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We characterise the asymptotic smile and term structure of implied volatility in the Heston model at small maturities and all strikes. Using saddlepoint methods we derive a small-maturity expansion formula for call option prices, which we then transform into a closed-form expansion (including the leading-order and correction terms) for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and all Heston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order result of~\cite{FJ09I} which found the relationship between the zero-expiry smile and the diffusion coefficients. We solve for in/out-of-the-money and at-the-money cases; in the latter case our proof involves subleading-order saddlepoint approximation along a suitable path of integration.
We investigate the worst-case behavior of the simplex algorithm on linear programs
with 3 variables, that is, on 3-dimensional simple polytopes. Among the
pivot rules that we consider, the “random edge” rule yields the best asymptotic
behavior as well as the most complicated analysis. All other rules turn out to be
much easier to study, but also produce worse results: Most of them show essentially
worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is
known to be subexponential without dimension restriction, as well as Zadeh’s deterministic
history-dependent rule, for which no non-polynomial instances in general
dimensions have been found so far.
The background for the general mathematical link between utility and information
theory investigated in this paper is a simple financial market model with two kinds of small
traders: less informed traders and insiders whose extra information is represented by an
enlargement of the other agents' filtration. The expected logarithmic utility increment,
i.e. the difference of the insider's and the less informed trader's expected logarithmic
utility is described in terms of the information drift, i.e. the drift one has to eliminate
in order to perceive the price dynamics as a martingale from the insider's perspective.
On the one hand, we describe the information drift in a very general setting by natural
quantities expressing the probabilistic better informed view of the world. This on the
other hand allows us to identify the additional utility by entropy related quantities known
from information theory. In particular in a complete market in which the insider has some
fixed additional information during the entire trading interval, its utility increment can
be represented by the Shannon information of his extra knowledge. For general markets,
and in some particular examples, we provide estimates of maximal utility by information
inequalities.
The Real Multiple Dual
(2009)
In this paper we present a dual representation for the multiple stopping
problem, hence multiple exercise options. As such it is a natural generalization of the
method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping
problem for American options. We consider this representation as the real dual as it is
solely expressed in terms of an infimum over martingales rather than an infimum over
martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple
dual representation we present three Monte Carlo simulation algorithms which require
only one degree of nesting.
The PSurface Library
(2010)
We describe psurface, a C++ library that allows to store and access piecewise linear mappings between simplicial surfaces in $\R^2$ and $\R^3$. These mappings are stored in a graph data structure and can be constructed explicitly, by projection, or by surface simplification. Piecewise linear maps can be used, e.g., to construct boundary
approximations for finite element grids, and grid intersections for domain decomposition methods. In computer graphics the mappings allow to build level-of-detail representations as well as texture- and bump maps. We document the data structures and algorithms used and show how \psurface is used in the numerical analysis framework Dune
and the visualization software Amira.
Relaying is a protocol extension for cellular wireless computer networks; in order to utilize radio resources more efficiently, several hops are allowed within one cell. This paper investigates the principle potential of relaying by casting transmission scheduling as a mathematical optimization problem, namely, a linear program. We analyze the throughput gains showing that, irrespective of the concrete scheduling algorithm, performance gains of up to 30\% on average for concrete example networks are achievable.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The perturbation and ADAE index of a degenerated hyperbolic system modelling a heat exchanger
(2007)
The heat exchanger in a heat pump can be modelled by the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.