Probabilistic aspects of financial risk

Problems arising in Finance have become a significant source of new developments in Stochastic Analysis. We discuss some recent case studies, in particular some decomposition and representation theorems which are motivated by problems of hedging derivatives and of intertemporal consumption choice

Efficient hedging: Cost versus shortfall risk

An investor faced with a contingent claim may eliminate risk by (super-)hedging in a financial market. As this is often quite expensive, we study partial hedges, which require less capital and reduce the risk. In a previous paper we determined quantile hedges which succeed with maximal probability, given a capital constraint. Here we look for strategies which minimize the shortfall risk defined as the expectation of the shortfall weighted by some loss function. The resulting efficient hedges allow the investor to interpolate in a systematic way between the extremes of no hedge and a perfect (super-)hedge, depending on the accepted level of shortfall risk

Quantile Hedging

In a complete financial market every contingent claim can be hedged perfectly. In an incomplete market it is possible to stay on the safe side by superhedging. But such strategies may require a large amount of initial capital. Here we study the question what an investor can do who is unwilling to spend that much, and who is ready to use a hedging strategy which succeeds with high probability

Optional decompositions under constraints

Motivated by a hedging problem in mathematical finance, El Karoui and Quenez [7] and Kramkov [14] have developed optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We investigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to di_erent classes of equivalent measures. As an application, we extend results of Karatzas and Cvitanic [3] on hedging problems with constrained portfolios

Optional Decomposition and Lagrange Multipliers

Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand ф such that the difference X−ф•S is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption

On weak Brownian motions of arbitrary order

We show the existence, for any k E N, of processes which have the same k-marginals as Brownian motion, although they are not Brownian motions. For k = 4, this proves a conjecture of Stoyanov. The law P' of such a weak Brownian motion of order k can be constructed to be equivalent to Wiener measure P' on c [O, 1]. On the other hand, there are weak Brownian motions of arbitrary order whose law is singular to Wiener measure. We also show that, for any e > 0, there are weak Brownian motions whose law coincides with wiener measure outside of any interval of length e

Canonical decomposition of linear transformations of two independent Brownian motions

Motivated by the Kyle-Back model of 'insider trading', we consider certain classes of linear transformations of two independent Brownian motions and study their canonical decomposition as semimartingales in their own filtration. In particular we characterize those transformations which generate again a Brownian motion

The Financial Crisis and the Systemic Failure of Academic Economics

The economics profession appears to have been unaware of the long build-up to the current worldwide financial crisis and to have significantly underestimated its dimensions once it started to unfold. In our view, this lack of understanding is due to a misallocation of research efforts in economics. We trace the deeper roots of this failure to the profession’s focus on models that, by design, disregard key elements driving outcomes in real-world markets. The economics profession has failed in communicating the limitations, weaknesses, and even dangers of its preferred models to the public. This state of affairs makes clear the need for a major reorientation of focus in the research economists undertake, as well as for the establishment of an ethical code that would ask economists to understand and communicate the limitations and potential misuses of their models.financial crisis, academic moral hazard, ethic responsibility of researchers