내재변동성 계산의 수치적 해법 = Numerical Methods for the Implied Volatility

다국어 초록 (Multilingual Abstract)

The factors that determine value of options are price of underlying assets, their exercise prices, risk-free interest rate, remaining period until the time of expiration, and volatility of prices of underlying assets. All the factors can be observed in the market except for volatility. Generally historical volatility which is statistically calculated from the data of past underlying assets is used to the option pricing models. The other way to measure volatility is to put transaction prices of options in current market into the Black-Scholes Formulas, and use the outcome of calculated volatility, which is called implied volatility.
Black-Scholes option pricing model is applied on the premise that the price of underlying assets, hypothetically, falls into regular volatility of normal distribution. In the real market data, the implied volatility measured by the market price of option can be different from the actual outcome of implied volatility. In 1973, the Chicago Board of Option Exchange has started there used to be a phenomenon called volatility smile transaction but after the great fall of stock market in 1987, the implied volatility parameterized by exercise price shown on a graph was shaped more of frowning rather than smiling since the curve is downshifted toward the right.
In this thesis, by using Black-Scholes option pricing model, numerical methods for implied volatility has been materialized, and each efficiency of the outcomes from the solution were analyzed. Bisection, false position, Newton, and secant methods were used to calculated the volatility of options numerically. Newton method showed the highest in efficiency among those solutions. Furthermore, it was also studied to see on a graph whether the implied volatility according to their exercise price was the smile type or the frown one. As the result, the graph of implied volatility based on the exercise price of call options and put options in KOSPI 200 at the end of November, 2007 was contoured to be rather frown shape which was curved downward toward the right.

목차 (Table of Contents)

• Ⅰ. 서론 = 1
• Ⅱ. Black-Scholes 방정식 = 3
• 1. 옵션 = 3
• 2. Black-Scholes 옵션가격결정모형 = 9
• 3. Black-Scholes 공식 = 17
• 4. 변동성(volatility) = 18
• Ⅲ. 방정식의 해를 구하는 수치적 해법 = 20
• 1. 이분법(Bisection Method) = 20
• 2. 가위치법(False Position Method) = 23
• 3. Newton 방법 = 25
• 4. 할선법(Secant Method) = 30
• Ⅳ. 내재변동성 구하기 = 33
• 1. Newton 방법을 이용한 내재변동성 계산 = 33
• 2. KOSPI 200 옵션 2007년 11월물을 이용한 내재변동성 계산 = 38
• 참고문헌 = 50
• ABSTRACT = 51