| Summary: | In this study, we first present a time-fractional L<inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>e</mi> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L<inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>e</mi> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>vy-time fractional diffusion equation of European-style options. Further, we introduce a more general model based on the L<inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>e</mi> <mo stretchy="false">^</mo> </mover> </semantics> </math> </inline-formula>vy-time fractional diffusion equation and review some recent findings associated with risk-neutral free European option pricing.
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