Abstracts

Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution.
We rely on numerical schemes and provide an unbiased, fast, and accurate Monte Carlo simulation based on the so-called ``Brownian bridge technique''. It proceeds as follows: First, the jump-instants of the process in consideration as well as the process immediately before and after the jump times are simulated. In between these generated points, one has a pure diffusion with fixed endpoints. Here, the so-called Brownian bridge probabilities provide an analytical expression for the first- passage time on a given threshold.
This simulation technique turns out to be (1) unbiased and (2) significantly faster than the standard Monte-Carlo simulation. We show how the Brownian bridge technique can be adapted to a large variety of exotic double barrier products. Those products are very flexible and thus allow investors to adapt to their specific hedging needs or speculative views. However, those contracts can hardly be traded without a fast and reliable pricing technique. Analytical solutions reach their limitations as they are often not flexible enough to adapt to complicated payoff streams and/or jump size distributions. To provide this flexibility for the Brownian bridge technique, we extend the existing algorithms and (1) allow to price double barrier derivatives that trigger different events depending on which barrier was hit first (a feature that is required to price, e.g., corridor bonus certificates) and (2) allow to evaluate payoff streams that depend on the first-passage time (a feature important in, e.g., structural credit risk models). Furthermore, (3) we show that time dependent barriers can easily be treated (a feature that is relevant for, e.g., window or step double barrier options). Finally, we discuss the implementation and show that -- in contrast to most alternative techniques -- the Brownian bridge algorithms are easy to understand and implement.

In this article we study the construction of coherent or convex risk measures defined either on a Orlicz heart, either on an Orlicz space, with respect to a Young loss function. The Orlicz heart is taken as a subset of $L^{0}(\omega, \mathcal{F}, \mu)$ endowed with the pointwise partial ordering. We define set-valued risk measures related to this partial ordering. We also derive monetary risk measures both by this class of set-valued risk measures and by the class of set-valued risk measures which has the properties mentioned in Jouini et al. (2004) and we compare their properties. We also use random measures related to heavy-tailed distributions in order to define monetary risk measures on Orlicz spaces, whose properties are also compared to the previous ones. The final part of the article is devoted to portfolio optimization problems which use these risk measures in practice and moreover with elements of their statistics.

This paper introduces variants of Strangles, called Euro-American or hybrid Strangles, and promotes a new numerical pricing technique. We highlight and compare properties of European, American and hybrid Strangles with pricing and hedging in mind. The new quadrature approach can deal with coupled integral equation systems locating early exercise boundaries of existing finite-lived contracts. The method is efficient, accurate and fast for pricing early exercisable Strangles. Other comparative advantages are that it avoids the non-monotonic gradient problem faced by precursors and allow users to control for errors. We then investigate the hedging of all Strangles, derive analytical expressions for some Greek parameters and finally stress how these parameters can differ (or not) one another.

The centenial lowest interest rates and the amplitude of the most rapid descent since 15 september 2008 , as well as their crawling behaviour since 5 years seem to have created a surprise. The Bank Risk Manager aims to rely on fundamentals of interest modelling like « Economie et Intérêt, Allais, 1947 », the « John Taylor’s rule », 1993. The data mining of long-term time-series was performed on LIBOR and swap interest rates with daily data for a dozen of maturities with 20 years history for some major currencies : CHF ; USD ; Euro (10 Y), GBP, Yen, and with yearly or other frequencies for Swiss Francs mortgage rates and other rates since 1850. Findings of such data mining include the incontestable downward wavy trends since 20 years from now, which fit into the 160 years record of Swiss mortgage rates, showing now the third local long term minimum, in addition to those of 1890 and 1950. The date-maturity interest rates maps and their statistics depict the populations which vary according to maturity, from three different populations for LIBOR to one population for long term swaps. The bizarre moves on GBP can reveal the recent fraud on LIBOR. Adjustment to asymetric thick tails distributions could be achieved. In accordance with Bachelier’s finding, the daily yields (or moves) do not show time-memory with the autocorrelation computations, this is the contrary for the memory of the I.R. values themselves. The statistical autocorrelation of the signs and of the absolute daily moves reveal a one-week memory of the absolute values of change. The series of consecutive draw-up and draw-downs behave differently as medium terme memory was depicted by the statistics on the number of consecutive bullish or bearish days. Herealso the absolute daily moves show an autocorrelation structure. SOM Self-Organizing Maps maps confirm the clustering of time-periods , and ANN Artificial Neural Networks achieve an adapted fitting of trends and disturbancies, the MLP MultiLayer Perceptron stick on long-term trends during temptative forecasts. As a conclusion, Data Mining Methods with spatial statistics extract the usefull information for model development, out of the half-million figures analyzed.