Abstract:

This dissertation consists of three essays. The first essay focuses on implied volatility estimation. A refined approach to estimating the implied volatility is introduced for options priced in the classic framework developed by Black and Scholes(1973) and Merton(1973). It extends the formula developed in Corrado and Miller's paper(1996, 2004) in which the formula works well for the Index options with the present value of strike price close to index price. The refined approach provides more accurate implied volatility estimates over a wider range of moneyness.

The second essay focuses on the well known volatility smile phenomenon. Numerous explanations for the volatility smile/skew phenomenon and extensions of or alternatives to the Black-Scholes model have been offered in the literature. The inconsistence between the alternative models and data makes it worthwhile to re-think the well-known Black-Scholes model. In this essay, I provide a new perspective of the volatility smile/skew and non-flat term structure phenomenon through carefully studying the mathematical structure of the Black-Scholes formula. Though it has been conjectured that the presence of measurement error can be of substantial impact on volatility estimation, I show rigorously the level of sensitivity of price error to the volatility estimation for the options at different strikes and maturities. In the presence of price error, my results show that the degree of the bias of the implied volatility obtained from the option market price is directly related to the option's moneyness and maturity and such bias can be minimized using option with appropriate strike and maturity from a mathematical point of view. I conclude that observed volatility smile/skew or non-flat term structure phenomenon does not necessarily violate the log-normal return and constant volatility assumption required by the Black-Scholes model in the presence of price error.

In the third essay, the focus is on the volatility forecasting--examining the informational content of the implied volatility, and more specifically, answering the question: which implied volatility provide the best measure of the future volatility? The volatility smile/skew phenomenon makes it unclear which implied volatility provides the best measure of the market volatility expectation over the remaining life of the options. Due to its liquidity and less sensitivity to price error, at-the-money implied volatility is often considered a good measure of future volatility. In this essay, I compare the predictive power of the implied volatilities from at-the-money option and option with the highest vega(defined as optimal option), and show that the advantage of using optimal option can increase as forecasting horizon increases.