Pricing European option under the time-changed mixed Brownian-fractional Brownian model

T-stability of the Euler method for impulsive stochastic differential equations driven by fractional Brownian motion

“…Modestly using the Taylor's formula to the right side of (14) and choosing fitting h k , we can have…”

“…From (26), such h k can fulfil (15) and (14). Therefore the Euler-Maruyama method (2) for the equation (1) is T-stable.…”

“…In fact, quite a lot of observed natural and social phenomena, such as the fluctuation of stock price and the rate of return in Financial markets, exhibit properties of "biased random walk" and "decile-shaped fat tail". It is suggested that fractional Brownian motion (fBm) with H ∈ (0, 1/2) ∪ (1/2, 1)(see [2,7,8,14]), which owns the self-similarity and the long-term memory, provides better modeling and description of such phenomena. In particular, such fractional equations are simple and effective, which were discussed early in [1,17,27,31].…”

Option Pricing under the CEV Model in a Composite-diffusive Regime

“…is a strictly increasing - stable Levy process (see, [8,9]). Based on the work of Magdziarz, scholars( [11][12][13][14]) generalize the subdiffusuve Brownian motion model.Specially, Liang et al [10] generalize Magdziarz's model to a compositediffusive regime. They introduced a composite-diffusive geometric Brownian motion…”

“…From theorem 2, we can see that ) (t c cannot be determined analytically, therefore, the above pricing formula can not to be use directly. Hence we turn into find the asymptotic representation of the solution of the PDE problem (13) and (7). .…”