A semilinear Black and Scholes partial differential equation for valuing American options

In the paper, we propose a new calculation scheme for American options in the framework of a forward backward stochastic differential equation (FBSDE). The wellknown decomposition of an American option price with that of a European option of the same maturity and the remaining early exercise premium can be cast into the form of a decoupled non-linear FBSDE. We numerically solve the FBSDE by applying an interacting particle method recently proposed by Fujii & Takahashi (2012d), which allows one to perform a Monte Carlo simulation in a fully forward-looking manner. We perform the fourth-order analysis for the Black-Scholes (BS) model and the thirdorder analysis for the Heston model. The comparison to those obtained from existing tree algorithms shows the effectiveness of the particle method.

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“…For more rigorous treatment, see the related proof in [32,23] as well as [2]. The case for a Put option can be shown similarly.…”

Section: Propositionmentioning

confidence: 95%

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

Fujii

Sato

Takahashi

2014

Asia-Pac Financ Markets

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“…The function V in (1.1) is the value function of an optimal stopping problem for which the dynamic programming principle holds [24]. Using the dynamic programming principle, we proved in [7] that V uniquely solves (in a viscosity solution sense) the following semilinear Black and Scholes equation set in a fixed domain: where g is the payoff function in (1.2). One should note that in (1.3) there is no free boundary that needs to be computed as part of the solution nor are there "side constraints" that need to be satisfied by the solution (as opposed to the quasi-variational formulation).…”

Section: Introductionmentioning

confidence: 99%

“…In [7], we presented a new approach to determining the value of an American option. The function V in (1.1) is the value function of an optimal stopping problem for which the dynamic programming principle holds [24].…”

Section: Introductionmentioning

confidence: 99%

“…One should note that in (1.3) there is no free boundary that needs to be computed as part of the solution nor are there "side constraints" that need to be satisfied by the solution (as opposed to the quasi-variational formulation). We refer to [7] for the motivation behind the semilinear Black and Scholes equation (1.3) and its rigorous derivation from (1.1) via the dynamic programming principle. We also refer to [7] for an overview of relevant literature.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

Benth

Karlsen

Reikvam

2004

Interfaces Free Bound.

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In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of "easy to implement" numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of "predictor-corrector type" for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor-corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.

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Finite Difference Methods for Early Exercise Options

Toivanen

2010

Encyclopedia of Quantitative Finance

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Usually, options with early exercise possibility like American and Bermudan options have to be priced numerically. In this article, approaches based on partial differential operators and their finite difference discretization are considered. The early exercise possibility leads to an inequality constraint for the price of the option. Linear complementarity problem (LCP) and other formulations incorporating this constraint are discussed. Fairly standard finite difference discretizations can be used for the underlying partial differential operator. Several solution and approximation methods including the projected successive over relaxation (PSOR) method and penalty methods are described. As an example, an American put option is priced using different methods under the Black–Scholes model.

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A semilinear Black and Scholes partial differential equation for valuing American options

Cited by 25 publications

References 26 publications

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

Finite Difference Methods for Early Exercise Options

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