Computation in an Asymptotic Expansion Method

Abstract

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 λ-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.

The Time-Dependent FX-SABR Model: Efficient Calibration based on Effective Parameters

“…Second, we p It is well known that Hagan's formulas are biased for extreme strikes and large times to maturity, see e.g. (Islah 2009, Oblój 2008, Paulot 2009, Takahashi et al 2007). In Antonov et al (2013) it is pointed out that for maturities larger than 10 years, the error in implied volatility can be 1% or more, even for ATM values.…”

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

“…, 5 and x ′ denotes the transpose of x. H n (x; Σ) denotes the Hermite polynomial of degree n and Σ =  T 0 |q 1t | 2 dt. For the derivation and more general results, see Section 3 in [30].…”

Pricing Derivatives using the Asymptotic Expansion Approach: Credit Migration Models with Stochastic Credit Spreads

“…This technique has been applied to mathematical statistics by Yoshida (1992) and was first applied to mathematical finance by Takahashi (1999). See also Takahashi (2001, 2003a,b), Takahashi and Takehara (2008) and Takahashi et al (2009). This approach is very flexible and used to evaluate many kinds of contingent claims, such as options on default-free bonds, options on stocks, and options in stochastic interest and volatility environment.…”

“…These conditional expectations are calculated using the following lemma. The proof of this lemma is given by Takahashi et al (2009). Lemma 6.1 {w t } is an n-dimensional Wiener process and the vector x is a k-dimensional vector.…”