Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options ( Revised in December 2008; subsequently published in "International Journal of Theoretical and Applied Finance", Vol.11-4,pp.381-401. )

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[1993]/Bates[1996] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

"An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates"

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on the third order asymptotic expansion scheme; we do not model a foreign exchange rate's variance such as in Heston[1993], but its volatility that follows a general time-inhomogeneous Markovian process, and we allow the correlations among all the factors, that is domestic and foreign interest rates, a spot foreign exchange rate and its volatility. Finally, we provide numerical examples and apply the pricing formula to the calibration of volatility surfaces in the JPY/USD option market.

"Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options"

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 [1993] / Bates [1996] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

Asymptotic Expansion Approaches in Finance: Applications to Currency Options

This chapter presents a basic of the methodology so-called an asymptotic expansion approach, and applies this method to approximation of prices of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. The scheme enables us to derive closed-form approximation formulas for pricing currency options even with high flexibility of the underlying model; we do not model a foreign exchange rate's variance such as in Heston [27], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. At the end of this chapter some numerical examples are provided and the pricing formula is applied to the calibration of volatility surfaces in the JPY/USD option market.

A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options ( Revised in April 2008, January 2009 and April 2010; forthcoming in "International Journal of Theoretical and Applied Finance". )

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. Moreover, the hybrid scheme develops Fourier transform method with an asymptotic expansion to utilize closed-form characteristic functions obtainable in parts of a model. Our scheme 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 closed-form characteristic functions. Finally, a series of numerical examples shows the validity of our scheme.

"Asymptotic Expansion Approaches in Finance: Applications to Currency Options"

This chapter presents a basic of the methodology so-called an asymptotic expansion approach, and applies this method to approximation of prices of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. The scheme enables us to derive closed-form approximation formulas for pricing currency options even with high flexibility of the underlying model; we do not model a foreign exchange rate's variance such as in Heston [27], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. At the end of this chapter some numerical examples are provided and the pricing formula is applied to the calibration of volatility surfaces in the JPY/USD option market.

APPLICATION OF A HIGH-ORDER ASYMPTOTIC EXPANSION SCHEME TO LONG-TERM CURRENCY OPTIONS

Recently, not only academic researchers but also many practitioners have used the methodology so-called ``an asymptotic expansion method" in their proposed techniques for a variety of financial issues. e.g. pricing or hedging complex derivatives under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory (Watanabe [1987], Yoshida [1992a,b]) in Malliavin calculus and essentially based on the framework initiated by Kunitomo and Takahashi [2003], Takahashi [1995,1999] in a financial context. In practical applications, it is desirable to investigate the accuracy and stability of the method especially with expansion up to high orders in situations where the underlying processes are highly volatile as seen in recent financial markets. After Takahashi [1995,1999] and Takahashi and Takehara [2007] had provided explicit formulas for the expansion up to the third order, Takahashi, Takehara and Toda [2009] develops general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe them in a simple setting to illustrate thier key idea, and to demonstrate their effectiveness apply them to pricing long-term currency options under a cross-currency Libor market model and a general stochastic volatility of a spot exchange rate with maturities up to twenty years.

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

This paper presents a new computational scheme for an asymptotic expansion method of an arbitrary order. The asymptotic expansion method in finance initiated by Kunitomo and Takahashi [1992], Yoshida [1992b] and Takahashi [1995], [1999] is a widely applicable methodology for an analytic approximation of expectation of a certain functional of diffusion processes. Hence, not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under high-dimensional underlying stochastic environments. In practical applications of the expansion, a crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [1995], [1999] and Takahashi and Takehara [2007] provided explicit formulas for those conditional expectations necessary for the asymptotic expansion up to the third order. This paper presents the new method for computing an arbitrary-order expansion in a general diffusion-type stochastic environment, which is powerful especially for high-order expansions: We develops a new calculation algorithm for computing coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations directly. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for a lambda-SABR model up to the fifth order.

"Computation in an Asymptotic Expansion Method"

An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida[68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for expansions up to the fourth order explicitly. The second one develops a new calculation algorithm for computing the coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for ă-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.

"Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options"

Recently, not only academic researchers but also many practitioners have used the methodology so-called "an asymptotic expansion method" in their proposed techniques for a variety of financial issues. e.g. pricing or hedging complex derivatives under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory(Watanabe [1987], Yoshida [1992a,b]) in Malliavin calculus and essentially based on the framework initiated by Kunitomo and Takahashi [2003], Takahashi [1995,1999] in a financial context. In practical applications, it is desirable to investigate the accuracy and stability of the method especially with expansion up to high orders in situations where the underlying processes are highly volatile as seen in recent financial markets. After Takahashi [1995,1999] and Takahashi and Takehara [2007] had provided explicit formulas for the expansion up to the third order, Takahashi, Takehara and Toda [2009] develops general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe them in a simple setting to illustrate thier key idea, and to demonstrate their effectiveness apply them to pricing long-term currency options under a cross-currency Libor market model and a general stochastic volatility of a spot exchange rate with maturities up to twenty years.