12,638 research outputs found
Elementary Introduction to Stochastic Finance in Discrete Time
This article gives an elementary introduction to stochastic finance (in discrete time). A formalization of random variables is given and some elements of Borel sets are considered. Furthermore, special functions (for buying a present portfolio and the value of a portfolio in the future) and some statements about the relation between these functions are introduced. For details see: [8] (p. 185), [7] (pp. 12, 20), [6] (pp. 3-6).Ludwig Maximilians University of Munich, GermanyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2 edition, 2004.Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990
An overview of probabilistic and time series models in finance
In this paper, we partially review probabilistic and time series models in finance. Both discrete and continuous .time models are described. The characterization of the No- Arbitrage paradigm is extensively studied in several financial market contexts. As the probabilistic models become more and more complex to be realistic, the Econometrics needed to estimate them are more difficult. Consequently, there is still much research to be done on the link between probabilistic and time series models.Modeling Text Databases.- An Overview of Probabilistic and Time Series Models in Finance.- Stereological Estimation of the Rose of Directions.- Approximations for Multiple Scan Statistics.- Krawtchouk Polynomials and Krawtchouk Matrices.- An Elementary Rigorous Introduction to Exact Sampling.- On the Different Extensions of the Ergodic Theorem of Information Theory.- Dynamic Stochastic Models for Indexes and Thesauri.- Stability and Optimal Control.- Statistical Distances Based on Euclidean Graphs.- Implied Volatility.- On the Increments of the Brownian Sheet.- Compound Poisson Approximation.- Penalized Model Selection for Ill-posed Linear Problems.- The Arov-Grossman Model.- Recent Results in Geometric Analysis.- Dependence or Independence of the Sample Mean.- Optimal Stopping Problems for Time-Homogeneous Diffusions.- Criticality in Epidemics.- Acknowledgments.- Reference.- Index
Utility Maximization under Model Uncertainty in Discrete Time
We give a general formulation of the utility maximization problem under
nondominated model uncertainty in discrete time and show that an optimal
portfolio exists for any utility function that is bounded from above. In the
unbounded case, integrability conditions are needed as nonexistence may arise
even if the value function is finite.Comment: 18 page
Drift dependence of optimal trade execution strategies under transient price impact
We give a complete solution to the problem of minimizing the expected
liquidity costs in presence of a general drift when the underlying market
impact model has linear transient price impact with exponential resilience. It
turns out that this problem is well-posed only if the drift is absolutely
continuous. Optimal strategies often do not exist, and when they do, they
depend strongly on the derivative of the drift. Our approach uses elements from
singular stochastic control, even though the problem is essentially
non-Markovian due to the transience of price impact and the lack in Markovian
structure of the underlying price process. As a corollary, we give a complete
solution to the minimization of a certain cost-risk criterion in our setting
Time and foreign exchange markets
The definition of time is still an open question when one deals with high
frequency time series. If time is simply the calendar time, prices can be
modeled as continuous random processes and values resulting from transactions
or given quotes are discrete samples of this underlying dynamics. On the
contrary, if one takes the business time point of view, price dynamics is a
discrete random process, and time is simply the ordering according which prices
are quoted in the market. In this paper we suggest that the business time
approach is perhaps a better way of modeling price dynamics than calendar time.
This conclusion comes out from testing probability densities and conditional
variances predicted by the two models against the experimental ones. The data
set we use contains the DEM/USD exchange quotes provided to us by Olsen &
Associates during a period of one year from January to December 1998. In this
period 1,620,843 quotes entries in the EFX system were recorded
Growth Optimal Investment and Pricing of Derivatives
We introduce a criterion how to price derivatives in incomplete markets,
based on the theory of growth optimal strategy in repeated multiplicative
games. We present reasons why these growth-optimal strategies should be
particularly relevant to the problem of pricing derivatives. We compare our
result with other alternative pricing procedures in the literature, and discuss
the limits of validity of the lognormal approximation. We also generalize the
pricing method to a market with correlated stocks. The expected estimation
error of the optimal investment fraction is derived in a closed form, and its
validity is checked with a small-scale empirical test.Comment: 21 pages, 5 figure
Mixtures in non stable Levy processes
We analyze the Levy processes produced by means of two interconnected classes
of non stable, infinitely divisible distribution: the Variance Gamma and the
Student laws. While the Variance Gamma family is closed under convolution, the
Student one is not: this makes its time evolution more complicated. We prove
that -- at least for one particular type of Student processes suggested by
recent empirical results, and for integral times -- the distribution of the
process is a mixture of other types of Student distributions, randomized by
means of a new probability distribution. The mixture is such that along the
time the asymptotic behavior of the probability density functions always
coincide with that of the generating Student law. We put forward the conjecture
that this can be a general feature of the Student processes. We finally analyze
the Ornstein--Uhlenbeck process driven by our Levy noises and show a few
simulation of it.Comment: 28 pages, 3 figures, to be published in J. Phys. A: Math. Ge
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