Comparing Implied Volatility of B-S Model and Dupire Style Local Volatility in Option Pricing

This paper mainly studies the relationships between the implied volatilities inferred by the BlackScholes model and volatilities derived by the Dupire style local volatility model. By studying the difference between the implied volatility by B-S model and Dupire style local volatilities, we can search for models that have similar volatilities to those of Black-Scholes models, and find the more realistic volatility and also applies plausible price process that do not depend on a constant volatility, unlike the ones of the Black-Scholes models. Also Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) attribute this behavior to the fact that the Black-Scholes constant volatility assumption is violated in real situation. We derive Dupire style local volatility in call price and Dupire style local volatility in terms of implied volatility. Then we compare implied volatility from B-S option pricing, Dupire style local volatility which is derived local volatility in call price and local volatility in terms of implied volatility by simulation. Our results suggest that Dupire sytle local volatility can be considered as alternative volatility to normal volatility or implied volatility in pricing option.