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

Zbl 0983.60054
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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
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