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Local
scale invariance and contingent claim pricing
J.K.Hoogland and C.D.D.Neumann
IJTAF, vol.4 no. 1 (2001), p1-21 |
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Prices of tradables can
only be expressed relative to each other at any instant of time. This
fundamental fact should therefore also hold for contingent claims,
i.e. tradable instruments, whose prices depend on the prices of other
tradables. We show that this property induces a 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 possess a
manifest gauge invariance. A unique solution can only be given when
we impose restrictions on the drifts and 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.
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Local
scale invariance and contingent claim pricing II: path-dependent contingent
claims
J.K.Hoogland and C.D.D.Neumann
IJTAF, vol.4 no. 1 (2001), p23-43 |
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This article is the second
one in a series on the use of local scale 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.
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On
the structure of Gaussian pricing models
and Gaussian Markov functional models
C.D.D.Neumann
working paper |
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This article investigates the structure
of Gaussian pricing models (that is, models in which future returns
are normally distributed). Although much is already known about such
models, this article differs in that it is based on a formulation
of the theory of derivative pricing in which numeraire invariance
is manifest, extending earlier work on this subject. The focus on
symmetry properties leads to a deeper insight in the structure of
these models. The central idea is the construction of the most general
class of derived Gaussian tradables given a set of underlying tradables
which are themselves Gaussian. These derived tradables are called
"generalized power tradables" and they correspond to portfolios
in which the fraction of total value invested in each asset is a deterministic
function of time. Applying this theory to Gaussian HJM models, the
new tradables give an explicit description of the interdependence
of bonds implicit in such models. Given this structure, a simple condition
is derived under which these models allow a description in terms of
an M-factor Markov functional model, as introduced by Hunt, Kennedy
and Pelsser. Finally, conditions are derived under which these Gaussian
Markov functional models are time homogeneous (bond volatilities depending
only on the time to maturity). This result is linked to recent results
by Björk and Gombani.
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Asians
and cash dividends: exploiting symmetries in pricing theory
J.K.Hoogland and C.D.D.Neumann
working paper |
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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.
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Tradable
schemes
J.K.Hoogland and C.D.D.Neumann
working paper |
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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.
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Symmetries
in jump-diffusion models with applications in option pricing and credit
risk
J.K.Hoogland,
C.D.D.Neumann and M.H. Vellekoop
IJTAF, vol. 6, no. 2 (2003) p135-172
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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.
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Converting
the Reset
D.
Bloch, J.K.Hoogland and C.D.D.Neumann
working paper |
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We give a simple algorithm
to incorporate the effects of resets in convertible bond prices, without
having to add an extra factor to take into account the value of the
reset. Furthermore we show that the effect of a notice period, and
additional make-whole features, can be treated in a straightforward
and simple manner. Although we present these results with the stockprice
driven by geometric Brownian and a deterministic interest term structure,
our results can be extended to more general cases, e.g. stochastic
interest rates.
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Perturbative
BPS-algebras in superstring theory
C.D.D.Neumann
Nuclear Physics B 499:596 (1997) |
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This paper investigates
the algebraic structure that exists on perturbative BPS states in
the superstring, compactified on the product of a circle and a Calabi-Yau
fourfold. This structure was defined in a recent article by Harvey
and Moore. It is shown that for a toroidal compactification this algebra
is related to a generalized Kac-Moody algebra. The BPS algebra itself
is not a Lie-algebra. However, it turns out to be possible to construct
a Lie algebra with the same graded dimensions, in terms of a half-twisted
model. The dimensions of these algebras are related to the elliptic
genus of the transverse part of the string algebra. Finally, the construction
is applied to an orbifold compactification of the superstring.
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The
elliptic genus of Calabi-Yau 3- and 4-folds, product formulae and
generalized Kac-Moody algebras
C.D.D.Neumann
Journal of Geometry And Physics 29:5 (1999) |
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In this paper the elliptic
genus for a general Calabi-Yau fourfold is derived. The recent work
of Kawai calculating N=2 heterotic string one-loop threshold corrections
with a Wilson line turned on is extended to a similar computation
where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all
cases there seems to be a generalized Kac-Moody algebra involved,
whose denominator formula appears in the result.
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