• Journal of applied mathematics & informatics
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• v.41 no.1
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• pp.33-50
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• 2023

Foreign exchange options are derivative financial instruments that can exchange one currency for another at a prescribed exchange rate on a specified date. In this study, we examine the analytic formulas for vulnerable foreign exchange options based on multi-scale stochastic volatility driven by two diffusion processes: a fast mean-reverting process and a slow mean-reverting process. In particular, we take advantage of the asymptotic analysis and the technique of the Mellin transform on the partial differential equation (PDE) with respect to the option price, to derive approximated prices that are combined with a leading order price and two correction term prices. To verify the price accuracy of the approximated solutions, we utilize the Monte Carlo method. Furthermore, in the numerical experiments, we investigate the behaviors of the vulnerable foreign exchange options prices in terms of model parameters and the sensitivities of the stochastic volatility factors to the option price.

• The Korean Journal of Applied Statistics
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• v.16 no.2
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• pp.217-224
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• 2003

The volatility in the financial data is usually measured by conditional variance. Two main streams for gauging conditional variance are stochastic volatility (SV) model and autoregressive type approach (GARCH). This article is conducting comparative study between SV and GARCH through the Korean Stock Prices Index (KOSPI) data. It is seen that SV model is slightly better than GARCH(1,1) in analyzing KOSPI data.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.20 no.2
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• pp.123-135
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• 2016

This article deals with the pricing of Asian options under a constant elasticity of variance (CEV) model as well as a stochastic elasticity of variance (SEV) model. The CEV and SEV models are underlying asset price models proposed to overcome shortcomings of the constant volatility model. In particular, the SEV model is attractive because it can characterize the feature of volatility in risky situation such as the global financial crisis both quantitatively and qualitatively. We use an asymptotic expansion method to approximate the no-arbitrage price of an arithmetic average Asian option under both CEV and SEV models. Subsequently, the zero and non-zero constant leverage effects as well as stochastic leverage effects are compared with each other. Lastly, we investigate the SEV correction effects to the CEV model for the price of Asian options.

• 제어로봇시스템학회:학술대회논문집
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• 1991.10b
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• pp.1858-1863
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• 1991

The theory of stock option pricing has, recently, attracted attention of many researchers interested not only in finance but also in statistics and control theory. In this field, the problem of estimating stock return volatility is, above all, of great importance in calculating actual stock option value. In this paper, we assume that the stock market is represented by the stochastic volatility model which is the same as that of Hull and White. Then, we propose an approximation function of option value. It is a type of Black-Sholes option formula in which the first and the second order moments of logarithmic stock value are modified in a special form from the original model. Finally, an algorithm of estimating the parameters of the stochastic volatility model is given, and parameters are estimated by using Nikkei 225 index option data.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.361-388
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• 2023

In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al. [17] and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.

• The Korean Journal of Applied Statistics
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• v.24 no.1
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• pp.83-92
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• 2011

This paper examines empirically Durham's (2008) asset pricing models to the KOSPI200 index. This model Incorporates the VKOSPI index as a proxy for 1 month integrated volatility. This approach uses option prices to back out implied volatility states with an explicitly speci ed risk-neutral measure and risk premia estimated from the data. The application uses daily observations of the KOSPI200 and VKOSPI indices from January 2, 2003 to September 24, 2010. The empirical results show that non-affine model perform better than affine model.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.180-193
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• 2023

Approximating the implied volatilities and estimating the model parameters are important topics in quantitative finance. This study proposes an approximation formula for short-maturity near-the-money implied volatilities in stochastic volatility models. A general second-order nonlinear PDE for implied volatility is derived in terms of time-to-maturity and log-moneyness from the Feyman-Kac formula. Using regularity conditions and the Taylor expansion, an approximation formula for implied volatility is obtained for short-maturity nearthe-money call options in two stochastic volatility models: Heston model and SABR model. In addition, we proposed a novel numerical method to estimate model parameters. This method reduces the number of model parameters that should be estimated. Generating sample data on log-moneyness, time-to-maturity, and implied volatility, we estimate the model parameters fitting the sample data in the above two models. Our method provides parameter estimates that are close to true values.

• East Asian mathematical journal
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• v.33 no.1
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• pp.23-36
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• 2017

We study an optimization problem for HARA utility function under a multi-scale Heston's stochastic volatility model. We investigate a practical strategy that do not depend on the incorporated factor which is unobservable in the market.

• The Korean Journal of Applied Statistics
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• v.35 no.4
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• pp.501-515
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• 2022

Despite the stylized statistical features of returns of financial returns such as fat-tailed distribution and leverage effect, no stochastic volatility models that can explicitly capture these features have been presented in the existing frequentist approach. we propose an approximate parameterization of stochastic volatility models that can explicitly capture the fat-tailed distribution and leverage effect of financial returns and a maximum likelihood estimation of the model using Langrock et al. (2012)'s hidden Markov model approximation in a frequentist approach. Through extensive simulation experiments and an empirical analysis, we present the statistical evidences validating the efficacy and accuracy of proposed parameterization.

• Journal of the Korean Mathematical Society
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• v.57 no.5
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• pp.1167-1186
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• 2020

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possesses the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirm that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps.