Three Essays on Local Volatility Models

Abstract

In financial engineering, the Black-Scholes model is the most popular and basic model for pricing derivatives. A lot of research papers, however, show that the Black-Scholes model fails to explain features of the implied volatility surface such as volatility smile and the volatility skew. Local volatility models, introduced in 1994 by Dupire and Derman and Kani, treat volatility of underlying asset as a deterministic function of asset level and time. Implied volatility represents the market's expectation while local volatility represents average over all possible smile-consistent instantaneous volatilities in a stochastic volatility world. The advantages of using local volatility model are that they are smile-consistent one factor markovian model and they make possible to price exotic options consistently with the known market prices. They are also relatively easy to estimate compare to stochastic volatility models.We can recover local volatility surface from implied volatility surface using Dupire formula. There are some challenges in modelling the implied volatility surfaces: They must be arbitrage free. And they must have functional flexibility but not over-fitting. In addition, it can be helpful, if confidence interval information on the estimated volatility surfaces is provided. For these purposes, local smoothing method manipulating the bandwidth for estimating implied volatility surfaces is proposed in this thesis. Also, estimation method for local volatility model with jumps is provided. Additional jump term makes calibration difficult. Calibrating with reasonable starting points can solve this problem at least in some part.To verify the performance of the proposed methods, we conduct simulations on both model-generated option prices data and real market option prices data. Using local smoothing with no-arbitrage constraints, we estimate implied volatility and local volatility. Then we value the KOSPI 200 index options, ELW, and barrier options. The estimation and prediction results show that local volatility function recovered from the proposed method improves the pricing performance compared with volatility surfaces from other existing estimation methods. Also we show that proposed calibration method for local volatility jump diffusion model works well in estimation and prediction.