Financial markets with memory I: Dynamic models

Anh, Vo V. & Inoue, Akihiko (2005) Financial markets with memory I: Dynamic models. Stochastic Analysis and Applications, 23(2), pp. 275-300.


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(t) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y(t) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show that the process Y(t) has a good MA(∞)-type representation. The existence of such simultaneous good AR(∞) and MA(∞) 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.

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22 citations in Scopus
21 citations in Web of Science®
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ID Code: 4403
Item Type: Journal Article
Refereed: Yes
Additional URLs:
ISSN: 1532-9356
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2005 Taylor & Francis
Copyright Statement: First published in Stochastic Analysis and Applications 23(2):pp. 275-300.
Deposited On: 01 Jun 2006 00:00
Last Modified: 29 Feb 2012 13:17

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