90. Implied Volatility: Theory and Empirical Method

The estimation of the implied volatility is the one of most important topics in option pricing research. The main purpose of this chapter is to review the different theoretical methods used to estimate implied standard deviation and to show how the implied volatility can be estimated in empirical work. The OLS method for estimating implied standard deviation is first introduced, and the formulas derived by applying a Taylor series expansion method to Black–Scholes option pricing model are also described. Three approaches of estimating implied volatility are derived from one, two, and three options, respectively. Regarding to these formulas with the remainder terms, the accuracy of these formulas depends on how an underlying asset is close to the present value of exercise price in an option. The formula utilizing three options for estimating implied volatility is more accurate rather than other two approaches.

In empirical work, we use call options on S&P 500 index futures in 2010 and 2011 to illustrate how MATLAB can be used to deal with the issue of convergence in estimating implied volatility of future options. The results show that the time series of implied volatility significantly violates the assumption of constant volatility in Black–Scholes option pricing model. The volatility parameter in the option pricing model fluctuates over time and therefore should be estimated by the time series and cross-sectional model.