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.

"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.

Pricing Swaptions under the Libor Market Model of Interest Rates with Local-Stochastic Volatility Models

This paper presents a new approximation formula for pricing swaptions and caps/floors under the LIBOR market model of interest rates (LMM) with the local and affine-type stochastic volatility. In particular, two approximation methods are applied in pricing, one of which is so called gdrift-freezingh that fixes parts of the underlying stochastic processes at their initial values. Another approximation is based on an asymptotic expansion approach. An advantage of our method is that those approximations can be applied in a unified manner to a general class of local-stochastic volatility models of interest rates. To demonstrate effectiveness of our method, the paper takes CEVHeston LMM and Quadratic-Heston LMM as examples; it confirms sufficient flexibility of the models for calibration in a caplet market and enough accuracies of the approximation method for numerical evaluation of swaption values under the models.

"Pricing Swaptions under the Libor Market Model of Interest Rates with Local-Stochastic Volatility Models"

This paper presents a new approximation formula for pricing swaptions and caps/floors under the LIBOR market model of interest rates (LMM) with the local and affine-type stochastic volatility. In particular, two approximation methods are applied in pricing, one of which is so called "drift-freezing" that fixes parts of the underlying stochastic processes at their initial values. Another approximation is based on an asymptotic expansion approach. An advantage of our method is that those approximations can be applied in a unified manner to a general class of local-stochastic volatility models of interest rates. To demonstrate effectiveness of our method, the paper takes CEVHeston LMM and Quadratic-Heston LMM as examples; it confirms sufficient flexibility of the models for calibration in a caplet market and enough accuracies of the approximation method for numerical evaluation of swaption values under the models.

"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.

On weak and strong convergence rate for the Heston stochastic volatility model

The Heston stochastic volatility model is one of the most fundamental models in mathematical finance. Recently, many numerical schemes have been developed for the Heston model. However, in the literature, there is no weak or strong convergence rate obtained for the full parameter regime. In this PhD thesis, we shall focus on the numerical scheme that simulates the variance process exactly and applies the stochastic trapezoidal rule to approximate the time integral of the variance process in the SDE of the logarithmic asset process. Our goal is to obtain the weak and strong convergence rates of such a numerical scheme for the Heston model. The weak convergence rate is of traditional interest, because it is an important measure on how fast the bias of a numerical scheme decays. We prove that the numerical scheme we consider converges at rate two for the whole parameter regime, and the test function can be any polynomial of the logarithmic asset process. The rate is consistent with the standard rate of the stochastic trapezoidal rule, although the Lipschitz assumption is not satisfied. The strong convergence analysis is meaningful in the framework of Multi-level Monte Carlo (MLMC). The MLMC can be regarded as a variance reduction technique for numerical schemes on SDEs, as long as there is a MLMC estimator with a good strong convergence rate. We establish efficient MLMC estimators, separately for the path-independent and path-dependent simulations. We are able to provide the strong convergence rates in both situations