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

Otto, Matthias
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
55.77%
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...

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

Feehan, Paul M. N.; Pop, Camelia
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
55.78%
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

## An application of Malliavin Calculus to Finance

Kohatsu-Higa, Arturo; Montero, Miquel
Tipo: Artigo de Revista Científica
Publicado em 29/11/2001 Português
Relevância na Pesquisa
55.67%
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

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

Pagès, Gilles; Wilbertz, Benedikt
Tipo: Artigo de Revista Científica
Publicado em 17/01/2011 Português
Relevância na Pesquisa
55.77%
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.

## Multifractal fluctuations in finance

Schmitt, F.; Schertzer, D.; Lovejoy, S.
Tipo: Artigo de Revista Científica
Publicado em 21/02/2001 Português
Relevância na Pesquisa
55.67%
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

## Some applications of first-passage ideas to finance

Chicheportiche, Rémy; Bouchaud, Jean-Philippe
Tipo: Artigo de Revista Científica
Publicado em 13/06/2013 Português
Relevância na Pesquisa
55.67%
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)

## Coupled continuous time random walks in finance

Meerschaert, Mark M.; Scalas, Enrico
Tipo: Artigo de Revista Científica
Publicado em 29/08/2006 Português
Relevância na Pesquisa
55.67%
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...

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

Pham, Huyen
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
55.67%
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.

## A New Kind of Finance

Maymin, Philip Z.
Tipo: Artigo de Revista Científica
Publicado em 04/10/2012 Português
Relevância na Pesquisa
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

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

Gou, Chengling
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

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

Gaß, Maximilian; Glau, Kathrin; Mair, Maximilian
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.

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

Manuge, D. J.
Tipo: Artigo de Revista Científica
Publicado em 12/03/2015 Português
Relevância na Pesquisa
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

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

Feehan, Paul M. N.; Pop, Camelia A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
55.78%
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

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

Homescu, Cristian
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

## Multilevel Monte Carlo methods for applications in finance

Giles, Mike; Szpruch, Lukasz
Tipo: Artigo de Revista Científica
Publicado em 06/12/2012 Português
Relevância na Pesquisa
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

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

Jasiulewicz, Helena; Kordecki, Wojciech
Tipo: Artigo de Revista Científica
Publicado em 20/06/2013 Português
Relevância na Pesquisa
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.

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

Nakayama, Yu
Tipo: Artigo de Revista Científica
Publicado em 23/06/2009 Português
Relevância na Pesquisa
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

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

McCauley, Joseph L.
Tipo: Artigo de Revista Científica
Publicado em 31/03/2008 Português
Relevância na Pesquisa
55.67%
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.

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

Bassetti, Federico
Tipo: Artigo de Revista Científica
Publicado em 08/07/2008 Português
Relevância na Pesquisa
55.69%
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.

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

Camargo, Sabrina; Queiros, Silvio M. Duarte; Anteneodo, Celia