• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.845-858
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• 2006

We present two approaches of the stochastic interest rate European option pricing model. One is a bond numeraire approach which is applicable to a nonzero value asset. In this approach, we assume log-normality of returns of the asset normalized by a bond whose maturity is the same as the expiration date of an option instead that of an asset itself. Another one is the expectation hypothesis approach for value zero asset which has futures-style margining. Bond numeraire approach allows us to calculate volatilities implied in options even though stochastic interest rate is considered.

• 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.

• The Pure and Applied Mathematics
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• v.27 no.1
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• pp.43-50
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• 2020

We investigate the domain of influence of the local volatility function on the solutions of the general Black-Scholes model. First, we generate the sample paths of underlying asset using the Monte Carlo simulation. Next, we define the inner and outer domains to find the effective volatility region. To confirm the effect of the inner domain, we use the root mean square error for the European call option prices, and then change the values of volatility in the proposed domain. The computational experiments confirm that there is an effective region which dominates the option pricing.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.1
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• pp.19-30
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• 2019

In this paper, we develop an accurate explicit finite difference method for the two-dimensional Black-Scholes equation with a hybrid boundary condition. In general, the correlation term in multi-asset options is problematic in numerical treatments partially due to cross derivatives and numerical boundary conditions at the far field domain corners. In the proposed hybrid boundary condition, we use a linear boundary condition at the boundaries where at least one asset is zero. After updating the numerical solution by one time step, we reduce the computational domain so that we do not need boundary conditions. To demonstrate the accuracy and efficiency of the proposed algorithm, we calculate option prices and their Greeks for the two-asset European call and cash-or-nothing options. Computational results show that the proposed method is accurate and is very useful for nonlinear boundary conditions.

• East Asian mathematical journal
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• v.35 no.1
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• pp.59-66
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• 2019

In this paper we study implicit-explicit (IMEX) methods combined with a semi-Lagrangian scheme to evaluate the prices of fixed strike arithmetic Asian options under jump-diffusion models. An Asian option is described by a two-dimensional partial integro-differential equation (PIDE) that has no diffusion term in the arithmetic average direction. The IMEX methods with the semi-Lagrangian scheme to solve the PIDE are discretized along characteristic curves and performed without any fixed point iteration techniques at each time step. We implement numerical simulations for the prices of a European fixed strike arithmetic Asian put option under the Merton model to demonstrate the second-order convergence rate.

• Journal of the Korean Data and Information Science Society
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• v.24 no.4
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• pp.669-677
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• 2013

Quasi-Monte Carlo method is known to have lower convergence rate than the standard Monte Carlo method. Quasi-Monte Carlo methods are using low discrepancy sequences as quasi-random numbers. They include Halton sequence, Faure sequence, and Sobol sequence. In this article, we compared standard Monte Carlo method, quasi-Monte Carlo methods and three scrambling methods of Owen, Faure-Tezuka, Owen-Faure-Tezuka in valuation of multi-asset European call option through simulations. Moro inversion method is used in generating random numbers from normal distribution. It has been shown that three scrambling methods are superior in estimating option prices regardless of the number of assets, volatility, and correlations between assets. However, there are no big differences between them.

• Journal of the Korean Data and Information Science Society
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• v.25 no.3
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• pp.513-522
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• 2014

Using the assumption that the price of a stock follows a geometric Brownian motion with constant volatility, Black and Scholes (BS) derived a formula that gives the price of a European call option on the stock as a function of the stock price, the strike price, the time to maturity, the risk-free interest rate, the dividend rate paid by the stock, and the volatility of the stock's return. However, implied volatilities of BS method tend to depend on the stock prices and the time to maturity in practice. To address this shortcoming, we estimate the implied volatility function as a function of the strike priceand the time to maturity for data consisting of the daily prices for KOSPI200 call options from January 2007 to May 2009 using support vector regression (SVR), the multiple additive regression trees (MART) algorithm, and ordinary least squaress (OLS) regression. In conclusion, use of MART or SVR in the BS pricing model reduced both RMSE and MAE, compared to the OLS-based BS pricing model.

• Environmental and Resource Economics Review
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• v.14 no.4
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• pp.943-964
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• 2005

This study estimated a wide range of stochastic process models using the frameworks of CKLS (1992) and Nowman and Wang (2001). For empirical analysis, the GMM estimation procedure is adopted for the monthly Brent crude oil prices from January 1996 to January 2005. Using the simulated price series, European call option premiums were calculated and compared each other. The empirical results suggest that the crude oil price has a strong dependency of volatility on the price level. Contrary to the results of previous related studies, it shows a weak tendency of mean reversion. In addition, the models provide different implications for pricing derivatives on crude oil.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.26 no.4
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• pp.296-309
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• 2022

Timer options are one of the contingent claims that, for given the variance budget, its payoff depends on a random maturity in terms of the realized variance unlike the standard European vanilla option with a fixed time maturity. Since it was first launched by Société Générale Corporate and Investment Banking in 2007, the valuation of the timer options under several stochastic environment for the volatility has been conducted by many researches. In this study, we propose the pricing of timer power options combined with standard timer options and the index of the power to the underlying asset for the investors to actualize lower risks and higher returns at the same time under the uncertain markets. By using the asymptotic analysis, we obtain the first-order approximation of timer power options. Moreover, we demonstrate that our solution has been derived accurately by comparing it with the solution from the Monte-Carlo method. Finally, we analyze the impact of the stochastic volatility with regards to various parameters on the timer power options numerically.

• Communications for Statistical Applications and Methods
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• v.15 no.4
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• pp.563-585
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• 2008

This paper presents how to improve the efficiency and accuracy in the pricing and sensitivity evaluation for derivatives, since the need for the evaluation of complicated derivatives is increased. The Monte Carlo(MC) simulation using the quasi random number instead of pseudo random number can improve the elapsed time and accuracy for the valuation of European-type derivatives. However, the quasi MC simulation method has its limit for applying it in the multi-dimensional case such as American-type and path-dependent options due to the increased correlation between dimensions as the dimension of random numbers is increased. In order to complement this problem, we develop a modified method in which correlation values are controlled to be below a pre-specified value. Thus, this method is applicable for the pricing of either derivatives ill which underlying assets or risk factors are several or derivatives having path-dependent or early redemption property. Furthermore, we illustrate that it is important to take an appropriate grid interval for the use of finite difference method(FDM) by applying the FDM to one example of non-symmetrical butterfly spreads.