The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Numerical studies on approximate option prices

근사적 옵션 가격의 수치적 비교

  • Received : 2017.01.20
  • Accepted : 2017.03.15
  • Published : 2017.04.30
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Abstract

In this paper, we compare several methods to approximate option prices: Edgeworth expansion, A-type and C-type Gram-Charlier expansions, a method using normal inverse gaussian (NIG) distribution, and an asymptotic method using nonlinear regression. We used two different types of approximation. The first (called the RNM method) approximates the risk neutral probability density function of the log return of the underlying asset and computes the option price. The second (called the OPTIM method) finds the approximate option pricing formula and then estimates parameters to compute the option price. For simulation experiments, we generated underlying asset data from the Heston model and NIG model, a well-known stochastic volatility model and a well-known Levy model, respectively. We also applied the above approximating methods to the KOSPI200 call option price as a real data application. We then found that the OPTIM method shows better performance on average than the RNM method. Among the OPTIM, A-type Gram-Charlier expansion and the asymptotic method that uses nonlinear regression showed relatively better performance; in addition, among RNM, the method of using NIG distribution was relatively better than others.

본 논문에서는 옵션의 가격을 결정하기 위해 사용될 수 있는 몇 가지 근사적인 방법들을 수치적으로 비교하였다. 헤르미트 다항식 계열의 Edgeworth 확장과 A-type Gram-Charlier 방법, C-type Gram-Charlier 방법, normal inverse gaussian (NIG) 분포를 이용하는 방법, 그리고 비선형 회귀를 이용한 점근적 근사방법이 그것이다. 이 방법들을 위험중립 확률측도 하에서 수익률의 분포함수를 근사하여 옵션가격을 계산하는 방식과 옵션의 근사가격식을 먼저 구하고 모수를 추정하여 가격을 계산하는 두 가지 방식을 사용하여 비교하였다. 모의실험에서는 확률변동성 모형에서 많이 사용되는 Heston 모형과 레비확률과정에서 좋은 적합도를 보이는 NIG 모형을 이용하여 자료를 생성하였고, 실제 자료로는 KOSPI200 콜옵션을 이용하였다. 모의실험과 실제 자료분석의 결과, 근사적 가격식을 먼저 구하는 방식이 좀 더 우수한 성능을 보였고 그 가운데 A-type Gram-Charlier와 비선형 회귀를 이용한 점근적 근사방법이 좋은 성능을 보였으며, 분포함수를 추정하여 옵션가격을 계산하는 경우 NIG분포를 이용하는 것이 상대적으로 좋은 결과를 보였다.

Keywords

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