Pricing currency options in a fractional Brownian motion with jumps

Xiao, Weilin; Zhang, Weiguo; Zhang, Xili; Ying-luo, Wang

doi:10.1016/j.econmod.2010.05.010

Cited by 72 publications

(29 citation statements)

References 31 publications

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“…However, it is unrealistic for the exchange rate to obey the geometric Brownian motion in the subsequent literatures. By modifying the G-K model, more and more methodologies for the currency option pricing have been proposed, such as Bollen and Rasieland [3], Carr and Wu [4], Sun [5], Swishchuk et al [6], Xiao et al [7], and Wang, Zhou, and Yang [8].…”

Section: Introductionmentioning

confidence: 99%

Pricing of European Currency Options with Uncertain Exchange Rate and Stochastic Interest Rates

Wang

2019

Discrete Dynamics in Nature and Society

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Suppose that the interest rates obey stochastic differential equations, while the exchange rate follows an uncertain differential equation; this paper proposes a new currency model. Under the proposed currency model, the pricing formula of European currency options is then derived. Some numerical examples recorded illustrate the quality of pricing formulas. Meanwhile, this paper analyzes the relationship between the pricing formula and some parameters.

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Section: Introductionmentioning

confidence: 99%

Pricing of European Currency Options with Uncertain Exchange Rate and Stochastic Interest Rates

Wang

2019

Discrete Dynamics in Nature and Society

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“…One of the first applications of FC in finance was through the fractional Brownian motion, which enables incorporating long-range auto-correlations, typically observed in finance [26,27], volatility modeling [28], and option pricing [29]. FC has been specifically used in many option pricing models [30][31][32][33], also in connection with the jump processes [34] or in pricing of more complicated types of options, as American options [31], double barrier options [35], or currency options [36]. These models have been also investigated by numerical methods [37,38], and some applications to implied volatility have been also discussed [39].In this paper, we focus on FC applications to option pricing, which is one of the most important tasks of financial mathematics.…”

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confidence: 99%

Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations

2019

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In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters. IntroductionFractional Calculus (FC) is nearly as old as conventional calculus. Many prominent mathematicians including Leibniz, Fourier, Laplace, Liouville, Riemann, Weyl, and Riesz suggested their own definitions of the fractional integrals and derivatives and studied their properties. Whereas the mathematical theory of FC was nearly completed a long time ago (see, e.g., [1] or the recent reference books [2,3]), only a few applications of FC outside mathematics were known until recently. During the last three decades, the situation changed completely, and currently, the majority of FC publications is devoted to modeling of a broad class of systems and processes using either the FC operators or the so-called fractional ordinary or partial differential equations. In particular, the FC models were successfully employed in physics [4,5], control theory [6], as well as in engineering, life, and social sciences [7,8], to mention only a few of the many application areas.The two probably most prominent and broadly-recognized FC applications are in linear viscoelasticity [9] and for describing anomalous transport processes [10]. In both cases, the FC models in the form of the fractional ODEs (linear viscoelasticity) and the fractional PDEs (anomalous transport processes), respectively, cannot be derived from "first principles". Instead, they are introduced as interpolations between several models formulated in terms of the ODEs or PDEs, respectively. In the case of linear viscoelasticity, the basic FC model interpolates between the Hooke model (elasticity, ODE Mathematics 2019, 7, 796 2 of 23 of the zero order) and the Newton model (viscosity, ODE of the first order). The FC model for the slow anomalous diffusion interpolates between a stationary state (time-independent Poisson equation) and the diffusion equation (first order equation with respect to time), whereas the fast diffusion is modeled by the fractional diffusion-wave PDE that interpolates between the diffusion equation (first order equation with respect to time) and the wave equation (second order equation with respect to time).Of course, this kind of model needs an additional justification...

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“…So, Kou (2002) proposed a jump-diffusion process with Poisson jump to match the abnormal fluctuation of stock price. Cai et al (2010), Cai and Kou (2011) and Xiao et al (2010) also considered the problem of pricing options under a jump-diffusion environment in a larger setting. Some scholars proposed a mixed fractional Brownian motion version of an option pricing Merton model (Mounir, 2006;Sun and Yan, 2012), and Sun and Yan (2012) assumed that the value of the firm obeys a geometric mixed fractional Brownian motion, the result shows that the mixed-fractional model that simulates credit risk pricing is a reasonable one.…”

Section: Introductionmentioning

confidence: 99%

Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

Yang

2017

J. Finan. Eng.

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This study presents an efficient method for pricing the American fractional lookback option in the case where the stock price follows a mixed jump diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral, a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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Pricing currency options in a fractional Brownian motion with jumps

Cited by 72 publications

References 31 publications

Pricing of European Currency Options with Uncertain Exchange Rate and Stochastic Interest Rates

Pricing of European Currency Options with Uncertain Exchange Rate and Stochastic Interest Rates

Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations

Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

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