197 research outputs found
Mean Exit Time and Survival Probability within the CTRW Formalism
An intense research on financial market microstructure is presently in
progress. Continuous time random walks (CTRWs) are general models capable to
capture the small-scale properties that high frequency data series show. The
use of CTRW models in the analysis of financial problems is quite recent and
their potentials have not been fully developed. Here we present two (closely
related) applications of great interest in risk control. In the first place, we
will review the problem of modelling the behaviour of the mean exit time (MET)
of a process out of a given region of fixed size. The surveyed stochastic
processes are the cumulative returns of asset prices. The link between the
value of the MET and the timescale of the market fluctuations of a certain
degree is crystal clear. In this sense, MET value may help, for instance, in
deciding the optimal time horizon for the investment. The MET is, however, one
among the statistics of a distribution of bigger interest: the survival
probability (SP), the likelihood that after some lapse of time a process
remains inside the given region without having crossed its boundaries. The
final part of the article is devoted to the study of this quantity. Note that
the use of SPs may outperform the standard "Value at Risk" (VaR) method for two
reasons: we can consider other market dynamics than the limited Wiener process
and, even in this case, a risk level derived from the SP will ensure (within
the desired quintile) that the quoted value of the portfolio will not leave the
safety zone. We present some preliminary theoretical and applied results
concerning this topic.Comment: 10 pages, 2 figures, revtex4; corrected typos, to appear in the APFA5
proceeding
Activity autocorrelation in financial markets. A comparative study between several models
We study the activity, i.e., the number of transactions per unit time, of
financial markets. Using the diffusion entropy technique we show that the
autocorrelation of the activity is caused by the presence of peaks whose time
distances are distributed following an asymptotic power law which ultimately
recovers the Poissonian behavior. We discuss these results in comparison with
ARCH models, stochastic volatility models and multi-agent models showing that
ARCH and stochastic volatility models better describe the observed experimental
evidences.Comment: 15 pages, 4 figure
Statistical analysis and stochastic interest rate modelling for valuing the future with implications in climate change mitigation
High future discounting rates favor inaction on present expending while lower
rates advise for a more immediate political action. A possible approach to this
key issue in global economy is to take historical time series for nominal
interest rates and inflation, and to construct then real interest rates and
finally obtaining the resulting discount rate according to a specific
stochastic model. Extended periods of negative real interest rates, in which
inflation dominates over nominal rates, are commonly observed, occurring in
many epochs and in all countries. This feature leads us to choose a well-known
model in statistical physics, the Ornstein-Uhlenbeck model, as a basic
dynamical tool in which real interest rates randomly fluctuate and can become
negative, even if they tend to revert to a positive mean value. By covering 14
countries over hundreds of years we suggest different scenarios and include an
error analysis in order to consider the impact of statistical uncertainty in
our results. We find that only 4 of the countries have positive long-run
discount rates while the other ten countries have negative rates. Even if one
rejects the countries where hyperinflation has occurred, our results support
the need to consider low discounting rates. The results provided by these
fourteen countries significantly increase the priority of confronting global
actions such as climate change mitigation. We finally extend the analysis by
first allowing for fluctuations of the mean level in the Ornstein-Uhlenbeck
model and secondly by considering modified versions of the Feller and lognormal
models. In both cases, results remain basically unchanged thus demonstrating
the robustness of the results presented.Comment: 29 pages, 5 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1311.406
Uncertain growth and the value of the future
For environmental problems such as global warming future costs must be
balanced against present costs. This is traditionally done using an exponential
function with a constant discount rate, which reduces the present value of
future costs. The result is highly sensitive to the choice of discount rate and
has generated a major controversy as to the urgency for immediate action. We
study analytically several standard interest rate models from finance and
compare their properties to empirical data. From historical time series for
nominal interest rates and inflation covering 14 countries over hundreds of
years, we find that extended periods of negative real interest rates are
common, occurring in many epochs in all countries. This leads us to choose the
Ornstein-Uhlenbeck model, in which real short run interest rates fluctuate
stochastically and can become negative, even if they revert to a positive mean
value. We solve the model in closed form and prove that the long-run discount
rate is always less than the mean; indeed it can be zero or even negative,
despite the fact that the mean short term interest rate is positive. We fit the
parameters of the model to the data, and find that nine of the countries have
positive long run discount rates while five have negative long-run discount
rates. Even if one rejects the countries where hyperinflation has occurred, our
results support the low discounting rate used in the Stern report over higher
rates advocated by others.Comment: 8 pages, 4 figure
Theory of Second and Higher Order Stochastic Processes
This paper presents a general approach to linear stochastic processes driven
by various random noises. Mathematically, such processes are described by
linear stochastic differential equations of arbitrary order (the simplest
non-trivial example is , where is not a Gaussian white
noise). The stochastic process is discretized into time-steps, all possible
realizations are summed up and the continuum limit is taken. This procedure
often yields closed form formulas for the joint probability distributions.
Completely worked out examples include all Gaussian random forces and a large
class of Markovian (non-Gaussian) forces. This approach is also useful for
deriving Fokker-Planck equations for the probability distribution functions.
This is worked out for Gaussian noises and for the Markovian dichotomous noise.Comment: 35 pages, PlainTex, accepted for publication in Phys Rev. E
Extreme times in financial markets
We apply the theory of continuous time random walks to study some aspects of
the extreme value problem applied to financial time series. We focus our
attention on extreme times, specifically the mean exit time and the mean
first-passage time. We set the general equations for these extremes and
evaluate the mean exit time for actual data.Comment: 6 pages, 3 figure
Stochastic volatility and leverage effect
We prove that a wide class of correlated stochastic volatility models exactly
measure an empirical fact in which past returns are anticorrelated with future
volatilities: the so-called ``leverage effect''. This quantitative measure
allows us to fully estimate all parameters involved and it will entail a deeper
study on correlated stochastic volatility models with practical applications on
option pricing and risk management.Comment: 4 pages, 2 figure
Integrated random processes exhibiting long tails, finite moments and 1/f spectra
A dynamical model based on a continuous addition of colored shot noises is
presented. The resulting process is colored and non-Gaussian. A general
expression for the characteristic function of the process is obtained, which,
after a scaling assumption, takes on a form that is the basis of the results
derived in the rest of the paper. One of these is an expansion for the
cumulants, which are all finite, subject to mild conditions on the functions
defining the process. This is in contrast with the Levy distribution -which can
be obtained from our model in certain limits- which has no finite moments. The
evaluation of the power spectrum and the form of the probability density
function in the tails of the distribution shows that the model exhibits a 1/f
spectrum and long tails in a natural way. A careful analysis of the
characteristic function shows that it may be separated into a part representing
a Levy processes together with another part representing the deviation of our
model from the Levy process. This allows our process to be viewed as a
generalization of the Levy process which has finite moments.Comment: Revtex (aps), 15 pages, no figures. Submitted to Phys. Rev.
Discounting the Distant Future
If the historical average annual real interest rate is m \u3e 0, and if the world is stationary, should consumption in the distant future be discounted at the rate of m per year? Suppose the annual real interest rate r ( t ) reverts to m according to the Ornstein Uhlenbeck (OU) continuous time process dr ( t ) = α[ m – r ( t )] dt + kdw ( t ), where w is a standard Wiener process. Then we prove that the long run rate of interest is r ∞ = m – k 2 /2α 2 . This confirms the Weitzman-Gollier principle that the volatility and the persistence of interest rates lower long run discounting. We fit the OU model to historical data across 14 countries covering 87 to 318 years and estimate the average short rate m and the long run rate r ∞ for each country. The data corroborate that, when doing cost benefit analysis, the long run rate of discount should be taken to be substantially less than the average short run rate observed over a very long history
Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model
We study the pricing problem for a European call option when the volatility
of the underlying asset is random and follows the exponential
Ornstein-Uhlenbeck model. The random diffusion model proposed is a
two-dimensional market process that takes a log-Brownian motion to describe
price dynamics and an Ornstein-Uhlenbeck subordinated process describing the
randomness of the log-volatility. We derive an approximate option price that is
valid when (i) the fluctuations of the volatility are larger than its normal
level, (ii) the volatility presents a slow driving force toward its normal
level and, finally, (iii) the market price of risk is a linear function of the
log-volatility. We study the resulting European call price and its implied
volatility for a range of parameters consistent with daily Dow Jones Index
data.Comment: 26 pages, 6 colored figure
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