• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.271-287
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• 2008

This paper deals with American call and put options. Determining the fair price and the free boundary of an American option is a very difficult problem since they depends on each other. This paper presents numerical algorithms of finite element method based on the three-level scheme to compute both the price and the free boundary. One algorithm is designed for American call options and the other one for American put options. These algorithms are formulated on the system of the Jamshidian equation for the option price and the free boundary. Here, the Jamshidian equation is of a kind of the nonhomogeneous Black-Scholes equations. We prove the existence and uniqueness of the numerical solution by the Lax-Milgram lemma and carried out extensive numerical experiments to compare with various methods.

• Korean Journal of Mathematics
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• v.25 no.2
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• pp.229-246
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• 2017

This paper concerns barrier option of American type where the underlying price is monitored during only part of the option's life. Analytic valuation formulas of the American partial barrier options are obtained by approximation method. This approximation method is based on barrier options along with exponential early exercise policies. This result is an extension of Jun and Ku [10] where the exercise policies are constant.

• 제어로봇시스템학회:학술대회논문집
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• 2001.10a
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• pp.106.1-106
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• 2001

American options no doubt is invariably more popular than European options, due to the fact that it gives the owner the option to exercise a contract before and up to the expiration date, unlike an European option, which only allows the owner to exercise a contract on the date of expiration. Owing to its popularity, many methods like the binomial numerical method and the pseudo American method have been devised for computing of the value of the American options. The aim of this research is to develop an effective 3-dimensional visualization for American option portfolio based on the Geske-Roll-Whaley model. It is obvious that it is extremely tedious and unadvisable for researchers to interprte chunks of data by looking at graphs or pie charts, which are simple but not effective for analyzing important dta. Hence, the generation of the Geske-Roll-Whaley ...

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.791-812
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• 2011

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.1
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• pp.1-21
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• 2015

This paper presents two algorithms based on the Jamshidian equation which is from the Black-Scholes partial differential equation. The first algorithm is for American call options and the second one is for American put options. They compute numerically free boundary and then option price, iteratively, because the free boundary and the option price are coupled implicitly. By the upwind finite-difference scheme, we discretize the Jamshidian equation with respect to asset variable s and set up a linear system whose solution is an approximation to the option value. Using the property that the coefficient matrix of this linear system is an M-matrix, we prove several theorems in order to formulate a bisection method, which generates a sequence of intervals converging to the fixed interval containing the free boundary value with error bound h. These algorithms have the accuracy of O(k + h), where k and h are step sizes of variables t and s, respectively. We prove that they are unconditionally stable. We applied our algorithms for a series of numerical experiments and compared them with other algorithms. Our algorithms are efficient and applicable to options with such constraints as r > d, $r{\leq}d$, long-time or short-time maturity T.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.769-784
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• 2013

We consider the modulus-based successive overrelaxation method for the linear complementarity problems from the discretization of Black-Scholes American options model. The $H_+$-matrix property of the system matrix discretized from American option pricing which guarantees the convergence of the proposed method for the linear complementarity problem is analyzed. Numerical experiments confirm the theoretical analysis, and further show that the modulus-based successive overrelaxation method is superior to the classical projected successive overrelaxation method with optimal parameter.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.2
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• pp.11-20
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• 2000

This paper presents a formula that relates the optimal exercise boundaries of American call and put options on futures contract. It is shown that the geometric mean of the optimal exercise boundaries for call and put written on the same futures contract with the same exercise price is equal to the exercise price which is time invariant. The paper also investigates the properties of American calls and puts on futures contract.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.397-406
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• 2013

We present an improved binomial method for pricing European- and American-type Asian options based on the arithmetic average of the prices of the underlying asset. At each node of the tree we propose a simple algorithm to choose the representative averages among all the effective averages. Then the backward valuation process and the interpolation are performed to compute the price of the option. The simulation results for European and American Asian options show that the proposed method gives much more accurate price than other recent lattice methods with less computational effort.

• Journal of Korean Institute of Industrial Engineers
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• v.38 no.4
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• pp.244-248
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• 2012

We present a simple numerical method for pricing American put options under a constant elasticity of variance (CEV) model. Our analysis is done in a general framework where only the risk-neutral transition density of the underlying asset price is given. We obtain an integral equation of early exercise premium. By exploiting a modification of the integral equation, we propose a novel and simple numerical iterative valuation method for American put options.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.413-426
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• 2011

Two types of new methods with variable time steps are proposed in order to valuate binary options efficiently. Type I changes adaptively the size of the time step at each time based on the magnitude of the local error, while Type II combines two uniform meshes. The new methods are hybrid finite difference methods, namely starting the computation with a fully implicit finite difference method for a few time steps for accuracy then performing a ${\theta}$-method during the rest of computation for efficiency. Numerical experiments for standard European vanilla, binary and American options show that both Type I and II variable time step methods are much more efficient than the fully implicit method or hybrid methods with uniform time steps.