Model Calibration
The next important step is the calibration of the financial model used to value the
VSTOXX options to available market data. For an in-depth discussion of this topic and
example code in Python see Hilpisch (2015), in particular Chapter 11.
Relevant Market Data
The first step when calibrating a model is to decide on the relevant market data to be used.
For the example, let us assume the following:
Pricing date shall be 31 March 2014.
Option maturity shall be October 2014.
The following Python code defines the pricing_date and maturity, reads the
initial_value for the VSTOXX from the respective DataFrame object, and also reads the
corresponding value forward for the VSTOXX future with the appropriate maturity.
In [ 24 ]: pricing_date = dt.datetime( 2014 , 3 , 31 )
# last trading day in March 2014
maturity = third_fridays[ 10 ]
# October maturity
initial_value = vstoxx_index[‘V2TX’][pricing_date]
# VSTOXX on pricing_date
forward = vstoxx_futures[(vstoxx_futures.DATE == pricing_date)
& (vstoxx_futures.MATURITY == maturity)][‘PRICE’].values[ 0 ]
Out of the many options quotes in the data set, we take only those that are:
From the pricing date
For the right maturity date
For call options that are less than 20% out-of-the-money or in-the-money
We therefore have:
In [ 25 ]: tol = 0.20
option_selection = \
vstoxx_options[(vstoxx_options.DATE == pricing_date)
& (vstoxx_options.MATURITY == maturity)
& (vstoxx_options.TYPE == ‘C’)
& (vstoxx_options.STRIKE > ( 1 - tol) * forward)
& (vstoxx_options.STRIKE < ( 1 + tol) * forward)]
This leaves the following option quotes for the calibration procedure:
In [ 26 ]: option_selection
Out[26]: DATE EXP_YEAR EXP_MONTH TYPE STRIKE PRICE MATURITY
46482 2014-03-31 2014 10 C 17 4.85 2014-10-17
46483 2014-03-31 2014 10 C 18 4.30 2014-10-17
46484 2014-03-31 2014 10 C 19 3.80 2014-10-17
46485 2014-03-31 2014 10 C 20 3.40 2014-10-17
46486 2014-03-31 2014 10 C 21 3.05 2014-10-17
46487 2014-03-31 2014 10 C 22 2.75 2014-10-17
46488 2014-03-31 2014 10 C 23 2.50 2014-10-17
46489 2014-03-31 2014 10 C 24 2.25 2014-10-17
46490 2014-03-31 2014 10 C 25 2.10 2014-10-17
Option Modeling
For the calibration of the square_root_diffusion model, the options selected before have
