27 research outputs found
A Non-Gaussian Option Pricing Model with Skew
Closed form option pricing formulae explaining skew and smile are obtained
within a parsimonious non-Gaussian framework. We extend the non-Gaussian option
pricing model of L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) to
include volatility-stock correlations consistent with the leverage effect. A
generalized Black-Scholes partial differential equation for this model is
obtained, together with closed-form approximate solutions for the fair price of
a European call option. In certain limits, the standard Black-Scholes model is
recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and
Ross. Alternative methods of solution to that model are thereby also discussed.
The model parameters are partially fit from empirical observations of the
distribution of the underlying. The option pricing model then predicts European
call prices which fit well to empirical market data over several maturities.Comment: 37 pages, 11 ps figures, minor changes, typos corrected, to appear in
Quantitative Financ
Individual and collective stock dynamics: intra-day seasonalities
We establish several new stylised facts concerning the intra-day
seasonalities of stock dynamics. Beyond the well known U-shaped pattern of the
volatility, we find that the average correlation between stocks increases
throughout the day, leading to a smaller relative dispersion between stocks.
Somewhat paradoxically, the kurtosis (a measure of volatility surprises)
reaches a minimum at the open of the market, when the volatility is at its
peak. We confirm that the dispersion kurtosis is a markedly decreasing function
of the index return. This means that during large market swings, the
idiosyncratic component of the stock dynamics becomes sub-dominant. In a
nutshell, early hours of trading are dominated by idiosyncratic or sector
specific effects with little surprises, whereas the influence of the market
factor increases throughout the day, and surprises become more frequent.Comment: 9 pages, 7 figure
Goodness-of-Fit tests with Dependent Observations
We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit
(GoF) tests and propose a generalisation to identically distributed, but
dependent univariate random variables. We show that the dependence leads to a
reduction of the "effective" number of independent observations. The
generalised GoF tests are not distribution-free but rather depend on all the
lagged bivariate copulas. These objects, that we call "self-copulas", encode
all the non-linear temporal dependences. We introduce a specific, log-normal
model for these self-copulas, for which a number of analytical results are
derived. An application to financial time series is provided. As is well known,
the dependence is to be long-ranged in this case, a finding that we confirm
using self-copulas. As a consequence, the acceptance rates for GoF tests are
substantially higher than if the returns were iid random variables.Comment: 26 page
Liquidity and the multiscaling properties of the volume traded on the stock market
We investigate the correlation properties of transaction data from the New
York Stock Exchange. The trading activity f(t) of each stock displays a
crossover from weaker to stronger correlations at time scales 60-390 minutes.
In both regimes, the Hurst exponent H depends logarithmically on the liquidity
of the stock, measured by the mean traded value per minute. All multiscaling
exponents tau(q) display a similar liquidity dependence, which clearly
indicates the lack of a universal form assumed by other studies. The origin of
this behavior is both the long memory in the frequency and the size of
consecutive transactions.Comment: 7 pages, 3 figures, submitted to Europhysics Letter
Volatility Effects on the Escape Time in Financial Market Models
We shortly review the statistical properties of the escape times, or hitting
times, for stock price returns by using different models which describe the
stock market evolution. We compare the probability function (PF) of these
escape times with that obtained from real market data. Afterwards we analyze in
detail the effect both of noise and different initial conditions on the escape
time in a market model with stochastic volatility and a cubic nonlinearity. For
this model we compare the PF of the stock price returns, the PF of the
volatility and the return correlation with the same statistical characteristics
obtained from real market data.Comment: 12 pages, 9 figures, to appear in Int. J. of Bifurcation and Chaos,
200
Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
We construct a N-dimensional Gaussian landscape with multiscale, translation
invariant, logarithmic correlations and investigate the statistical mechanics
of a single particle in this environment. In the limit of high dimension N>>1
the free energy of the system and overlap function are calculated exactly using
the replica trick and Parisi's hierarchical ansatz. In the thermodynamic limit,
we recover the most general version of the Derrida's Generalized Random Energy
Model (GREM). The low-temperature behaviour depends essentially on the spectrum
of length scales involved in the construction of the landscape. If the latter
consists of K discrete values, the system is characterized by a K-step Replica
Symmetry Breaking solution. We argue that our construction is in fact valid in
any finite spatial dimensions . We discuss implications of our results
for the singularity spectrum describing multifractality of the associated
Boltzmann-Gibbs measure. Finally we discuss several generalisations and open
problems, the dynamics in such a landscape and the construction of a
Generalized Multifractal Random Walk.Comment: 25 pages, published version with a few misprints correcte
Probability distribution of returns in the exponential Ornstein-Uhlenbeck model
We analyze the problem of the analytical characterization of the probability
distribution of financial returns in the exponential Ornstein-Uhlenbeck model
with stochastic volatility. In this model the prices are driven by a Geometric
Brownian motion, whose diffusion coefficient is expressed through an
exponential function of an hidden variable Y governed by a mean-reverting
process. We derive closed-form expressions for the probability distribution and
its characteristic function in two limit cases. In the first one the
fluctuations of Y are larger than the volatility normal level, while the second
one corresponds to the assumption of a small stationary value for the variance
of Y. Theoretical results are tested numerically by intensive use of Monte
Carlo simulations. The effectiveness of the analytical predictions is checked
via a careful analysis of the parameters involved in the numerical
implementation of the Euler-Maruyama scheme and is tested on a data set of
financial indexes. In particular, we discuss results for the German DAX30 and
Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data
and the theoretical description.Comment: 26 pages, 9 figures and 3 tables. New section with real data analysis
and related references added, some minor typos corrected. Accepted for
publication on JSTA
On a generalised model for time-dependent variance with long-term memory
The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of
stochastic time series with time-dependent variance like it appears on a wide
broad of systems besides economics in which ARCH was born. Although the ARCH
process captures the so-called "volatility clustering" and the asymptotic
power-law probability density distribution of the random variable, it is not
capable to reproduce further statistical properties of many of these time
series such as: the strong persistence of the instantaneous variance
characterised by large values of the Hurst exponent (H > 0.8), and asymptotic
power-law decay of the absolute values self-correlation function. By means of
considering an effective return obtained from a correlation of past returns
that has a q-exponential form we are able to fix the limitations of the
original model. Moreover, this improvement can be obtained through the correct
choice of a sole additional parameter, . The assessment of its validity
and usefulness is made by mimicking daily fluctuations of SP500 financial
index.Comment: 6 pages, 4 figure
Minding impacting events in a model of stochastic variance
We introduce a generalisation of the well-known ARCH process, widely used for
generating uncorrelated stochastic time series with long-term non-Gaussian
distributions and long-lasting correlations in the (instantaneous) standard
deviation exhibiting a clustering profile. Specifically, inspired by the fact
that in a variety of systems impacting events are hardly forgot, we split the
process into two different regimes: a first one for regular periods where the
average volatility of the fluctuations within a certain period of time is below
a certain threshold and another one when the local standard deviation
outnumbers it. In the former situation we use standard rules for
heteroscedastic processes whereas in the latter case the system starts
recalling past values that surpassed the threshold. Our results show that for
appropriate parameter values the model is able to provide fat tailed
probability density functions and strong persistence of the instantaneous
variance characterised by large values of the Hurst exponent is greater than
0.8, which are ubiquitous features in complex systems.Comment: 18 pages, 5 figures, 1 table. To published in PLoS on
On the concentration of large deviations for fat tailed distributions, with application to financial data
Large deviations for fat tailed distributions, i.e. those that decay slower
than exponential, are not only relatively likely, but they also occur in a
rather peculiar way where a finite fraction of the whole sample deviation is
concentrated on a single variable. The regime of large deviations is separated
from the regime of typical fluctuations by a phase transition where the
symmetry between the points in the sample is spontaneously broken. For
stochastic processes with a fat tailed microscopic noise, this implies that
while typical realizations are well described by a diffusion process with
continuous sample paths, large deviation paths are typically discontinuous. For
eigenvalues of random matrices with fat tailed distributed elements, a large
deviation where the trace of the matrix is anomalously large concentrates on
just a single eigenvalue, whereas in the thin tailed world the large deviation
affects the whole distribution. These results find a natural application to
finance. Since the price dynamics of financial stocks is characterized by fat
tailed increments, large fluctuations of stock prices are expected to be
realized by discrete jumps. Interestingly, we find that large excursions of
prices are more likely realized by continuous drifts rather than by
discontinuous jumps. Indeed, auto-correlations suppress the concentration of
large deviations. Financial covariance matrices also exhibit an anomalously
large eigenvalue, the market mode, as compared to the prediction of random
matrix theory. We show that this is explained by a large deviation with excess
covariance rather than by one with excess volatility.Comment: 38 pages, 12 figure