Abstract
Stochastic volatility models play an important role in finance modeling. Under a mixed fractional Brownian motion environment, we study the continuity and estimates of a solution to a kind of stochastic differential equations with double volatility terms. Besides, we propose to price the vulnerable option with the discretization method and present the results using a Monte Carlo simulation.
Highlights
Most of the existing literature on financial models assumes that the volatility of assets is constant
To model the volatility smile effectively, one solution is using the stochastic volatility under two cases: (1) the function of stochastic processes is used to describe the volatility [3, 4], and (2) the additional Brownian motion is introduced to describe the stochastic parts of stochastic volatility (SV) models
Wang et al [10] extend the framework of Siu et al [11] and focus on currency options under a two-factor Markov-modulated stochastic volatility jump-diffusion model
Summary
Most of the existing literature on financial models assumes that the volatility of assets is constant. To model the volatility smile effectively, one solution is using the stochastic volatility under two cases: (1) the function of stochastic processes is used to describe the volatility [3, 4], and (2) the additional Brownian motion is introduced to describe the stochastic parts of stochastic volatility (SV) models. All the existing SV models mentioned are based on the Brownian motion with increments following the independent norm distribution. M1H,1(t), M1H,2(t), M2H,1(t) and M2H,2(t) are mfBm processes, which, together with relative conclusions, will be defined, and α is an elastic constant We call this model a mixed factional CEV model. A fractional Brownian motion (fBm) {BH(t), t ≥ 0} with Hurst parameter H is a continuous and centered Gaussian process with covariance [16].
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