A Hybrid Asymptotic Expansion Scheme:an Application to Long-term Currency Options

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. 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 validity of our scheme. This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for valuation of European derivatives. Specifically, we apply it to pricing longterm currency options under a market model of interest rates and a general stochastic volatility model with jumps of spot exchange rates. Our scheme is very effective for models which admits correlations among all factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle 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 calculated instantly and thus can be used for calibration or explicit approximations of Greeks. 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. It also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable to make full use of analytical approximations by the asymptotic expansion and the closed-form characteristic functions. Finally, a series of numerical examples shows the validity of our scheme.