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

2007 ◽  
Author(s):  
Akihiko Takahashi ◽  
Kohta Takehara
Keyword(s):  
Asymptotic Expansion ◽  
Currency Options ◽  
Expansion Scheme

A HYBRID ASYMPTOTIC EXPANSION SCHEME: AN APPLICATION TO LONG-TERM CURRENCY OPTIONS

2010 ◽  
Vol 13 (08) ◽  
pp. 1179-1221 ◽  
Author(s):  
AKIHIKO TAKAHASHI ◽  
KOHTA TAKEHARA
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Market Model ◽  
Characteristic Functions ◽  
Approximation Formula ◽  
Currency Options ◽  
Analytical Approximations ◽  
Volatility Model ◽  
Expansion Scheme

This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for 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.

scholarly journals An asymptotic expansion of forward-backward SDEs with a perturbed driver

2015 ◽  
Vol 02 (02) ◽  
pp. 1550020 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Error Estimate ◽  
Arbitrary Order ◽  
Nonlinear Pricing ◽  
Small Noise ◽  
Small Diffusion ◽  
Noise Perturbation ◽  
Backward Sdes ◽  
Expansion Scheme

Motivated by nonlinear pricing in finance, this paper presents a mathematical validity of an asymptotic expansion scheme for a system of forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. In particular, we represent the coefficients of the expansion of the FBSDE up to an arbitrary order, and obtain the error estimate of the expansion with respect to the driver and the small noise perturbation.

A micromechanical investigation of interfacial transport processes. I. Interfacial conservation equations

1993 ◽  
Vol 345 (1675) ◽  
pp. 165-207 ◽  
Keyword(s):  
Mass Transfer ◽  
Asymptotic Expansion ◽  
Control Volume ◽  
Transport Processes ◽  
Matched Asymptotic Expansion ◽  
Immiscible Fluid ◽  
Conservation Equations ◽  
Interfacial Transport ◽  
Fluid Systems ◽  
Expansion Scheme

Macroscale interfacial conservation equations are derived for transport processes occurring in immiscible fluid—fluid systems possessing moving and deforming interfaces via a rigorous matched asymptotic expansion scheme from the more exact, continuous (‘diffuse’) microscale equations underlying them . A surface-fixed coordinate system is developed for the parameterization of the interface, alleviating approximations which result when either a material or a space-fixed control volume is used to investigate systems undergoing interphase mass transfer.

FOURIER TRANSFORM METHOD WITH AN ASYMPTOTIC EXPANSION APPROACH: AN APPLICATION TO CURRENCY OPTIONS

2008 ◽  
Vol 11 (04) ◽  
pp. 381-401 ◽  
Author(s):  
AKIHIKO TAKAHASHI ◽  
KOHTA TAKEHARA
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Interest Rates ◽  
Jump Diffusion ◽  
Characteristic Functions ◽  
Currency Options ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Libor Market Model ◽  
Jump Diffusion Model

This paper develops a Fourier transform method with an asymptotic expansion approach for option pricing. The method is applied to European currency options with a libor market model of interest rates and jump-diffusion stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas of the characteristic functions of log-prices of the underlying assets and the prices of currency options based on a third order asymptotic expansion scheme; we use a jump-diffusion model with a mean-reverting stochastic variance process such as in Heston [7]/Bates [1] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

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