Tag - Mathematical finance

Andreas Hamel: From multivariate quantiles to set-valued risk measures: a set optimization approach to financial models with frictions

5th September, 2018

Some questions in mathematics are not answered for quite some time, but just sidestepped. One of those questions is the following: What is the quantile of a multi-dimensional random variable? The 'sidestepping' in this case produced so-called depth functions and depth regions, and the most prominent among them is the halfspace depth invented by Tukey in 1975, a very popular tool in statistics. When it comes to the definition of multivariate quantiles, depth functions replace cummulative distribution functions, and depth regions provide potential candidates for quantile vectors. However, Tukey depth functions, for example, do not share all features with (univariate) cdf's and do not even generalize them.

On the other hand, the naive definition of quantiles via the joint distribution function turned out to be not very helpful for statistical purposes, although it is still in use to define multivariate V@Rs (Embrechts and others) as well as stochastic dominance orders (Müller/Stoyan and others).

The crucial point and an obstacle for substantial progress for a long time is the missing (total) order for the values of a multi-dimensional random variable. On the other hand, (non-total) orders appear quite natural in financial models with proportional transaction costs (a.k.a. the Kabanov market) in form of solvency cones.

We propose new concepts for multivariate ranking functions with features very close to univariate cdf's and for set-valued quantile functions which, at the same time, generalize univariate quantiles as well as Tukey's halfspace depth regions. Our constructions are designed to deal with general vector orders for the values of random variables, and they produce unambigious lower and upper multivariate quantiles, multivariate value-at-risks as well as a multivariate first order stochastic dominance relation. Financial applications to markets with frictions are discussed as well as many other examples and pictures which show the interesting geometric features of the new quantile sets.

Ilya Molchanov: Set-valued risk measures of non-convex portfolios

5th September, 2018

Non-convex random sets of admissible positions naturally arise in the setting of fixed transaction costs or when only a finite range of possible transactions is considered. The talk defines set-valued risk measures in such cases and explores the situations when they return convex result, namely, when Lyapunov’s theorem applies. The case of fixed transaction costs is analysed in greater details.

Daniel Balint: Discounting invariant FTAP for large financial markets

4th September, 2018

For large financial markets as introduced in Kramkov and Kabanov 94, there are several existing absence-of-arbitrage conditions in the literature. They all have in common that they depend in a crucial way on the discounting factor. We introduce a new concept, generalizing NAA1 (K&K 94) and NAA (Rokhlin 08), which is invariant with respect to discounting. We derive a dual characterization by a contiguity property (FTAP).We investigate connections to the in finite time horizon framework (as for example in Karatzas and Kardaras 07) and illustrate negative result by counterexamples.

Jocelyne Bion Nadal: Approximation and calibration of laws of solutions to stochastic differential equations

4th September, 2018

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.

This new distance W̃2 is defined similarly to the classical Wasserstein distance W̃2 but the set of couplings is restricted to the set of laws of solutions of 2d-dimensional stochastic differential equations. We prove that this new distance W̃2 metrizes the weak topology. Furthermore this distance W̃2 is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.