First let us import prices of S&P 500 call options available on October 27, 2006.
This data can be stored in a DataFrame.
Extract data from this DataFrame.
Value of the underlying, risk-free rate and dividend yield.
Extract exercise dates and times for which data is available.
Implied volatilities for options maturing in December 2006.
Extract a subset of the DataFrame corresponding to the observations in December 2006:
Compute the implied volatility for these dates and plot the results:
Implied volatilities for options maturing in December 2007.
Extract a subset of the DataFrame corresponding to the observations in December 2007:
Compute the implied volatility for these dates and plot the results:
We will use the following model for the volatility surface.
We can compute the corresponding Black-Scholes price as a function of strike and maturity.
We can use non-linear fitting routines from the statistics data to find the values of that best fit our data. Construct a matrix of parameters and a vector of the corresponding value of the objective function.
Here is the corresponding implied volatility function.
Here is another way to estimate these parameters.
Here is the corresponding implied volatility function.
We can compare both fits with the actual implied volatilities.