Asymptotic Expansion Approach in Finance

2018

Stochastics

Self Cite

This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.

Section: Evaluation Schemementioning

confidence: 99%

“…It has been shown, in various numerical examples, that the first few terms of expansion are enough to achieve accurate approximation for option pricing with typical volatilities ranging from 10% to 20% and maturities up to a few years. See a review [50] for the details and a comprehensive list of literature.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansion for forward-backward SDEs with jumps

Fujii

2018

Stochastics

Self Cite

“…Although there is no closed-form solution except this special case, asymptotic expansion technique can be applied to derive an analytical approximate solution. See, for example, Takahashi (2015) [22] and references therein.…”

Section: Fx Optionmentioning

confidence: 99%

A general framework for the benchmark pricing in a fully collateralized market

Fujii

2016

J. Finan. Eng.

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Collateralization with daily margining has become a new standard in the post-crisis market. Although there appeared vast literature on a so-called multi-curve framework, a complete picture of a multi-currency setup with cross-currency basis can be rarely found since our initial attempts. This work gives its extension regarding a general framework of interest rates in a fully collateralized market. It gives a new formulation of the currency funding spread which is better suited for the general dependence. In the last half, it develops a discretization of the HJM framework with a fixed tenor structure, which makes it implementable as a traditional Market Model.

“…Although the method was extended to be applied to a jump-diffusion model by Kunitomo and Takahashi [24] and Takahashi [25,26], they concentrated on approximation of only bond prices or/and plain-vanilla option prices under a local volatility jump-diffusion model, and did not derive higher order expansions than the first order for the option pricing. Subsequently, Takahashi and Takehara [27] found a scheme for pricing plain-vanilla options in a jump-diffusion with stochastic volatility model.…”

Section: Introductionmentioning

confidence: 99%

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

Shiraya

Journal of Computational and Applied Mathematics

2016

Self Cite

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.