arXiv subjects

Marc Potters

Publications and source records attributed to Marc Potters.

53 records · Page 3Linked to original sources

arXiv2001

The leverage effect in financial markets: retarded volatility and market panic

We investigate quantitatively the so-called leverage effect, which corresponds to a negative correlation between past returns and future volatility. For individual stocks, this correlation is moderate and decays exponentially over 50 days, while for stock indices, it is much stronger but decays faster. For individual stocks, the magnitude of this correlation has a universal value that can be rationalized in terms of a new `retarded' model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect.

cond-mat↗

arXiv2000

Option Pricing and Hedging with Temporal Correlations

We consider the problem of option pricing and hedging when stock returns are correlated in time. Within a quadratic-risk minimisation scheme, we obtain a general formula, valid for weakly correlated non-Gaussian processes. We show that for Gaussian price increments, the correlations are irrelevant, and the Black-Scholes formula holds with the volatility of the price increments on the scale of the re-hedging. For non-Gaussian processes, further non trivial corrections to the `smile' are brought about by the correlations, even when the hedge is the Black-Scholes Delta-hedge. We introduce a compact notation which eases the computations and could be of use to deal with more complicated models.

cond-mat↗

arXiv2000

Hedged Monte-Carlo: low variance derivative pricing with objective probabilities

We propose a new `hedged' Monte-Carlo (HMC) method to price financial derivatives, which allows to determine simultaneously the optimal hedge. The inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading, and for the very same reason reduces considerably the variance of our HMC scheme as compared to previous methods. The explicit accounting of the hedging cost naturally converts the objective probability into the `risk-neutral' one. This allows a consistent use of purely historical time series to price derivatives and obtain their residual risk. The method can be used to price a large class of exotic options, including those with path dependent and early exercise features.

cond-mat↗

arXiv1999

Worse fluctuation method for fast Value-at-Risk estimates

We show how one can actually take advantage of the strongly non-Gaussian nature of the fluctuations of financial assets to simplify the calculation of the Value-at-Risk of complex non linear portfolios. The resulting equations are not hard to solve numerically, and should allow fast VaR and $Δ$VaR estimates of large portfolios, where {\it by construction} the influence of rare events is taken into account reliably. Our method can be seen as a correctly probabilized `scenario' calculation (or `stress-testing').

cond-mat↗

arXiv1999

Apparent multifractality in financial time series

We present a exactly soluble model for financial time series that mimics the long range volatility correlations known to be present in financial data. Although our model is `monofractal' by construction, it shows apparent multiscaling as a result of a slow crossover phenomenon on finite time scales. Our results suggest that it might be hard to distinguish apparent and true multifractal behavior in financial data. Our model also leads to a new family of stable laws for sums of correlated random variables.

cond-mat↗

arXiv1999

Theory of Financial Risk: Basic notions in probability

Risk control has become one of the major concern of financial institutions. The need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of financial markets is clearly expressed, in particular for derivative markets. Classical theories, however, are based on simplified assumptions -- such as Gaussian statistics -- and lead to a systematic (and sometimes dramatic) underestimation of real risks. We summarize a few basic notions in probability theory which are useful in a financial context: Statistics of Extremes, Sums of Random Variables, Central Limit Theorems and Deviations, Correlations and Dependence, and a brief introduction to Random Matrix Theory. This self-contained text corresponds to the first chapter of our book, `Theory of Financial Risk'.

cond-mat↗

arXiv1998

Noise Dressing of Financial Correlation Matrices

We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management.

cond-mat↗

arXiv1998

Back to basics: historical option pricing revisited

We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of arbitrary probability density. We discuss in particular how, in the Gaussian limit, the Black-Scholes results are recovered, including the fact that the average return of the underlying stock disappears from the price (and the hedging strategy). We compare this theory to real option prices and find these reflect in a surprisingly accurate way the subtle statistical features of the underlying asset fluctuations.

cond-mat↗

arXiv1998

Are Financial Crashes Predictable?

We critically review recent claims that financial crashes can be predicted using the idea of log-periodic oscillations or by other methods inspired by the physics of critical phenomena. In particular, the October 1997 `correction' does not appear to be the accumulation point of a geometric series of local minima.

cond-mat.stat-mech↗

arXiv1998

Rational Decisions, Random Matrices and Spin Glasses

We consider the problem of rational decision making in the presence of nonlinear constraints. By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio optimisation problem. We show that the number of ``optimal'' solutions is generically exponentially large: rationality is thus de facto of limited use. In addition, this problem is related to spin glasses with Lévy-like (long-ranged) couplings, for which we show that the ground state is not exponentially degenerate.

cond-mat.dis-nn↗

arXiv1997

Missing Information and Asset Allocation

When the available statistical information is imperfect, it is dangerous to follow standard optimisation procedures to construct an optimal portfolio, which usually leads to a strong concentration of the weights on very few assets. We propose a new way, based on generalised entropies, to ensure a minimal degree of diversification.

cond-mat.stat-mech↗

arXiv1997

Financial markets as adaptative ecosystems

We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset fluctuations: `fat tails' and correlations in the scale of fluctuations. These aspects, although not included in the pricing models, are very precisely reflected in the price fixed by the market as a whole. Financial markets thus behave as rather efficient adaptive systems.

cond-mat↗

arXiv1997

Scaling in stock market data: stable laws and beyond

The concepts of scale invariance, self-similarity and scaling have been fruitfully applied to the study of price fluctuations in financial markets. After a brief review of the properties of stable Levy distributions and their applications to market data we indicate the shortcomings of such models and describe the truncated Levy flight as an alternative model for price movements. Furthermore, studying the dependence structure of the price increments shows that while their autocorrelation function decreases rapidly to zero, the correlation of their squares and absolute values shows a slow power law decay, indicating persistence in the scale of fluctuations, a property which can be related to the anomalous scaling of the kurtosis. In the last section we review, in the light of these empirical facts, recent attempts to draw analogies between scaling in financial markets and in turbulent flows.

cond-mat.stat-mech↗

arXiv1996

Comment on ``Turbulent cascades in foreign exchange markets''

Recently, Ghashghaie et al. have shown that some statistical aspects of fully developed turbulence and exchange rate fluctuations exhibit striking similarities (Nature 381, 767 (1996)). The authors then suggested that the two problems might be deeply connected, and speculated on the existence of an `information cascade' which would play the role in finance of the well known Kolmogorov energy cascade in turbulence. Here we want to convince the reader that the two problems differ on a fundamental aspect, namely, correlations.

cond-mat↗

arXiv1995

On the number of metastable states in spin glasses

In this letter, we show that the formulae of Bray and Moore for the average logarithm of the number of metastable states in spin glasses can be obtained by calculating the partition function with $m$ coupled replicas with the symmetry among these explicitly broken according to a generalization of the `two-group' ansatz. This equivalence allows us to find solutions of the BM equations where the lower `band-edge' free energy equals the standard static free energy. We present these results for the Sherrington-Kirkpatrick model, but we expect them to apply to all mean-field spin glasses.

cond-mat↗

arXiv1995

Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e.\ the fully-frustrated model on an infinite dimensional hypercube and the so-called sine-model. We use the mean-field (or {\sc tap}) equations which we derive by resuming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the {\sc tap} equations. We find that the configurational entropy (or complexity) is extensive in the range $T_{\mbox{\tiny RSB}}<T<T_{\mbox{\tiny M}}$. Finally we present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of {\sc tap} solutions.

cond-mat↗

arXiv1994

Statistical Mechanics and Visual Signal Processing

The nervous system solves a wide variety of problems in signal processing. In many cases the performance of the nervous system is so good that it apporaches fundamental physical limits, such as the limits imposed by diffraction and photon shot noise in vision. In this paper we show how to use the language of statistical field theory to address and solve problems in signal processing, that is problems in which one must estimate some aspect of the environment from the data in an array of sensors. In the field theory formulation the optimal estimator can be written as an expectation value in an ensemble where the input data act as external field. Problems at low signal-to-noise ratio can be solved in perturbation theory, while high signal-to-noise ratios are treated with a saddle-point approximation. These ideas are illustrated in detail by an example of visual motion estimation which is chosen to model a problem solved by the fly's brain. In this problem the optimal estimator has a rich structure, adapting to various parameters of the environment such as the mean-square contrast and the correlation time of contrast fluctuations. This structure is in qualitative accord with existing measurements on motion sensitive neurons in the fly's brain, and we argue that the adaptive properties of the optimal estimator may help resolve conlficts among different interpretations of these data. Finally we propose some crucial direct tests of the adaptive behavior.

cond-mat↗