1,792 research outputs found
Diffusion of Context and Credit Information in Markovian Models
This paper studies the problem of ergodicity of transition probability
matrices in Markovian models, such as hidden Markov models (HMMs), and how it
makes very difficult the task of learning to represent long-term context for
sequential data. This phenomenon hurts the forward propagation of long-term
context information, as well as learning a hidden state representation to
represent long-term context, which depends on propagating credit information
backwards in time. Using results from Markov chain theory, we show that this
problem of diffusion of context and credit is reduced when the transition
probabilities approach 0 or 1, i.e., the transition probability matrices are
sparse and the model essentially deterministic. The results found in this paper
apply to learning approaches based on continuous optimization, such as gradient
descent and the Baum-Welch algorithm.Comment: See http://www.jair.org/ for any accompanying file
Option Pricing under Fast-varying and Rough Stochastic Volatility
Recent empirical studies suggest that the volatilities associated with
financial time series exhibit short-range correlations. This entails that the
volatility process is very rough and its autocorrelation exhibits sharp decay
at the origin. Another classic stylistic feature often assumed for the
volatility is that it is mean reverting. In this paper it is shown that the
price impact of a rapidly mean reverting rough volatility model coincides with
that associated with fast mean reverting Markov stochastic volatility models.
This reconciles the empirical observation of rough volatility paths with the
good fit of the implied volatility surface to models of fast mean reverting
Markov volatilities. Moreover, the result conforms with recent numerical
results regarding rough stochastic volatility models. It extends the scope of
models for which the asymptotic results of fast mean reverting Markov
volatilities are valid. The paper concludes with a general discussion of
fractional volatility asymptotics and their interrelation. The regimes
discussed there include fast and slow volatility factors with strong or small
volatility fluctuations and with the limits not commuting in general. The
notion of a characteristic term structure exponent is introduced, this exponent
governs the implied volatility term structure in the various asymptotic
regimes.Comment: arXiv admin note: text overlap with arXiv:1604.0010
Forward transition rates
The idea of forward rates stems from interest rate theory. It has natural
connotations to transition rates in multi-state models. The generalization from
the forward mortality rate in a survival model to multi-state models is
non-trivial and several definitions have been proposed. We establish a
theoretical framework for the discussion of forward rates. Furthermore, we
provide a novel definition with its own logic and merits and compare it with
the proposals in the literature. The definition turns the Kolmogorov forward
equations inside out by interchanging the transition probabilities with the
transition intensities as the object to be calculated.Comment: Revision of manuscript. The manuscript now contains a section on
'Forward-thinking and actuarial practice'. Furthermore, we have corrected
typos and re-written certain sentences to improve readability and accurac
The History of the Quantitative Methods in Finance Conference Series. 1992-2007
This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.
USLV: Unspanned Stochastic Local Volatility Model
We propose a new framework for modeling stochastic local volatility, with
potential applications to modeling derivatives on interest rates, commodities,
credit, equity, FX etc., as well as hybrid derivatives. Our model extends the
linearity-generating unspanned volatility term structure model by Carr et al.
(2011) by adding a local volatility layer to it. We outline efficient numerical
schemes for pricing derivatives in this framework for a particular four-factor
specification (two "curve" factors plus two "volatility" factors). We show that
the dynamics of such a system can be approximated by a Markov chain on a
two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by
direct (Kroneker) products of values of pairs of curve and volatility factors,
respectively. The resulting Markov chain dynamics on such partly "folded" state
space enables fast pricing by the standard backward induction. Using a
nonparametric specification of the Markov chain generator, one can accurately
match arbitrary sets of vanilla option quotes with different strikes and
maturities. Furthermore, we consider an alternative formulation of the model in
terms of an implied time change process. The latter is specified
nonparametrically, again enabling accurate calibration to arbitrary sets of
vanilla option quotes.Comment: Sections 3.2 and 3.3 are re-written, 3 figures adde
Response to worrying trends in econophysics
This article is a response to the recent “Worrying Trends in Econophysics” critique written by four respected theoretical economists [1]. Two of the four have written books and papers that provide very useful critical analyses of the shortcomings of the standard textbook economic model, neo-classical economic theory [2,3] and have even endorsed my book [4]. Largely, their new paper reflects criticism that I have long made [4,5,6,7,] and that our group as a whole has more recently made [8]. But I differ with the authors on some of their criticism, and partly with their proposed remedy.General equilibrium; uncertainty; conservation laws; money nonconservation; nonintegrability of dynamical systems; financial markets; stochastic processes
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