The fractional and mixed-fractional CEV model

Abstract

The continuous observation of the financial markets has identified some 'stylized facts' which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Itô's calculus and the Fokker-Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.

The sub-fractional CEV model

“…Since the ratio a(τ )/b(τ ) is time-independent (constant), the PDE (10) could be solved using the Feller's lemma with time varying coefficients [1,16]. Thus:…”

“…Using the same arguments supplied in [1], and defining z s (t) = k s (t)E 2−α , the European Call price at the inception time is given by:…”

“…Following the procedure given by Araneda [1] to fractional case, and the Itô calculus for sfBm [12], we derive the sub-fractional Fokker-Planck equation and the transition probability density function for the sub-fractional CEV (sfCEV) is obtained, leading to the price formula for an European Call option in terms of the non-central chi-squared distribution and the M-Whittaker function.…”

On the discrete-time simulation of the rough Heston model

“…Many of the rough volatility models consist in replacing the Brownian motions in the classical models by fractional Brownian motions, which leads to some SDEs or more general stochastic system driven by fractional Brownian motions, see e.g. [13,14,11,6,20,10] among many others. Another important way to model the rough volatility process is to use a stochastic Volterra equation, such as the rough Heston model introduced by El Euch and Rosenbaum [16]:…”

Arbitrage opportunities in sub-fractional Black-Scholes model

“…文献 [31,32] 进一步利用文献 [33,34] 定义的关于次分 数 Brown 运动的随机积分, 研究了金融衍生品的定价问题. 近年来, 文献 [35] 研究了混合分数 Brown 运动下常方差弹性系数模型的期权定价问题. 文献 [36] 采用偏微分方程方法, 进一步研究了混合次分 数 Brown 运动下常方差弹性系数模型的欧式期权定价问题.…”