• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제27권3호
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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.

• 응용통계연구
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• 제35권1호
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• pp.93-104
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• 2022

S&P 500과 RUSSELL 2000, DJIA, Nasdaq 100 4가지 미국 주가지수의 실현변동성(realized volatility, RV)을 예측하는데 있어서 사람들의 관심 지표로 삼을 수 있는 인터넷 검색량(search volume, SV) 지수와 내재변동성(implied volatility, IV)를 이용하여 LSTM 딥러닝(deep learning) 방법으로 RV의 예측력을 높이고자하였다. SV을 이용한 LSTM 방법의 실현변동성 예측력이 기존의 기본적인 vector autoregressive (VAR) 모형, vector error correction (VEC)보다 우수하였다. 또한, 최근 제안된 RV와 IV의 공적분 관계를 이용한 vector error correction heterogeneous autoregressive (VECHAR) 모형보다도 전반적으로 예측력이 더 높음을 확인하였다.

• 재무관리연구
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• 제22권2호
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• pp.91-105
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• 2005

본 연구는 합리적 기대가설에 입각하여 내재변동성의 장기변화율과 단기변화율사이의 관계에 대해 Campa and Chang(1995)의 모형을 토대로 분석하였다. 합리적기대가설이란 옵션시장 참가자들이 기대하는 장기내재변동성이 미래단기의 내재변동성을 결정한다는 가설이다. 1999년 1월부터 2003년 12월까지 한국의 KOSPI 200 주가지수옵션시장에서 장기내재변동성과 단기내 재변동성과의 관계를 분석해 본 결과 우리나라 주가지수 옵션시장에서는 기대가설을 지지하지 않는다는 결론을 도출하게 되었다. 하지만 기대가설은 그 자체가 내포하고 있는 무시할 수 없는 중요한 의미를 지닌다는 것을 발견하였다. 첫째, 장기내재변동성 변화율이 기대 미래단기내재변동성 변화율과 같은 방향으로 움직인다는 것과 둘째, 단기변동성에 대한 장기변동성의 증가가 다음 장기변동성에 있어서는 증가 혹은 감소를 동반한다는 것을 분석하게 되었다. 즉, 어떠한 규칙적인 사이클을 찾을 수는 없었다는 것이다.

• 재무관리논총
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• 제3권1호
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• pp.213-231
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• 1996

In this study we conduct a specification test of at-the-money option volatility. Results show that the implied volatility estimate recovered from the Black-Scholes European option pricing model is nearly indistinguishable from the implied volatility estimate obtained from the Barone-Adesi and Whaley's American option pricing model. This study also investigates whether the use of Black-Scholes implied volatility estimates in American put pricing model significantly affect the prediction the prediction of American put option prices. Results show that, at long as the possibility of early exercise is carefully controlled in calculation of implied volatilities prediction of American put prices is not significantly distorted. This suggests that at-the-money option implied volatility estimates are robust across option pricing model.

• Management Science and Financial Engineering
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• 제20권1호
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• pp.21-26
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• 2014

A lot of researches have been conducted to estimate the volatility smile effect shown in the option market. This paper proposes a method to approximate an implied volatility function, given noisy real market option data. To construct an implied volatility function, we use Gaussian Processes (GPs). Their output values are implied volatilities while moneyness values (the ratios of strike price to underlying asset price) and time to maturities are as their input values. To show the performances of our proposed method, we conduct experimental simulations with Korean Equity-Linked Warrant (ELW) market data as well as toy data.

• 한국수학사학회지
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• 제35권1호
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• pp.1-18
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• 2022

This paper presents a statistical machine learning method that generates the implied volatility surface under the rareness of the market data. We apply the practitioner's Black-Scholes model and Gaussian process regression method to construct a Bayesian inference system with observed volatilities as a prior information and estimate the posterior distribution of the unobserved volatilities. The variance instead of the volatility is the target of the estimation, and the radial basis function is applied to the mean and kernel function of the Gaussian process regression. We present two types of Gaussian process regression methods and empirically analyze them.

• 대한수학회지
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• 제43권4호
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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.

• 한국경영과학회지
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• 제39권4호
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• pp.137-152
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• 2014

In this paper, I suggest several techniques to estimate covariance matrix and compare the performance of the global minimum variance portfolio (GMVP) in terms of out of sample mean standard deviation and return. As a result, the return differences among the GMVPs are insignificant. The mean standard deviation of the GMVP using historical covariance is sensitive to the estimation window and the number of assets in the portfolio. Among the model covariance, the GMVP using constant systematic risk ratio model or using short sale restriction shows the best performance. The performance difference between the GMVPs using historical covariance and model covariance becomes insignificant as the historical covariance is estimated with longer estimation window. Lastly, the implied volatilities from ELW prices do not lead to superior performance to the historical variance.

• Journal of the Korean Data and Information Science Society
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• 제25권3호
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• pp.513-522
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• 2014

옵션가격의 결정에 있어서 실제 변동성은 사후에 알 수 있는 정보이므로 대용값으로 내재변동성을 가장 많이 사용하는데 본 연구에서는 동일한 기초자산을 가진 옵션의 잔존만기와 행사가격을 이용하여 내재변동성을 추정하고자 한다. KOSPI200 옵션 데이터와 서포트벡터회귀, 나무모형 및 회귀모형을 통해 모형의 설명력을 평균제곱근오차 (RMSE)와 평균절대오차 (MAE)를 사용하여 살펴보았다. 그 결과 서포트벡터회귀와 MART의 성능이 최소제곱회귀보다 우수한 것으로 나타났으며, 서포트벡터회귀와 MART의 성능은 거의 비슷하였다.

• 유통과학연구
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• 제13권5호
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• pp.53-60
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• 2015

Purpose - The purpose of this paper is to shed light on the importance of the slope and curvature of the volatility curve implied in option prices in the KOSPI 200 options index. A number of studies examine the implied volatility curve, however, these usually focus on cross-sectional characteristics such as the volatility smile. Contrary to previous studies, we focus on time-series characteristics; we investigate correlation dynamics among slope, curvature, and level of the implied volatility curve to capture market information embodied therein. Our study may provide useful implications for investors to utilize current market expectations in managing portfolios dynamically and efficiently. Research design, data, and methodology - For our empirical purpose, we gathered daily KOSPI200 index option prices executed at 2:50 pm in the Korean Exchange distribution market during the period of January 2, 2004 and January 31, 2012. In order to measure slope and curvature of the volatility curve, we use approximated delta distance; the slope is defined as the difference of implied volatilities between 15 delta call options and 15 delta put options; the curvature is defined as the difference between out-of-the-money (OTM) options and at-the-money (ATM) options. We use generalized method of moments (GMM) and the seemingly unrelated regression (SUR) method to verify correlations among level, slope, and curvature of the implied volatility curve with statistical support. Results - We find that slope as well as curvature is positively correlated with volatility level, implying that put option prices increase in a downward market. Further, we find that curvature and slope are positively correlated; however, the relation is weakened at deep moneyness. The results lead us to examine whether slope decreases monotonically as the delta increases, and it is verified with statistical significance that the deeper the moneyness, the lower the slope. It enables us to infer that when volatility surges above a certain level due to any tail risk, investors would rather take long positions in OTM call options, expecting market recovery in the near future. Conclusions - Our results are the evidence of the investor's increasing hedging demand for put options when downside market risks are expected. Adding to this, the slope and curvature of the volatility curve may provide important information regarding the timing of market recovery from a nosedive. For financial product distributors, using the dynamic relation among the three key indicators of the implied volatility curve might be helpful in enhancing profit and gaining trust and loyalty. However, it should be noted that our implications are limited since we do not provide rigorous evidence for the predictability power of volatility curves. Meaning, we need to verify whether the slope and curvature of the volatility curve have statistical significance in predicting the market trough. As one of the verifications, for instance, the performance of trading strategy based on information of slope and curvature could be tested. We reserve this for the future research.