• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제15권1호
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• pp.1-17
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• 2011

A new method for option pricing based on the trinomial tree method is introduced. The new method calculates the local average of option prices around a node at each time, instead of computing prices at each node of the trinomial tree. Local averaging has a smoothing effect to reduce oscillations of the tree method and to speed up the convergence. The option price and the hedging parameters are then obtained by the compact scheme and the Richardson extrapolation. Computational results for the valuation of European and American vanilla and barrier options show superiority of the proposed scheme to several existing tree methods.

• Journal of the Korean Statistical Society
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• 제36권2호
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• pp.237-256
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• 2007

This paper studies the problem of option pricing in an incomplete market. The market incompleteness comes from the discontinuity of the underlying asset price process which is, in particular, assumed to be a compound Poisson process. To find a reasonable price for a European contingent claim, we first find the unique minimal martingale measure and get a price by taking an expectation of the payoff under this measure. To get a closed-form price, we use an asymptotic expansion. In case where the minimal martingale measure is a signed measure, we use a sequence of martingale measures (probability measures) that converges to the equivalent martingale measure in the limit to compute the price. Again, we get a closed form of asymptotic option price. It is the Black-Scholes price and a correction term, when the distribution of the return process has nonzero skewness up to the first order.

• 재무관리연구
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• 제21권2호
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• pp.211-233
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• 2004

본 연구는 기초자산의 수익률이 정규분포가 아닌 급첨분포(leptokurtic distribution)를 따른다고 가정할 경우 옵션의 가격식을 도출한다. 두 정규분포의 확률밀도함수의 선형 결합으로 첨도가 3이 아닌 급첨분포의 확률밀도함수를 모델링하고 이를 이용하여 Black- Scholes 공식의 확장된 형태인 옵션 가격 공식을 유도한다. 본 논문에서 제시한 급첨분포에 의한 옵션가격모형은 변동성 스마일 성질을 설명할 뿐만 아니라 기존의 실증연구에서 제기된 Black-Scholes 옵션가격의 과대 및 과소평가 현상을 설명한다. 마지막으로 본 가격식의 모델적합성을 검증하기 위하여 KOSOI 200 지수옵션의 시장가격으로부터 내재변동성과 내재첨도를 추정한다.

• 대한수학회논문집
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• 제29권3호
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• pp.489-496
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• 2014

We drive a closed-form expression for the price of a European quanto call option in the double square root stochastic volatility model.

• 대한수학회논문집
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• 제31권2호
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• pp.415-422
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• 2016

We derive a closed-form expression for the price of a European quanto call option when both foreign and domestic interest rates follow the Vasicek's short rate model.

• 대한수학회보
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• 제56권2호
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• pp.351-364
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• 2019

We derive full closed-form expressions for the prices of European symmetric power quanto call options with four different forms of terminal payoffs under the assumption of the classical lognormal asset price and exchange rate model.

• 응용통계연구
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• 제24권4호
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• pp.721-733
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• 2011

We consider the state price densities that are implicit in financial asset prices. In the pricing of an option, the state price density is proportional to the second derivative of the option pricing function and this relationship together with no arbitrage principle imposes restrictions on the pricing function such as monotonicity and convexity. Since the state price density is a proper density function and most of the shape constraints are caused by this, we propose to estimate the state price density directly by specifying candidate densities in a flexible nonparametric way and applying methods of regularization under extra constraints. The problem is easy to solve and the resulting state price density estimates satisfy all the restrictions required by economic theory.

• Industrial Engineering and Management Systems
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• 제10권2호
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• pp.170-177
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• 2011

We present an improved binomial method for pricing financial deriva-tives by using cell averages. After non-overlapping cells are introduced around each node in the binomial tree, the proposed method calculates cell averages of payoffs at expiry and then performs the backward valuation process. The price of the derivative and its hedging parameters such as Greeks on the valuation date are then computed using the compact scheme and Richardson extrapolation. The simulation results for European and American barrier options show that the pro-posed method gives much more accurate price and Greeks than other recent lattice methods with less computational effort.

• 대한수학회보
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• 제48권4호
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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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• 제18권1호
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• pp.61-74
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• 2014

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.