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## Stochastic relaxational dynamics applied to finance: towards non-equilibrium option pricing theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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55.77%

#Condensed Matter - Statistical Mechanics#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities

Non-equilibrium phenomena occur not only in physical world, but also in
finance. In this work, stochastic relaxational dynamics (together with path
integrals) is applied to option pricing theory. A recently proposed model (by
Ilinski et al.) considers fluctuations around this equilibrium state by
introducing a relaxational dynamics with random noise for intermediate
deviations called ``virtual'' arbitrage returns. In this work, the model is
incorporated within a martingale pricing method for derivatives on securities
(e.g. stocks) in incomplete markets using a mapping to option pricing theory
with stochastic interest rates. Using a famous result by Merton and with some
help from the path integral method, exact pricing formulas for European call
and put options under the influence of virtual arbitrage returns (or
intermediate deviations from economic equilibrium) are derived where only the
final integration over initial arbitrage returns needs to be performed
numerically. This result is complemented by a discussion of the hedging
strategy associated to a derivative, which replicates the final payoff but
turns out to be not self-financing in the real world, but self-financing {\it
when summed over the derivative's remaining life time}. Numerical examples are
given which underline the fact that an additional positive risk premium (with
respect to the Black-Scholes values) is found reflecting extra hedging costs
due to intermediate deviations from economic equilibrium.; Comment: 21 pages...

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## Degenerate elliptic operators in mathematical finance and Holder continuity for solutions to variational equations and inequalities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

55.78%

#Mathematics - Analysis of PDEs#Mathematics - Probability#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#35J70, 35J86, 49J40, 35R45 (Primary) 35R35, 49J20, 60J60 (Secondary)

The Heston stochastic volatility process, which is widely used as an asset
price model in mathematical finance, is a paradigm for a degenerate diffusion
process where the degeneracy in the diffusion coefficient is proportional to
the square root of the distance to the boundary of the half-plane. The
generator of this process with killing, called the elliptic Heston operator, is
a second-order, degenerate-elliptic partial differential operator whose
coefficients have linear growth in the spatial variables and where the
degeneracy in the operator symbol is proportional to the distance to the
boundary of the half-plane. With the aid of weighted Sobolev spaces, we prove
supremum bounds, a Harnack inequality, and H\"older continuity near the
boundary for solutions to variational equations defined by the elliptic Heston
operator, as well as H\"older continuity up to the boundary for solutions to
variational inequalities defined by the elliptic Heston operator. In
mathematical finance, solutions to obstacle problems for the elliptic Heston
operator correspond to value functions for perpetual American-style options on
the underlying asset.; Comment: 68 pages, 3 figures

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## An application of Malliavin Calculus to Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/11/2001
Português

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In this article, we give a brief informal introduction to Malliavin Calculus
for newcomers. We apply these ideas to the simulation of Greeks in Finance.
First to European-type options where formulas can be computed explicitly and
therefore can serve as testing ground. Later we study the case of Asian options
where close formulas are not available. The Greeks are computed through Monte
Carlo simulation.; Comment: 12 pages, 3 figures, coference proceedins

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## GPGPUs in computational finance: Massive parallel computing for American style options

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/01/2011
Português

Relevância na Pesquisa

55.77%

#Quantitative Finance - Computational Finance#Mathematics - Probability#Quantitative Finance - Pricing of Securities

The pricing of American style and multiple exercise options is a very
challenging problem in mathematical finance. One usually employs a Least-Square
Monte Carlo approach (Longstaff-Schwartz method) for the evaluation of
conditional expectations which arise in the Backward Dynamic Programming
principle for such optimal stopping or stochastic control problems in a
Markovian framework. Unfortunately, these Least-Square Monte Carlo approaches
are rather slow and allow, due to the dependency structure in the Backward
Dynamic Programming principle, no parallel implementation; whether on the Monte
Carlo levelnor on the time layer level of this problem. We therefore present in
this paper a quantization method for the computation of the conditional
expectations, that allows a straightforward parallelization on the Monte Carlo
level. Moreover, we are able to develop for AR(1)-processes a further
parallelization in the time domain, which makes use of faster memory structures
and therefore maximizes parallel execution. Finally, we present numerical
results for a CUDA implementation of this methods. It will turn out that such
an implementation leads to an impressive speed-up compared to a serial CPU
implementation.

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## Multifractal fluctuations in finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/02/2001
Português

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55.67%

#Condensed Matter - Statistical Mechanics#Condensed Matter - Disordered Systems and Neural Networks#Quantitative Finance - Statistical Finance

We consider the structure functions S^(q)(T), i.e. the moments of order q of
the increments X(t+T)-X(t) of the Foreign Exchange rate X(t) which give clear
evidence of scaling (S^(q)(T)~T^z(q)). We demonstrate that the nonlinearity of
the observed scaling exponent z(q) is incompatible with monofractal additive
stochastic models usually introduced in finance: Brownian motion, Levy
processes and their truncated versions. This nonlinearity corresponds to
multifractal intermittency yielded by multiplicative processes. The
non-analycity of z(q) corresponds to universal multifractals, which are
furthermore able to produce ``hyperbolic'' pdf tails with an exponent q_D >2.
We argue that it is necessary to introduce stochastic evolution equations which
are compatible with this multifractal behaviour.; Comment: 4 pages, 2 figures

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## Some applications of first-passage ideas to finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/06/2013
Português

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55.67%

#Quantitative Finance - Statistical Finance#Physics - Data Analysis, Statistics and Probability#Statistics - Applications

Many problems in finance are related to first passage times. Among all of
them, we chose three on which we contributed personally. Our first example
relates Kolmogorov-Smirnov like goodness-of-fit tests, modified in such a way
that tail events and core events contribute equally to the test (in the
standard Kolmogorov-Smirnov, the tails contribute very little to the measure of
goodness-of-fit). We show that this problem can be mapped onto that of a random
walk inside moving walls. The second example is the optimal time to sell an
asset (modelled as a random walk with drift) such that the sell time is as
close as possible to the time at which the asset reaches its maximum value. The
last example concerns optimal trading in the presence of transaction costs. In
this case, the optimal strategy is to wait until the predictor reaches (plus or
minus) a threshold value before buying or selling. The value of this threshold
is found by mapping the problem onto that of a random walk between two walls.; Comment: 30 pages. To appear in the special volume "First-Passage Phenomena
and Their Applications", Eds. R. Metzler, G. Oshanin, S. Redner. World
Scientific (2013)

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## Coupled continuous time random walks in finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/08/2006
Português

Relevância na Pesquisa

55.67%

#Physics - Data Analysis, Statistics and Probability#Physics - Physics and Society#Quantitative Finance - Statistical Finance

Continuous time random walks (CTRWs) are used in physics to model anomalous
diffusion, by incorporating a random waiting time between particle jumps. In
finance, the particle jumps are log-returns and the waiting times measure delay
between transactions. These two random variables (log-return and waiting time)
are typically not independent. For these coupled CTRW models, we can now
compute the limiting stochastic process (just like Brownian motion is the limit
of a simple random walk), even in the case of heavy tailed (power-law) price
jumps and/or waiting times. The probability density functions for this limit
process solve fractional partial differential equations. In some cases, these
equations can be explicitly solved to yield descriptions of long-term price
changes, based on a high-resolution model of individual trades that includes
the statistical dependence between waiting times and the subsequent
log-returns. In the heavy tailed case, this involves operator stable space-time
random vectors that generalize the familiar stable models. In this paper, we
will review the fundamental theory and present two applications with
tick-by-tick stock and futures data.; Comment: 7 pages, 2 figures. Paper presented at the Econophysics Colloquium...

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## Some applications and methods of large deviations in finance and insurance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

55.67%

#Mathematics - Probability#Quantitative Finance - Statistical Finance#60F10, 62P05, 65C05, 91B28, 91B30

In these notes, we present some methods and applications of large deviations
to finance and insurance. We begin with the classical ruin problem related to
the Cramer's theorem and give en extension to an insurance model with
investment in stock market. We then describe how large deviation approximation
and importance sampling are used in rare event simulation for option pricing.
We finally focus on large deviations methods in risk management for the
estimation of large portfolio losses in credit risk and portfolio performance
in market investment.

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## A New Kind of Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/2012
Português

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55.67%

Finance has benefited from the Wolfram's NKS approach but it can and will
benefit even more in the future, and the gains from the influence may actually
be concentrated among practitioners who unintentionally employ those principles
as a group.; Comment: 13 pages; Forthcoming in "Irreducibility and Computational
Equivalence: 10 Years After Wolfram's A New Kind of Science," Hector Zenil,
ed., Springer Verlag, 2013

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## The Quantitative Relations between Stock Prices and Quantities of Tradable Stock Shares and Its Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/03/2005
Português

Relevância na Pesquisa

55.69%

This paper analyzes the quantitative relations between stock prices and
quantities of tradable stock shares in Chinese stock markets at six time points
by means of Exploratory Data Analysis (EDA) method. It is found the resulting
formulae have the same structure but different parameters. This paper also uses
these relationships in order to analyse the feasibility of policies for Chinese
Government to sell the state-owned shares in Chinese stock markets.; Comment: 13 pages,7 figures, 3 tables

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## Magic points in finance: Empirical integration for parametric option pricing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

55.67%

We propose an interpolation method for parametric option pricing tailored to
the persistently recurring task of pricing liquid financial instruments. The
method supports the acceleration of such essential tasks of mathematical
finance as model calibration, real-time pricing, and, more generally, risk
assessment and parameter risk estimation. We adapt the empirical magic point
interpolation method of Barrault et al. (2004) to parametric Fourier pricing.
For a large class of combinations of option types, models and free parameters
the approximation converges exponentially in the degrees of freedom and
moreover has explicit error bounds. Numerical experiments confirm our
theoretical findings and show a significant gain in efficiency, even for
examples beyond the scope of the theoretical results. This is especially
promising for further applications of the method.

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## L\'evy Processes For Finance: An Introduction In R

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/03/2015
Português

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55.67%

This brief manuscript provides an introduction to L\'evy processes and their
applications in finance as the random process that drives asset models.
Characteristic functions and random variable generators of popular L\'evy
processes are presented in R.; Comment: 18 pages, 9 figures

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## Degenerate-elliptic operators in mathematical finance and higher-order regularity for solutions to variational equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

55.78%

#Mathematics - Analysis of PDEs#Mathematics - Probability#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#35J70, 49J40, 35R45 (Primary), 60J60 (Secondary)

We establish higher-order weighted Sobolev and Holder regularity for
solutions to variational equations defined by the elliptic Heston operator, a
linear second-order degenerate-elliptic operator arising in mathematical
finance. Furthermore, given $C^\infty$-smooth data, we prove
$C^\infty$-regularity of solutions up to the portion of the boundary where the
operator is degenerate. In mathematical finance, solutions to obstacle problems
for the elliptic Heston operator correspond to value functions for perpetual
American-style options on the underlying asset.; Comment: 55 pages, 1 figure. To appear in Advances in Differential Equations.
Incorporates final galley proof corrections corresponding to published
version

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## Adjoints and Automatic (Algorithmic) Differentiation in Computational Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/07/2011
Português

Relevância na Pesquisa

55.67%

Two of the most important areas in computational finance: Greeks and,
respectively, calibration, are based on efficient and accurate computation of a
large number of sensitivities. This paper gives an overview of adjoint and
automatic differentiation (AD), also known as algorithmic differentiation,
techniques to calculate these sensitivities. When compared to finite difference
approximation, this approach can potentially reduce the computational cost by
several orders of magnitude, with sensitivities accurate up to machine
precision. Examples and a literature survey are also provided.; Comment: 23 pages

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## Multilevel Monte Carlo methods for applications in finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/12/2012
Português

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55.67%

Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.; Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with
arXiv:1106.4730 by other authors

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## Additive versus multiplicative parameters - applications in economics and finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/06/2013
Português

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55.69%

In this paper, we pay our attention to geometric parameters and their
applications in economics and finance. We discuss the multiplicative models in
which a geometric mean and a geometric standard deviation are more natural than
arithmetic ones. We give two examples from Warsaw Stock Exchange in 1995--2009
and from a bid of 52-week treasury bills in 1992--2009 in Poland as an
illustrative example. For distributions having applications in finance and
insurance we give their multiplicative parameters as well as their estimations.
We consider, among others, heavy-tailed distributions such as lognormal and
Pareto distribution, applied to modelling of large losses.

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## Gravity Dual for Reggeon Field Theory and Non-linear Quantum Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/06/2009
Português

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55.67%

We study scale invariant but not necessarily conformal invariant deformations
of non-relativistic conformal field theories from the dual gravity viewpoint.
We present the corresponding metric that solves the Einstein equation coupled
with a massive vector field. We find that, within the class of metric we study,
when we assume the Galilean invariance, the scale invariant deformation always
preserves the non-relativistic conformal invariance. We discuss applications to
scaling regime of Reggeon field theory and non-linear quantum finance. These
theories possess scale invariance but may or may not break the conformal
invariance, depending on the underlying symmetry assumptions.; Comment: 31 pages

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## ARCH and GARCH Models vs. Martingale Volatility of Finance Market Returns

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/03/2008
Português

Relevância na Pesquisa

55.67%

#Quantitative Finance - Statistical Finance#Physics - Data Analysis, Statistics and Probability#Physics - Physics and Society

ARCH and GARCH models assume either i.i.d. or (what economists lable as)
white noise as is usual in regression analysis while assuming memory in a
conditional mean square fluctuation with stationary increments. We will show
that ARCH/GARCH is inconsistent with uncorrelated increments, violating the
i.i.d. and white assumptions and finance data and the efficient market
hypothesis as well.

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## Quantitative comparisons between finitary posterior distributions and Bayesian posterior distributions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/07/2008
Português

Relevância na Pesquisa

55.69%

#Quantitative Finance - Statistical Finance#Mathematics - Probability#Mathematics - Statistics Theory#Statistics - Methodology#62C10, 62F15, 60G09

The main object of Bayesian statistical inference is the determination of
posterior distributions. Sometimes these laws are given for quantities devoid
of empirical value. This serious drawback vanishes when one confines oneself to
considering a finite horizon framework. However, assuming infinite
exchangeability gives rise to fairly tractable {\it a posteriori} quantities,
which is very attractive in applications. Hence, with a view to a
reconciliation between these two aspects of the Bayesian way of reasoning, in
this paper we provide quantitative comparisons between posterior distributions
of finitary parameters and posterior distributions of allied parameters
appearing in usual statistical models.

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## Bridging stylized facts in finance and data non-stationarities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

55.71%

Employing a recent technique which allows the representation of nonstationary
data by means of a juxtaposition of locally stationary patches of different
length, we introduce a comprehensive analysis of the key observables in a
financial market: the trading volume and the price fluctuations. From the
segmentation procedure we are able to introduce a quantitative description of a
group of statistical features (stylizes facts) of the trading volume and price
fluctuations, namely the tails of each distribution, the U-shaped profile of
the volume in a trading session and the evolution of the trading volume
autocorrelation function. The segmentation of the trading volume series
provides evidence of slow evolution of the fluctuating parameters of each
patch, pointing to the mixing scenario. Assuming that long-term features are
the outcome of a statistical mixture of simple local forms, we test and compare
different probability density functions to provide the long-term distribution
of the trading volume, concluding that the log-normal gives the best agreement
with the empirical distribution. Moreover, the segmentation of the magnitude
price fluctuations are quite different from the results for the trading volume,
indicating that changes in the statistics of price fluctuations occur at a
faster scale than in the case of trading volume.; Comment: 13 pages...

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