zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Financial markets with memory. I: Dynamic models. (English) Zbl 1108.91035
Summary: This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process $Y(\cdot)$ different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process $Y(\cdot)$ is defined by a continuous-time AR($\infty$)-type equation and may have either short or long memory. We show that the process $Y(\cdot)$ has a good MA($\infty$)-type representation. The existence of such simultaneous good AR($\infty$) and MA($\infty$) representations enables us to apply a new method for the calculation of relevant conditional expectations, whence to obtain various explicit results for problems such as portfolio optimization. The financial market defined by the above stock price process is complete, and if the coefficients are constant, then the prices of European calls and puts are given by the Black-Scholes formulas as in the Black-Scholes model. Unlike the latter, however, the model allows for differences between the historical and implied volatilities. The model includes a special case in which only two additional parameters are introduced to describe the memory of the market, compared with the Black-Scholes model. Analysis based on real market data shows that this simple model with two additional parameters is more realistic in capturing the memory effect of the market, while retaining the simplicity and usefulness of the Black-Scholes model.

91B28Finance etc. (MSC2000)
60G20Generalized stochastic processes
Full Text: DOI