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.

"Pricing Barrier and Average Options under Stochastic Volatility Environment"

This paper proposes a new approximation method of pricing barrier and average options under stochastic volatility environment by applying an asymptotic expansion approach. In particular, a high-order expansion scheme for general multi-dimensional diffusion processes is effectively applied. Moreover, the paper combines a static hedging method with the asymptotic expansion method for pricing barrier options. Finally, numerical examples show that the fourth or fifth-order asymptotic expansion scheme provides sufficiently accurate approximations under the ă-SABR and SABR models.

"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.

"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. An asymptotic expansion method in finance initiated by Kunitomo and Takahashi[9], Yoshida[34] and Takahashi [20], [21] is a widely applicable methodology for an analytic approximation of the expectation of a certain functional of diffusion processes and 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 highdimensional underlying stochastic environments. In practical applications of the expansion, the crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [20], [21] and Takahashi and Takehara [23] provided explicit formulas of 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 a high-order expansion: This 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. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for the -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.

Pricing Barrier and Average Options under Stochastic Volatility Environment

This paper proposes a new approximation method of pricing barrier and average options under stochastic volatility environment by applying an asymptotic expansion approach. In particular, a high-order expansion scheme for general multi-dimensional diffusion processes is effectively applied. Moreover, the paper combines a static hedging method with the asymptotic expansion method for pricing barrier options. Finally, numerical examples show that the fourth or fifth-order asymptotic expansion scheme provides sufficiently accurate approximations under the lambda-SABR and SABR models.

"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.

The Influence of Polyploidy and Genome Composition on Genomic Imprinting in Mice

Genomic imprinting is an epigenetic mechanism that switches the expression of imprinted genes involved in normal embryonic growth and development in a parent-of-origin-specific manner. Changes inDNAmethylation statuses from polyploidization are a well characterized epigenetic modification in plants. However, how changes in ploidy affect both imprinted gene expression and methylation status in mammals remains unclear. To address this, we used quantitative real time PCR to analyze expression levels of imprinted genes in mouse tetraploid fetuses. We used bisulfite sequencing to assess the methylation statuses of differentially methylated regions (DMRs) that regulate imprinted gene expression in triploid and tetraploid fetuses. The nine imprinted genes H19, Gtl2, Dlk1, Igf2r, Grb10, Zim1, Peg3, Ndn, and Ipw were all unregulated; in particular, the expression of Zim1 was more than 10-fold higher, and the expression of Ipw was repressed in tetraploid fetuses. The methylation statuses of four DMRs H19, intergenic (IG), Igf2r, and Snrpn in tetraploid and triploid fetuses were similar to those in diploid fetuses. We also performed allele-specific RT-PCR sequencing to determine the alleles expressing the three imprinted genes Igf2, Gtl2, and Dlk1 in tetraploid fetuses. These three imprinted genes showed monoallelic expression in a parent-of-origin-specific manner. Expression of non-imprinted genes regulating neural cell development significantly decreased in tetraploid fetuses, which might have been associated with unregulated imprinted gene expression. This study provides the first detailed analysis of genomic imprinting in tetraploid fetuses, suggesting that imprinted gene expression is disrupted, but DNA methylation statuses of DMRs are stable following changes in ploidy in mammals