竹内 敦司

Consider stochastic functional differential equations, whose coefficients depend on past histories. The solution determines a non-Markov process. In the present paper, we shall obtain the existence of smooth densities for joint distributions of solutions, under the uniformly elliptic condition on the diffusion coefficients, via the Malliavin calculus. As an application, we shall study the computations of the Greeks on options associated with the asset price dynamics models with delayed effects.

The purpose of this paper is to derive the Greeks formulas of Delta, Gamma, Vega and Theta for derivative securities with both continuous and discontinuous payoff structures under asset price dynamics described by stable and tempered stable processes with presentation of their practical simulation methods. Our approach is based on the representation of stable distributions using an exponential distribution whose scaling property with respect to the Girsanov transform is used in the Malliavin calculus framework on the Poisson space. Numerical results are presented to illustrate the effectiveness of our formulas in Monte Carlo simulations relative to the finite difference method.

Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument. © 2013 Akihiro Kitagawa and Atsushi Takeuchi.

Greeks formulas of Delta, Rho, Vega, and Gamma are derived in closed form for asset price dynamics described by gamma processes and Brownian motions time-changed by a gamma process. The model considered here includes many well-known models of practical interest, such as the variance gamma model and the Black-Scholes model. Our approach is based upon the Malliavin calculus for jump processes by making full use of a scaling property of gamma processes with respect to the Girsanov transform. The existence of their variance is investigated. Numerical results are provided to illustrate that the derived Greeks formulas have faster rate of convergence relative to the finite difference method.

Consider stochastic differential equations with jumps. The goal in this paper is to study the sensitivity of the solution with respect to the initial point, under the conditions on the Levy measure and the uniformly elliptic condition on the coefficients. The key tool is the martingale property based upon the Kolmogorov backward equation for the infinitesimal generator associated with the equation.

We consider jump-type stochastic differential equations with drift, diffusion, and jump terms. Logarithmic derivatives of densities for the solution process are studied, and Bismut-Elworthy-Li-type formulae are obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markov property of the process.

The main purpose of this paper is to derive unbiased Monte Carlo estimators of various sensitivity indices for an averaged asset price dynamics governed by the gamma Levy process. The key idea is to apply a scaling property of the gamma process with respect to the Esscher density transform parameter. Our framework covers not only the continuous Asian option, but also European, discrete Asian, average strike Asian, weighted average, spread options, and geometric average Asian options. Numerical results are provided to illustrate the effectiveness of our formulas in Monte Carlo simulations, relative to finite difference approximation. (C) 2009 Elsevier B.V. All rights reserved.

Consider the solution {X( t); t is an element of [- r, T]} of the following stochastic functional differential equation: dX( t) = {integral(0)(-r) rho(s) X(t + s) ds + A(0)( t, X(t))}dt + Sigma(m)(i=t) A(i) (t, X( t)) dW(i) ( t), where rho(t) is an R-valued function on [- r, 0], and {W( t); t is an element of [ 0, T]} is an m-dimensional Brownian motion. The main purpose is to study the smoothness of the probability density of X( T) with respect to the Lebesgue measure.

We study absolute continuity for the probability law of solutions to stochastic functional differential equations driven by Levy processes: dX(t) = A(0)(Xt-) dt + (m)Sigma(i=1) A(i) (Xt-) dL(i)(t), where X-t is the path defined by X-t(s) = X(t + s) for s is an element of [-r, 0]. We apply methods from the Malliavin calculus on the Poisson space to show that non-degenerate conditions on A(1),..., A(m) of the noise term guarantee the existence of densities.