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ABSTRACT: This paper derives a new semi closed-form approximation formula for pricing
an up-and-out barrier option under a certain type of stochastic volatility
model including SABR model by applying a rigorous asymptotic expansion method
developed by Kato, Takahashi and Yamada (2012). We also demonstrate the
validity of our approximation method through numerical examples.
02/2013;
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ABSTRACT: 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 Leandre's approach (Leandre, 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 time-homogenous local volatility and local-stochastic volatility models in finance, which include Heston (Heston, 1993) and (lambda-) SABR models (Hagan et.al., 2002, Labordere, 2008) as special cases. Some numerical examples are shown.
Derivatives eJournal. 02/2012;
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ABSTRACT: This paper develops a rigorous asymptotic expansion method with its numerical
scheme for the Cauchy-Dirichlet problem in second order parabolic partial
differential equations (PDEs). As an application, we propose a new
approximation formula for pricing a barrier option under stochastic volatility
environment including a log-normal type of SABR model
(Hagan-Kumar-Lesniewskie-Woodward(2002)).
02/2012;
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SIAM J. Financial Math. 01/2012; 3:95-136.
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[show abstract]
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ABSTRACT: This paper proposes a general approximation method for the solutions to second order parabolic partial differential equations (PDEs) by an extension of Leandre's approach and the Bismut identity in Malliavin calculus. We show two types of its applications, new approximations of derivatives prices and short-time asymptotic expansions of the heat kernel.In particular, we provide a new approximation formula for barrier option prices under a stochastic volatility model. We also derive short-time asymptotic expansions of the heat kernel under general time-homogenous local volatility and local-stochastic volatility models in finance which include Heston (Heston (1993)) and ($¥lambda$-) SABR models (Hagan et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.
Financial Engineering eJournal. 08/2011;
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ABSTRACT: 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 an asymptotic expansion approach. First, the paper derives an asymptotic expansion 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 approximation for pricing discrete barrier options with stochastic volatility models. Finally, it provides numerical examples for pricing double barrier call options with discrete monitoring under the Heston model.
Derivatives eJournal. 04/2010;
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[show abstract]
[hide abstract]
ABSTRACT: 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. Moreover, it provides numerical examples for pricing double barrier call options with discrete monitoring under the Heston model.
Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, CARF F-Series. 01/2010;
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[show abstract]
[hide abstract]
ABSTRACT: 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 an asymptotic expansion approach. First, the paper derives an asymptotic expansion 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 approximation for pricing discrete barrier options with stochastic volatility models. Finally, it provides numerical examples for pricing double barrier call options with discrete monitoring under the Heston model.
CIRJE, Faculty of Economics, University of Tokyo, CIRJE F-Series. 01/2010;
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[show abstract]
[hide abstract]
ABSTRACT: This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in a stochastic volatility model. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied, which provides an expansion formula for generalized Wiener functionals and the closed-form approximation formulas in stochastic volatility environment. In addition, it presents applications of the general formula to a local volatility expansion in the stochastic volatility model and expansions of option prices in the shifted log-normal and jump-diffusion models with stochastic volatilities. Finally, with an application of the Bismut identity the paper shows an expansion of the solution to the partial differential equation for pricing in a stochastic volatility model.
Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, CARF F-Series. 01/2009;
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[show abstract]
[hide abstract]
ABSTRACT: 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.
Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, CARF F-Series. 01/2009;
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[show abstract]
[hide abstract]
ABSTRACT: This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in a stochastic volatility model. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. It provides an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, it presents applications of the general formula to a local volatility expansion as well as to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some result of Malliavin calculus in jump-type models, this paper derives an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.
CIRJE, Faculty of Economics, University of Tokyo, CIRJE F-Series. 01/2009;
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[show abstract]
[hide abstract]
ABSTRACT: 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.
CIRJE, Faculty of Economics, University of Tokyo, CIRJE F-Series. 01/2009;
