×

Long range dependence, no arbitrage and the Black-Scholes formula. (English) Zbl 1016.91053

This paper studies a stock price model driven by the sum of a Wiener process and a continuous process \(Z\) having zero generalized quadratic variation. It uses forward integrals as in the author’s paper [Math. Nachr. 225, 145-183 (2001; Zbl 0983.60054)] to define self-financing portfolios. For contingent claims which are nice functions of the terminal stock price, it is then shown that one can construct a valuation function and a hedging portfolio from a solution of the usual Black-Scholes PDE. Under additional assumptions on \(Z\), it is also shown that the model contains no arbitrage strategies within a specified class. Particular examples yield models with long range dependence.

MSC:

91B28 Finance etc. (MSC2000)
60H05 Stochastic integrals
60G48 Generalizations of martingales

Citations:

Zbl 0983.60054
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alòs E., Taiwanese J. Math. 5 pp 609– (2001)
[2] DOI: 10.1086/260062 · Zbl 1092.91524
[3] DOI: 10.2307/3318626 · Zbl 1005.60053
[4] DOI: 10.1111/1467-9965.00025 · Zbl 0884.90045
[5] DOI: 10.1007/BF01195073 · Zbl 0792.60046
[6] DOI: 10.1016/0304-4149(95)93237-A · Zbl 0840.60052
[7] DOI: 10.1155/S104895339900012X · Zbl 0948.60047
[8] DOI: 10.1007/BF02214078 · Zbl 0855.60039
[9] DOI: 10.1007/s004400050171 · Zbl 0918.60037
[10] DOI: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0 · Zbl 0983.60054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.