Pricing Barrier and Bermudan Style Options Under Time-Changed Lévy Processes: Fast Hilbert Transform Approach

Zeng, Pingping; Kwok, Yue Kuen

doi:10.1137/130922495

Cited by 31 publications

(11 citation statements)

References 36 publications

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“…A c c e p t e d M a n u s c r i p t Remark 5 Although we propose a circulant preconditioner based on Strang's approximation in this paper, the other circulant preconditioner, such as preconditioner based on T. Chan approximation [18], can be used similarly in (33) and (34).…”

Section: Circulant Preconditionermentioning

confidence: 99%

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Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Wang

Chen

Ding

et al. 2015

International Journal of Computer Mathematics

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In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable (FMLS), CGMY and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives, is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.

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Section: Circulant Preconditionermentioning

confidence: 99%

“…Various transform methods have been developed for solving the problem, including Laplace transform [25], Hilbert transform [13,34] , and Fourier cosine transform [10,12]. In addition, Ballotta and Kyriakou [1] launched a novel Monte Carlo simulation method to price barrier options under the framework of the exponential Lévy process.…”

Section: Introductionmentioning

confidence: 99%

Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Wang

Chen

Ding

et al. 2015

International Journal of Computer Mathematics

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“…More details on the success of the fast Hilbert transform algorithms for pricing discrete barrier options, discrete lookback options and Bermudan options under Lévy processes or time-changed Lévy processes can be found in Feng and Linetsky (2008), Lin and Feng (2013), and Zeng and Kwok (2013).…”

Section: Review Of Hilbert Transformmentioning

confidence: 99%

Fast Hilbert Transform Algorithms for Pricing Discrete Timer Options Under Stochastic Volatility Models

2014

Self Cite

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Timer options are barrier style options in the volatility space. A typical timer option is similar to its European vanilla counterpart, except with uncertain expiration date. The finite-maturity timer option expires either when the accumulated realized variance of the underlying asset has reached a pre-specified level or on the mandated expiration date, whichever comes earlier. The challenge in the pricing procedure is the incorporation of the barrier feature in terms of the accumulated realized variance instead of the usual knock-out feature of hitting a barrier by the underlying asset price. We construct efficient and accurate fast Hilbert transform algorithms for pricing finite-maturity discrete timer options under different types of stochastic volatility processes. The stochastic volatility processes nest some popular stochastic volatility models, like the Heston model and 3/2 stochastic volatility model. The barrier feature associated with the accumulated realized variance can be incorporated effectively into the fast Hilbert transform procedure with the computational convenience of avoiding the nuisance of recovering the option values in the real domain at each monitoring time instant in order to check for the expiry condition. Our numerical tests demonstrate high level of accuracy of the fast Hilbert transform algorithms. We also explore the pricing properties of the timer options with respect to various parameters, like volatility of variance, correlation coefficient between the asset price process and instantaneous variance process, sampling frequency, and variance budget.

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“…Zeng and Kwok [13] extended the fast Hilbert transform approach for pricing barrier and Bermudian style options under time-changed Levy processes. The success of the fast Hilbert transform approach to compute the fair prices of barrier style derivatives in the Fourier domain lies in the mathematical identity that relates the Fourier transform of a price function multiplied by an indicator function to the Hilbert transform of the Fourier transform of the price function.…”

Section: Introductionmentioning

confidence: 99%

On Two Transform Methods for the Valuation of Contingent Claims

Nwozo¹,

Fadugba²

2015

JMF

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This paper presents two transform methods for pricing contingent claims namely the fast Fourier transform method and the fast Hilbert transform method. The fast Fourier transform method utilizes the characteristic function of the underlying instrument's price process. The fast Hilbert transform method is obtained by multiplying a square integrable function f by an indicator function associated with the barrier feature in the real domain. This is also obtained by taking the Hilbert transform in the Fourier domain. We derived closed-form solutions for European call options in a double exponential jump-diffusion model with stochastic volatility. We developed fast and accurate numerical solutions by means of the Fourier transform method. The comparison of the probability densities of the double exponential jump-diffusion model with stochastic volatility, the Black-Scholes model and the double exponential jump-diffusion model without stochastic volatility showed that the double exponential jump-diffusion model with stochastic volatility has better performance than the two other models with respect to pricing long term options. An analysis of the fast Fourier transform method revealed that the volatility of volatility σ and the correlation coefficient ρ have significant impact on option values. It was also observed that these parameters σ and ρ have effect on long-term option, stock returns and they are also negatively correlated with volatility. These negative correlations are important for contingent claims valuation. The fast Fourier transform method is useful for empirical analysis of the underlying asset price. This method can also be used for pricing contingent claims when the characteristic function of the return is known analytically. We considered the performance of the fast Hilbert transform method and Heston model for pricing finite-maturity discrete barrier style options under stochastic volatility and observed that the fast Hilbert transform method gives more accurate results than the Heston model as shown in Table 3.

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Pricing Barrier and Bermudan Style Options Under Time-Changed Lévy Processes: Fast Hilbert Transform Approach

Cited by 31 publications

References 36 publications

Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Circulant preconditioning technique for barrier options pricing under fractional diffusion models

Fast Hilbert Transform Algorithms for Pricing Discrete Timer Options Under Stochastic Volatility Models

On Two Transform Methods for the Valuation of Contingent Claims

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