Abstract
We study the pricing of European options when the underlying stock price is illiquid. Due to the lack of trades, the sample path followed by prices alternates between active and motionless periods that are replicable by a fractional jump–diffusion. This process is obtained by changing the time-scale of a jump–diffusion with the inverse of a Lévy subordinator. We prove that option prices are solutions of a forward partial differential equation in which the derivative with respect to time is replaced by a Dzerbayshan–Caputo (D–C) derivative. The form of the D–C derivative depends upon the chosen inverted Lévy subordinator. We detail this for inverted α stable and inverted Poisson subordinators. To conclude, we propose a numerical method to compute option prices for the two types of D–C derivatives.
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