Incorporating stochastic volatility and long memory into geometric Brownian motion model to forecast performance of Standard and Poor's 500 index

Alhagyan, Mohammed; Yassen, Mansour F.

doi:10.3934/math.2023945

Cited by 4 publications

(4 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Step 2: Choose ε 1 , ε 2 to approximate the pricing model (1) by applying model (10) Step 3: Compute the forward characteristic function by applying Formula (16) Step 4: Compute the cumulant c 1 , c 2 , c 4 by applying Theorem 2…”

Section: Algorithm 1 the Cos-based Algorithm For Pricing A Forward St...mentioning

confidence: 99%

“…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%

See 1 more Smart Citation

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

Zhang,

Xiao,

Yong

2024

Fractal Fract

View full text Add to dashboard Cite

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.

show abstract

Section: Algorithm 1 the Cos-based Algorithm For Pricing A Forward St...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

Zhang,

Xiao,

Yong

2024

Fractal Fract

View full text Add to dashboard Cite

show abstract

“…In real-world applications, these models are instrumental in helping financial analysts and institutions predict interest rate movements and manage associated risks. They serve as critical tools in the development of strategies for minimizing risks and maximizing returns, particularly in complex financial instruments like interest rate derivatives [11]. Moreover, these models underpin regulatory stress testing and scenario analysis, enabling financial firms to prepare for diverse economic conditions.…”

Section: Comparisons and Real-world Applicabilitymentioning

confidence: 99%

Application of Stochastic Processes in Financial Market Models

Zou,

Dong,

Yan

2024

AEMPS

View full text Add to dashboard Cite

This paper explores the significant role of stochastic processes in financial modeling, tracing the evolution from basic Brownian motion to sophisticated stochastic differential equations used in modern financial markets. Beginning with the historical development of Brownian motion, identified by Robert Brown and later mathematically modeled by Louis Bachelier for stock price fluctuations, the paper outlines its foundational influence on the Efficient Market Hypothesis and the random walk theory. The extension of these concepts in the Black-Scholes model for option pricing highlights the practical applications of these theories in predicting financial outcomes. The discussion progresses to geometric Brownian motion (GBM) and its crucial role in stock price modeling, emphasizing its use in Monte Carlo simulations for option pricing. The limitations of the Black-Scholes model and GBM in dealing with real market conditions such as stochastic volatility and jump-diffusion processes are addressed, showcasing the evolution of more complex models like the Heston model and GARCH. Interest rate models like the Vasicek and Cox-Ingersoll-Ross models are evaluated for their real-world applicability, particularly in scenarios of low and negative interest rates. This comprehensive review not only underscores the theoretical advancements in financial modeling but also its practical implications in contemporary financial markets.

show abstract

“…Many studies [1][2][3][4][5] suggest that asset price fluctuations exhibit "long memory". In addition, recent empirical studies [6][7][8][9] show that the roughness of the volatility process is observed.…”

Section: Introductionmentioning

confidence: 99%

Option Pricing with Fractional Stochastic Volatilities and Jumps

Zhang,

Yong,

Xiao

2023

Fractal Fract

View full text Add to dashboard Cite

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.

show abstract

Incorporating stochastic volatility and long memory into geometric Brownian motion model to forecast performance of Standard and Poor's 500 index

Cited by 4 publications

References 31 publications

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

Application of Stochastic Processes in Financial Market Models

Option Pricing with Fractional Stochastic Volatilities and Jumps

Contact Info

Product

Resources

About