"A Remark on Approximation of the Solutions to Partial Differential Equations in Finance"

This paper proposes a general approximation method for the solution to a second-order parabolic partial differential equation(PDE) widely used in finance through an extension of Léeandre's approach(Léandre (2006,2008)) and the Bismut identiy(e.g. chapter IX-7 of Malliavin (1997))] in Malliavin calculus. We present two types of its applications, approximations of derivatives prices and short-time asymptotic expansions of the heat kernel. In particular, we provide approximate formulas for option prices under local and stochastic volatility models. We also derive short-time asymptotic expansions of the heat kernel under general timehomogenous local volatility and local-stochastic volatility models in finance, which include Heston (Heston (1993)) and (λ-)SABR models (Hagan et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.

An Asymptotic Expansion with Malliavin Weights: An Application to Pricing Discrete Barrier Options

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula in Malliavin calculus is effectively applied in an asymptotic expansion approach. First, the paper derives an expansion formula for generalized Wiener functionals. After it is applied to pricing path-dependent derivatives with discrete monitoring, the paper presents an analytic (approximation) formula for valuation of discrete barrier options under stochastic volatility environment. To our knowledge, this paper is the first one that shows an analytical formula for pricing discrete barrier options with stochastic volatility models.

"An Asymptotic Expansion with Malliavin Weights: An Application to Pricing Discrete Barrier Options"

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula in Malliavin calculus is effectively applied in an asymptotic expansion approach. First, the paper derives an expansion formula for generalized Wiener functionals. After it is applied to pricing path-dependent derivatives with discrete monitoring, the paper presents an analytic (approximation) formula for valuation of discrete barrier options under stochastic volatility environment. To our knowledge, this paper is the first one that shows an analytical formula for pricing discrete barrier options with stochastic volatility models.

An Asymptotic Expansion with Push-Down of Malliavin Weights

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

An Asymptotic Expansion with Push-Down of Malliavin Weights

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

"A Semi-group Expansion for Pricing Barrier Options"

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develops a semi-group expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.

Pricing Discrete Barrier Options under Stochastic Volatility

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula and the duality formula in Malliavin calculus are effectively applied in pricing barrier options with discrete monitoring. To our knowledge, this paper is the first one that shows an analytical approximation for pricing discrete barrier options with stochastic volatility models. Furthermore, it provides numerical examples for pricing double barrier call options with discrete monitoring under Heston and ă-SABR models.

Early Rehabilitation with Weight-bearing Standing-shaking-board Exercise in Combination with Electrical Muscle Stimulation after Anterior Cruciate Ligament Reconstruction

The objective of early rehabilitation after anterior cruciate ligament (ACL) reconstruction is to increase the muscle strength of the lower extremities. Closed kinetic chain (CKC) exercise induces co-contraction of the agonist and antagonist muscles. The purpose of this study was to compare the postoperative muscle strength/mass of subjects who performed our new CKC exercise (new rehabilitation group:group N) from week 4, and subjects who received traditional rehabilitation alone (traditional rehabilitation group:group T). The subjects stood on the device and maintained balance. Then, low-frequency stimulation waves were applied to 2 points each in the anterior and posterior region of the injured thigh 3 times a week for 3 months. Measurement of muscle strength was performed 4 times (before the start, and then once a month). Muscle mass was evaluated in CT images of the extensor and flexor muscles of 10 knees (10 subjects) in each group. The injured legs of group N showed significant improvement after one month compared to group T. The cross-sectional area of the extensor muscles of the injured legs tended to a show a greater increase at 3 months in group N. This rehabilitation method makes it possible to contract fast-twitch muscles, which may be a useful for improving extensor muscle strength after ACL reconstruction