A General Computation Scheme for a High-Order Asymptotic Expansion Method

Takahashi, Akihiko; Takehara, Kohta; Toda, Masashi

doi:10.1142/s0219024912500446

Cited by 36 publications

(28 citation statements)

References 25 publications

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“…We also note that the higher order expansions can be derived in the similar manner, which is expected to provide more precise approximations as in the diffusion cases in [28,29].…”

Section: Resultsmentioning

confidence: 97%

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

Shiraya

Takahashi

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tThis paper develops a new approximation formula for pricing basket options in a localstochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models.Moreover, in numerical experiments, we provide approximate prices for basket options on the WTI futures and Brent futures based on the parameters through calibration to the plain-vanilla option prices, and confirm the validity of our approximation formula.

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“…We also note that the higher order expansions can be derived in the similar manner, which is expected to provide more precise approximations as in the diffusion cases in [28,29].…”

Section: Resultsmentioning

confidence: 97%

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

Shiraya

Takahashi

2016

Journal of Computational and Applied Mathematics

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“…Although there is no explicit expression of (3.3) for a generic stock process, it is always possible to obtain its approximation by asymptotic expansion (See [34,25,35,36] for the details of asymptotic expansion.). It allows us, at least approximately, to have an explicit expression of…”

Section: Th Ordermentioning

confidence: 99%

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

Fujii

Sato

Takahashi

2014

Asia-Pac Financ Markets

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In the paper, we propose a new calculation scheme for American options in the framework of a forward backward stochastic differential equation (FBSDE). The wellknown decomposition of an American option price with that of a European option of the same maturity and the remaining early exercise premium can be cast into the form of a decoupled non-linear FBSDE. We numerically solve the FBSDE by applying an interacting particle method recently proposed by Fujii & Takahashi (2012d), which allows one to perform a Monte Carlo simulation in a fully forward-looking manner. We perform the fourth-order analysis for the Black-Scholes (BS) model and the thirdorder analysis for the Heston model. The comparison to those obtained from existing tree algorithms shows the effectiveness of the particle method.

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“…To calculate the stochastic kernels of the SPTO, i.e., the transition density corresponding to a given stochastic differential equation, we apply the small disturbance asymptotic theory, which is an asymptotic expansion of the stochastic processes [27,28]. To apply this theory, we assume that the diffusion coefficients in Eqs.…”

Section: B Stochastic Phase Transition Operatormentioning

confidence: 99%

“…We introduce a Markov operator for an infinite relaxation rate using the small disturbance asymptotic theory [27,28], and this operator describes the stochastic dynamics around the limit cycle. We investigate the dynamics of the entire phase space without input impulses using our Markov operator for finite and infinite relaxation rates and analyze the response of the stochastic neuronal oscillator to impulsive inputs by examining the effects of the relaxation rate, intrinsic noise strength, and input impulse parameters.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics of a stochastic neuronal oscillator

Yamanobe

2013

Phys. Rev. E

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Nonlinear oscillators have been used to model neurons that fire periodically in the absence of input. These oscillators, which are called neuronal oscillators, share some common response structures with other biological oscillations such as cardiac cells. In this study, we analyze the dependence of the global dynamics of an impulse-driven stochastic neuronal oscillator on the relaxation rate to the limit cycle, the strength of the intrinsic noise, and the impulsive input parameters. To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. Previously, we derived the Markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane. Here, we construct a Markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle. In both cases, the response of the stochastic neuronal oscillator to time-varying impulses is described by a product of Markov operators. Furthermore, we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density. Specifically, we analyze the dynamics of the number of spikes per unit time based on the properties of the Markov operators. Each Markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions. This allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator, which we show to be based on the dependence of the oscillator on past activity. Our analysis shows how the duration of the past neuronal activity depends on the relaxation rate, the noise strength, and the impulsive input parameters.

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A General Computation Scheme for a High-Order Asymptotic Expansion Method

Cited by 36 publications

References 25 publications

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

Global dynamics of a stochastic neuronal oscillator

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