arXiv subjects

Jiri Hoogland

Publications and source records attributed to Jiri Hoogland.

14 recordsLinked to original sources

arXiv2025

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 $\sim 0.1%$ in about 10ms on a Pentium 233MHz computer and to $\sim 0.001%$ in a second. The scheme can also be used for market conform pricing, by fitting it to observed option prices.

cond-mat.stat-mech↗

arXiv2001

Symmetries in jump-diffusion models with applications in option pricing and credit risk

It is a well known fact that local scale invariance plays a fundamental role in the theory of derivative pricing. Specific applications of this principle have been used quite often under the name of `change of numeraire', but in recent work it was shown that when invoked as a fundamental first principle, it provides a powerful alternative method for the derivation of prices and hedges of derivative securities, when prices of the underlying tradables are driven by Wiener processes. In this article we extend this work to the pricing problem in markets driven not only by Wiener processes but also by Poisson processes, i.e. jump-diffusion models. It is shown that in this case too, the focus on symmetry aspects of the problem leads to important simplifications of, and a deeper insight into the problem. Among the applications of the theory we consider the pricing of stock options in the presence of jumps, and Levy-processes. Next we show how the same theory, by restricting the number of jumps, can be used to model credit risk, leading to a `market model' of credit risk. Both the traditional Duffie-Singleton and Jarrow-Turnbull models can be described within this framework, but also more general models, which incorporate default correlation in a consistent way. As an application of this theory we look at the pricing of a credit default swap (CDS) and a first-to-default basket option.

cond-mat↗

arXiv2000

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.

cond-mat.stat-mech↗

arXiv1999

Scaling invariance in finance 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 article (cond-mat/9906048), 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.

cond-mat↗

arXiv1999

Scale-invariance and contingent claim pricing

Prices of tradables can only be expressed relative to each other at any instant of time. This fundamental fact should therefore also hold for contigent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces local scale-invariance in the problem of pricing contingent claims. Due to this symmetry we do not require any martingale techniques to arrive at the price of a claim. If the tradables are driven by Brownian motion, we find, in a natural way, that this price satisfies a PDE. Both posses a manifest gauge-invariance. A unique solution can only be given when we impose restrictions on the drifts of volatilities of the tradables, i.e. the underlying market structure. We give some examples of the application of this PDE to the pricing of claims. In the Black-Scholes world we show the equivalence of our formulation with the standard approach. It is stressed that the formulation in terms of tradables leads to a significant conceptual simplification of the pricing-problem.

cond-mat↗

arXiv1997

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 $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit $N\to\infty$. 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.

physics.comp-ph↗

arXiv1997

Light Fermion Mass Effects in e^+e^- -> 4 Fermions

We investigated the effect of the light fermion masses on cross sections for e^+e^- into 4 fermions in the Fermion Loop scheme defined in hep-ph/9612260, and approximations to it. The effects are found to be very small, except of course in the collinear region of single W-boson production where the electron mass acts as the cut-off.

hep-ph↗

arXiv1996

Quasi-Monte Carlo, Discrepancies and Error Estimates

We discuss the problem of defining an estimate for the error in quasi-Monte Carlo integration. The key issue is the definition of an ensemble of quasi-random point sets that, on the one hand, includes a sufficiency of equivalent point sets, and on the other hand uses information on the degree of uniformity of the point set actually used, in the form of a discrepancy or diaphony. A few examples of such discrepancies are given. We derive the distribution of our error estimate in the limit of large number of points. In many cases, Gaussian central limits are obtained. We also present numerical results for the quadratic star-discrepancy for a number of quasi-random sequences.

physics.comp-ph↗

arXiv1996

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.

hep-ph↗

arXiv1996

Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers

This is basically a review of the field of Quasi-Monte Carlo intended for computational physicists and other potential users of quasi-random numbers. As such, much of the material is not new, but is presented here in a style hopefully more accessible to physicists than the specialized mathematical literature. There are also some new results: On the practical side we give important empirical properties of large quasi-random point sets, especially the exact quadratic discrepancies; on the theoretical side, there is the exact distribution of quadratic discrepancy for random point sets.

hep-ph↗

arXiv1996

Discrepancy-based error estimates for Quasi-Monte Carlo. II: Results in one dimension

The choice of a point set, to be used in numerical integration, determines, to a large extent, the error estimate of the integral. Point sets can be characterized by their discrepancy, which is a measure of its non-uniformity. Point sets with a discrepancy that is low with respect to the expected value for truly random point sets, are generally thought to be desirable. A low value of the discrepancy implies a negative correlation between the points, which may be usefully employed to improve the error estimate of a numerical integral based on the point set. We apply the formalism developed in a previous publication to compute this correlation for one-dimensional point sets, using a few different definitions of discrepancy.

hep-ph↗

arXiv1996

Definitions and upper bounds for unstable boson masses

We consider the problem of a proper definition of the mass of a heavy unstable boson. It is shown how various definitions are related by the renormalization-scheme independence of the full resummed amplitude. This is made explicit for the $W$ boson in the large $N_f$ limit, where we find an upper bound on the peak position of the lineshape and on the mass, provided a sensible definition of the mass is used. In more realistic models, we conjecture that under certain conditions the mass is similarly bounded. Using this approach an upper limit of about 1 TeV on the mass of the Minimal-Standard-Model Higgs boson is found, independent of any breakdown of perturbation theory.

hep-ph↗

arXiv1996

Discrepancy-based error estimates for Quasi-Monte Carlo. I: General formalism

We show how information on the uniformity properties of a point set employed in numerical multidimensional integration can be used to improve the error estimate over the usual Monte Carlo one. We introduce a new measure of (non-)uniformity for point sets, and derive explicit expressions for the various entities that enter in such an improved error estimate. The use of Feynman diagrams provides a transparent and straightforward way to compute this improved error estimate.

hep-ph↗

arXiv1995

Stable Calculations for Unstable Particles: Restoring Gauge Invariance

We discuss theoretical and phenomenological aspects of the use of boson propagators with energy-dependent widths in predictions for high-energy scattering processes. In general, gauge invariance is violated in such calculations. We discuss several approaches to restore gauge invariance, necessary for a reliable result. The most promising method is the addition of the relevant parts of the fermionic corrections, which fulfills all Ward identities. The numerical difference between this and other approaches is studied. A number of recommendations are given for LEP2 computations.

hep-ph↗