Multiscaling and rough volatility: An empirical investigation

Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.

Section: Introductionmentioning

confidence: 99%

Option Pricing with Fractional Stochastic Volatilities and Jumps

Zhang,

Yong,

Xiao

2023

“…All of the above studies are conducted under the models driven by standard Brownian motion which are Markovian or memoryless. However, many studies show that asset price fluctuations exhibit "long memory" [12][13][14][15][16][17][18] or "short memory" [19][20][21][22] which can be captured by stochastic volatility models driven by fractional Brownian motion with the Hurst index H ∈ (1/2, 1) or H ∈ (0, 1/2), respectively. In addition, jumps in the asset price were observed by Coqueret and Tavin [23], Jin and Hong [24], Bates [25], and Wang and Xia [26].…”

Section: Introductionmentioning

confidence: 99%

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

Zhang,

Xiao,

Yong

2024

This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.

“…This implies that the paths tend to be rougher, showing short-range dependency and can be effectively modelled using fractional Brownian motion with a Hurst parameter H < 1/2. For further insights, refer to studies such as Alos et al [3], Fukasawa [4], Gatheral et al [5], Livieri et al [6], Takaishi [7], Fukasawa [8], Brandi and Di Matteo [9], and related findings.…”

Section: Introductionmentioning

confidence: 99%

A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion

Mpanda

2024

This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter H∈(0,1). We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices.