CMA)

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Computing Greeks without derivatives

Abstract: In life insurance, the primary objective is to provide financial security to policy-holders. One way to do this is to manage investment guarantees whose prices can be obtained by Actuarial and Mathematical Finance methods. In Mathematical Finance the so-called ”Greeks” are quantities that measure the sensitivity of an option price or investment guarantee with respect to some model parameters involved. One of the most prominent applications of Malliavin calculus in Mathematical Finance is the representation of these Greeks as expectation functionals that does not involve the derivative of the pay-off function of the option. Since most financial pay-off functions are not smooth this representation yields a numerically effcient way to compute the Greeks. However, in the above mentioned representation of the Greeks the Ito diffusion modeling the price process of the underlying is assumed to have differentiable coeffcients. For example an extended Ornstein-Uhlenbeck process with switching mean reversion rate, an important model in electricity price modeling, is not included in this class of diffusions. In this presentation we demonstrate how to generalize this application of Malliavin calculus to price processes driven by Ito diffusions with irregular drift coeffcients. To this end, we study the general theoretical question of existence and Malliavin differentiability of strong solutions of stochastic differential equations (SDE’s) with irregular drift coeffcients. This approach yields the additional important result that the constructed strong solutions are even Malliavin differentiable. This insight finally enables us to represent greeks based on Ito diffusions with irregular drift coeffcients as expectation functionals that neither involve the derivative of the pay-off function nor the derivatives of the diffusion coefficients.

Jan 10: Jan Pedersen (Uni. Aarhus)

Time/place:
14.15 - 15.00 / seminar room B1036 (10th floor) in the Niels Henrik Abel building.

Title: Stochastic integration on the real line

Abstract: In this talk I will discuss the definition and the basic properties of integrals over the real line when the integrand is predictable and the integrator e.g. could be a two-sided Brownian motion (i.e. it is indexed by the real line). The first problem one encounters is that the two-sided Brownian motion is not a martingale in any filtration. Thus, it turns out that the right framework for the integrator is an increment martingale, or more generally an increment semimartingale, rather than a martingale or a semimartingale. The second problem is that sometimes the choice of filtration is a delicate matter. I discuss this in relation to representation of generalized Ornstein-Uhlenbeck processes. This talk is based on joint works with Andreas Basse-O'Connor and Svend-Erik Graversen.

Organizers:

Jukka Lempa, CMA
Fred Espen Benth, CMA
Giulia Di Nunno, CMA
An Ta Thi Kieu, CMA
Bernt Øksendal, CMA