A Hybrid Asymptotic Expansion Scheme: An Application to Long-Term Currency Options

CARF Working Paper Series No. CARF-F-116

45 Pages Posted: 18 Jun 2009

See all articles by Akihiko Takahashi

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Kohta Takehara

University of Tokyo - Graduate School of Economics

Date Written: December 23, 2007

Abstract

This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates.

Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained.

An asymptotic expansion approach provides a closed-form approximation formula for their values, which can be calculated in a moment and thus can be used for calibration or for an explicit approximation of Greeks of options. Moreover, this scheme develops Fourier transform method with an asymptotic expansion as well as with closed-form characteristic functions obtainable in parts of a model, extending the method proposed by Takehara and Takahashi [2008] to be applicable to a general class of models.

It also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of the closed-form characteristic functions. Finally, a series of numerical examples shows the effectiveness of our scheme.

Keywords: Currency option, libor market model, stochastic volatility, asymptotic expansion, Monte Carlo simulation

Suggested Citation

Takahashi, Akihiko and Takehara, Kohta, A Hybrid Asymptotic Expansion Scheme: An Application to Long-Term Currency Options (December 23, 2007). CARF Working Paper Series No. CARF-F-116, Available at SSRN: https://ssrn.com/abstract=1421914 or http://dx.doi.org/10.2139/ssrn.1421914

Akihiko Takahashi (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

Kohta Takehara

University of Tokyo - Graduate School of Economics ( email )

Yayoi 1-1-1
Bunkyo-ku
Tokyo, Tokyo 113-8657
Japan