Abstract
In this paper, we first investigate the stochastic representation of the modified advection-dispersion equation, which is proved to be a subordinated stochastic process. Taking advantage of this result, we get the analytical solution and mean square displacement for the equation. Then, applying the subordinated Brownian motion into the option pricing problem, we obtain the closed-form pricing formula for the European option, when the underlying of the option contract is supposed to be driven by the subordinated geometric Brownian motion. At last, we compare the obtained option pricing models with the classical Black–Scholes ones.
Highlights
The diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion [1]
We aim to evaluate the price of an European option when the underlying of the option contract is supposed to be driven by a subordinated geometric Brownian motion St X(Et), where the parent process X(τ) is given as dX(τ) μX(τ)dτ + σX(τ)dB(τ), (22)
Substituting σ2(t) into the above equation ends the proof. Noting that this option pricing formula is quite similar to the one obtained in [36], where the price of the underlying is supposed to be driven by a geometric fractional Brownian motion
Summary
The diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion [1]. When 〈x2〉 ∝ tc, the value c > 1 characterizes a superdiffusive process, c < 1 a subdiffusive one, and c 1 a normal diffusion. E FFPE for anomalous diffusion are obtained as particular cases of the Kolmogorov’s equation [5]. In [8], the authors introduced a simple and efficient method for computer simulation of sample paths of anomalous diffusion process described by the FFPE. It reveals that subdiffusion is a combination of two independent mechanisms: the first mechanism is the standard diffusion represented by some Itoprocess X(τ), and the second mechanism introduces the trapping events and is represented by the so-called inverse α-stable subordinator Sα(t). The probability density function (PDF) of the subordinated process X(Sα(t)) is the solution of the FFPE [9]
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