배당을 고려한 옵션과 바스켓옵션의 유사성에 대한 연구 = Dealing a Discrete Dividend with Basket Option Pricing Models

국문 초록 (Abstract)

옵션 가치를 평가할 때 배당을 고려하기 위해, 연속 배당 모형, 에스크로 모형, 선도모형, 그리고 구역별 로그정규분포모형 등 여러 가지 모형들이 개발되었다. 이 중에서 구역별 로그정규분포 모형이 가장 일관성 있게 현실을 반영한 것으로 여겨지고 있으나, 정확한 닫힌 꼴 공식은 개발되지 못했다. 이에 따라, 이 모형 아래에서의 옵션가격의 근사값에 대한 연구가 계속되고 있다. 이러한 가운데, 본 연구는 배당을 고려한 유럽형옵션이 바스켓옵션, 그중에서도 특히 스프레드옵션, 과 유사함을 밝힌다. 배당을 고려한 기존의 옵션 공식을 비교한 결과, Etore and Gobet(2012)가 가장 정확한 값을 제공하는 것으로 나타났다. 그런데 본 연구의 수치분석에 따르면, Li et al.(2008)의 스프레드옵션공식을 활용하여 배당을 고려해도 Etore and Gobet(2012)와 유사한 수준의 정확성을 나타낸다. 이러한 수치 분석결과는 향후에 두 상품의 연구가 함께 발전할 수 있음을 보여준다.

다국어 초록 (Multilingual Abstract)

For pricing options with discrete dividends, a continuous dividend model, an escrow model, a forward model, and a piecewise lognormal model have been developed. Among the models, the piecewise lognormal model is regarded as the most realistic with consistency. However, the model has no analytic closed form solution and therefore approximation analytic closed form solutions have been studied. Here, we show the similarity between European spread options, which are special cases of European basket options, and European options with a constant dividend. Our numerical study shows that an approximation of Etore and Gobet(2012) is the most accurate among option pricing formulas with the piecewise lognormal model. However, an approximation from spread option pricing formula of Li et al.(2008) is comparable to the approximation of Etore and Gobet(2012). These results show that an option pricing formula with a discrete dividend can be improved when we have an improved spread option pricing formula.

참고문헌 (Reference)

1 Posner, S., "Valuing exotic options by approximating the spd with higher moments" 7-, 1998

2 Merton, R. C., "Theory of rational option pricing" 141-183, 1973

3 Margrabe, W., "The value of an option to exchange one asset for another" 177-186, 1978

4 Black, F., "The pricing of options and corporate liabilities" 81 : 637-654, 1973

5 Bos, R., "Stock options dealing with discrete dividends" 16 : 109-112, 2003

6 Etoré, P., "Stochastic expansion for the pricing of call options with discrete dividends" 19 : 233-264, 2012

7 Levy, E., "Pricing european average rate currency options" 11 : 474-491, 1992

8 Ju, N., "Pricing asian and basket options via taylor expansion" 5 : 79-103, 2002

9 Hull, J. C., "Options, futures, and other derivatives" Pearson Education 2006

10 Lo, C.-L., "Moment-matching approximations for asian options" 21 : 103-122, 2014

11 Sahel, F., "Matching sensitivities to discrete dividends: A new approach for pricing vanillas" 2011 : 80-85, 2011

12 Bos, M., "Finessing fixed dividends" 15 : 157-158, 2002

13 Black, F., "Fact and fantasy in the use of options" 36-72, 1975

14 Vellekoop, M., "Efficient pricing of derivatives on assets with discrete dividends" 13 : 265-284, 2006

15 Kirk, E., "Correlation in the energy markets" 71-78, 1995

16 Li, M., "Closed-form approximations for spread option prices and greeks" 15 : 58-80, 2008

17 Veiga, C., "Closed formula for options with discrete dividends and its derivatives" 16 : 517-531, 2009

18 Bjerksund, P., "Closed form spread option valuation" 14 : 1785-1794, 2014

19 Haug, E.G., "Back to basics: A new approach to the discrete dividend problem" 9 : 37-47, 2003

20 Milevsky, M.A., "Asian options, the sum of lognormals, and the reciprocal gamma distribution" 33 : 409-422, 1998

21 Dai, T.-S., "Accurate approximation formulas for stock options with discrete dividends" 16 : 1657-1663, 2009

22 Milevsky, M.A., "A closed-form approximation for valuing basket options" 5 : 54-61, 1998

23 Borovkova, S., "A closed form approach to the valuation and hedging of basket and spread option" 14 : 8-24, 2007