Quadratic finite element and preconditioning methods for options pricing in the SVCJ model

Zhang, Ying Ying; Pang, Hong Kui; Feng, Liping; Jin, Xiao Qing

doi:10.21314/jcf.2014.287

Cited by 5 publications

(10 citation statements)

References 37 publications

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“…In contrast, the evaluation of the first-order approximation takes less than half of a second when the Mathematica function NIntegrate is used to compute the integral in (23). This computation time is also clearly lower to that of other numerical schemes such as finite elements [9] or boundary elements [10]. The first-order asymptotic expansion proposed in this paper therefore provides a fast way of obtaining barrier option prices which capture the common financial market phenomena of volatility randomness and mean reversion.…”

Section: Numerical Examplesmentioning

confidence: 94%

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Sousa

Cruzeiro

Guerra

2018

Journal of Computational and Applied Mathematics

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We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data.Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided.A natural way to address this issue is to introduce randomness in the volatility. For this reason, option pricing under stochastic volatility has been the subject of a great deal of research in recent years. Here we focus on the 2-hypergeometric stochastic volatility model, which was introduced by Da Fonseca and Martini [1] as a model which ensures that the volatility is strictly positive -this is an important property which is not present in some other well-established stochastic volatility models. In a very recent paper, Privault and She [2] demonstrated that, under this model, a closed-form asymptotic vanilla option pricing formula can be determined through a regular perturbation method. This is a notable result because their formulas are analytically very simple, which is rarely the case in models with stochastic volatility: as discussed by Zhu [3], the higher complexity of these models usually yields the need for rather sophisticated numerical implementations.The literature on barrier option pricing methods is extensive. Exact closed-form pricing formulas have been derived for only a few models other than that of Black and Scholes, none of which reproduces satisfactorily the market phenomena. (For explicit formulas under one-dimensional models see e.g. Davydov and Linestky [4], Hui and Lo [5]; for an explicit solution under the Heston stochastic volatility model see Lipton [6].) Given the unavailability of explicit formulas, to price barrier options under more complex models one needs to resort to numerical methods. The main approaches are the use of numerical partial differential equation (PDE) techniques and of Monte Carlo methods (we refer the reader to the books of Seydel [7] and of Glasserman [8]), which are often combined with other analytical or numerical techniques (for recent work see for instance Zhang et al. [9], Guardasoni and Sanfelici [10]). Unfortunately, the computational times are nowadays still largely incompatible with the demands of the financial industry.An alternative strategy for pricing under more general models is to derive approximate (or asymptotic) analytical solutions: this has been proposed not only for vanilla options (cf. Privault and She [2]) but also for barrier options, see e.g. Fouque et al. [11], Alos et al. [12]. These asymptotic methods are intrinsically computationally much less expen...

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Section: Numerical Examplesmentioning

confidence: 94%

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Sousa

Cruzeiro

Guerra

2018

Journal of Computational and Applied Mathematics

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“…We firstly recall the PIDE arising in the SVCJ model. For more details of the equation, we refer readers to [13,14,34]. Let u(t, x, y) denote the option price of a European-style option if at timeT −t the underlying asset price equals Ke x and its instantaneous variance equals (1 + y)θ, whereT refers to the maturity time, K is the strike price, and θ stands for the long-run variance level.…”

Section: Discretization Of the Pide In The Svcj Modelmentioning

confidence: 99%

“…Let u(t, x, y) denote the option price of a European-style option if at timeT −t the underlying asset price equals Ke x and its instantaneous variance equals (1 + y)θ, whereT refers to the maturity time, K is the strike price, and θ stands for the long-run variance level. According to [14,34], the option value function u(t, x, y) in the SVCJ model satisfies the following two-dimensional PIDE:…”

Section: Discretization Of the Pide In The Svcj Modelmentioning

confidence: 99%

“…is the drift parameter, κ is the rate of mean reversion, r is the risk-free interest rate, q is the dividend rate, λ is the intensity of compound Poisson process, and ν, µ J , ρ J , σ J are parameters related to jumps in return and variance. As stated in [14,34], the spatial domain Ω in (2.1) depends on the option that we investigate. In this paper, two types of options are considered, the European vanilla option and the double barrier option.…”

Section: Discretization Of the Pide In The Svcj Modelmentioning

confidence: 99%

“…Combined with the extrapolation approach, this method is remarkably fast and can achieve a polynomial accuracy in time direction. Zhang et al [34] further improve the efficiency of the extrapolation scheme by coupling with the quadratic finite element for spatial discretization and preconditioning techniques for resulting systems. It is known that the IMEX Euler based extrapolation scheme belongs to the time-stepping method.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

Pang

Sun

2014

E Asian J Appl Math

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The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

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A variable step‐size extrapolated Crank–Nicolson method for option pricing under stochastic volatility model with jump

Mao,

Tian,

Wang

2023

Math Methods in App Sciences

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This paper studies the numerical solution of the stochastic volatility model with jumps under European options. This model can be transformed into a partial integro‐differential equation (PIDE) with spatial mixed derivative terms. Due to the nonsmoothness of the initial function, the variable step‐size extrapolated Crank–Nicolson (CN) method, which explicitly discretizes the jump term and implicitly the rest, is proposed to solve this model. The finite difference method is used to discretize the spatial differential operator, the composite trapezoidal rule to calculate the jump integral, and then a linear system with a nine‐diagonal coefficient matrix is obtained, which is easy to solve. The stability of the variable step‐size extrapolated CN method is then proved. Based on realistic regularity assumptions on the data, the consistency error and global error bounds of the variable step‐size extrapolated CN method are derived in norm. Compared with the variable step‐size IMEX BDF2 method and the variable step‐size IMEX MP method, the numerical results show the efficiency of the proposed variable step‐size extrapolated CN method.

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Quadratic finite element and preconditioning methods for options pricing in the SVCJ model

Cited by 5 publications

References 37 publications

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

A variable step‐size extrapolated Crank–Nicolson method for option pricing under stochastic volatility model with jump

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