3,786 research outputs found
Gaussian limits for discrepancies. I: Asymptotic results
We consider the problem of finding, for a given quadratic measure of
non-uniformity of a set of points (such as star-discrepancy or
diaphony), the asymptotic distribution of this discrepancy for truly random
points in the limit . We then examine the circumstances under which
this distribution approaches a normal distribution. For large classes of
non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit
Theorem can be derived.Comment: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.st
Asians and cash dividends: Exploiting symmetries in pricing theory
In this article we present new results for the pricing of arithmetic Asian
options within a Black-Scholes context. To derive these results we make
extensive use of the local scale invariance that exists in the theory of
contingent claim pricing. This allows us to derive, in a natural way, a simple
PDE for the price of arithmetic Asians options. In the case of European average
strike options, a proper choice of numeraire reduces the dimension of this PDE
to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve
this PDE, finding a Laplace-transform representation for the price of average
strike options, both seasoned and unseasoned. This extends the results of Geman
and Yor, who discussed the case of average price options. Next we use symmetry
arguments to show that prices of average strike and average price options can
be expressed in terms of each other. Finally we show, again using symmetries,
that plain vanilla options on stocks paying known cash dividends are closely
related to arithmetic Asians, so that all the new techniques can be directly
applied to this case.Comment: 19 pages, no figure
Discrepancy-based error estimates for Quasi-Monte Carlo. III: Error distributions and central limits
In Quasi-Monte Carlo integration, the integration error is believed to be
generally smaller than in classical Monte Carlo with the same number of
integration points. Using an appropriate definition of an ensemble of
quasi-randompoint sets, we derive various results on the probability
distribution of the integration error, which can be compared to the standard
Central Limit theorem for normal stochastic sampling. In many cases, a Gaussian
error distribution is obtained.Comment: 15 page
Tradable Schemes
In this article we present a new approach to the numerical valuation of
derivative securities. The method is based on our previous work where we
formulated the theory of pricing in terms of tradables. The basic idea is to
fit a finite difference scheme to exact solutions of the pricing PDE. This can
be done in a very elegant way, due to the fact that in our tradable based
formulation there appear no drift terms in the PDE. We construct a mixed scheme
based on this idea and apply it to price various types of arithmetic Asian
options, as well as plain vanilla options (both european and american style) on
stocks paying known cash dividends. We find prices which are accurate to in about 10ms on a Pentium 233MHz computer and to in a
second. The scheme can also be used for market conform pricing, by fitting it
to observed option prices.Comment: 13 pages, 5 tables, LaTeX 2
Quantum field theory for discrepancies
The concept of discrepancy plays an important role in the study of uniformity
properties of point sets. For sets of random points, the discrepancy is a
random variable. We apply techniques from quantum field theory to translate the
problem of calculating the probability density of (quadratic) discrepancies
into that of evaluating certain path integrals. Both their perturbative and
non-perturbative properties are discussed.Comment: 26 page
Error in Monte Carlo, quasi-error in Quasi-Monte Carlo
While the Quasi-Monte Carlo method of numerical integration achieves smaller
integration error than standard Monte Carlo, its use in particle physics
phenomenology has been hindered by the abscence of a reliable way to estimate
that error. The standard Monte Carlo error estimator relies on the assumption
that the points are generated independently of each other and, therefore, fails
to account for the error improvement advertised by the Quasi-Monte Carlo
method. We advocate the construction of an estimator of stochastic nature,
based on the ensemble of pointsets with a particular discrepancy value. We
investigate the consequences of this choice and give some first empirical
results on the suggested estimators.Comment: 41 pages, 19 figure
Scale invariance and contingent claim pricing II: Path-dependent contingent claims
This article is the second one in a series on the use of scaling invariance in finance. In the first paper, we introduced a new formalism for the pricing of derivative securities, which focusses on tradable objects only, and which completely avoids the use of martingale techniques. In this article we show the use of the formalism in the context of path-dependent options. We derive compact and intuitive formulae for the prices of a whole range of well known options such as arithmetic and geometric average options, barriers, rebates and lookback options. Some of these have not appeared in the literature before. For example, we find rather elegant formulae for double barrier options with moving barriers, continuous dividends and all possible configurations of the barriers. The strength of the formalism reveals itself in the ease with which these prices can be derived. This allowed us to pinpoint some mistakes regarding geometric mean options, which frequently appear in the literature. Furthermore, symmetries such as put-call transformations appear in a natural way within the framework.contingent claim pricing, scale-invariance, homogeneity, partial differential equation
Tradable Schemes
In this article we present a new approach to the numerical valuation of derivative securities. The method is based on our previous work where we formulated the theory of pricing in terms of tradables. The basic idea is to fit a finite difference scheme to exact solutions of the pricing PDE. This can be done in a very elegant way, due to the fact that in our tradable based formulation there appear no drift terms in the PDE. We construct a mixed scheme based on this idea and apply it to price various types of arithmetic Asian options, as well as plain vanilla options (both european and american style) on stocks paying known cash dividends. We find prices which are accurate to ~0.1% in about 10ms on a Pentium 233MHz computer and to ~0.001% in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.contingent claim pricing, numeric methods, asian options, cash dividend, partial differential equation
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