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The fractional Black-Scholes equation. (English) Zbl 1058.91045

The Black-Scholes equation for the option of a European call involves the first derivative in time. Replacing this derivative by a fractional derivative of order \(\alpha\), \(0<\alpha<1\), leads to the fractional Black–Scholes equation. The author gives the complete solution to this equation with the help of Mellin, Laplace transformations and Green functions. He also finds the corresponding \(\Delta(\alpha)\), the Delta of call option, \(\Delta(\alpha)=\partial C(S,t)/\partial S\). Any information about \(\Delta(\alpha)\) gives some information on the order \(\alpha\).

MSC:

91B28 Finance etc. (MSC2000)
26A33 Fractional derivatives and integrals
62P05 Applications of statistics to actuarial sciences and financial mathematics
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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