relevant European call options. They only differentiate themselves by the relevant strike
price; everything else in the market environment is the same. We store the single valuation
objects in a dict object. As keys for the dict object, we take the index values of the
option quotes in the DataFrame object option_selection for unique identification:
In [ 40 ]: option_models = {}
for option in option_selection.index:
strike = option_selection[‘STRIKE’].ix[option]
me_vstoxx.add_constant(‘strike’, strike)
option_models[option] = \
valuation_mcs_european(
‘eur_call_%d’ % strike,
vstoxx_model,
me_vstoxx,
payoff_func)
A single step in the calibration routine makes the updating of all valuation objects and a
revaluation of all options necessary. For convenience, we put this functionality into a
separate function:
In [ 41 ]: def calculate_model_values(p0):
”’ Returns all relevant option values.
Parameters
===========
p0 : tuple/list
tuple of kappa, theta, volatility
Returns
=======
model_values : dict
dictionary with model values
”’
kappa, theta, volatility = p0
vstoxx_model.update(kappa=kappa,
theta=theta,
volatility=volatility)
model_values = {}
for option in option_models:
model_values[option] = \
option_models[option].present_value(fixed_seed=True)
return model_values
Providing a parameter tuple of kappa, theta, and volatility to the function
calculate_model_values gives back, ceteris paribus, model option values for all relevant
options:
In [ 42 ]: calculate_model_values((0.5, 27.5, vol_est))
Out[42]: {46482: 3.206401,
46483: 2.412354,
46484: 1.731028,
46485: 1.178823,
46486: 0.760421,
46487: 0.46249,
46488: 0.263662,
46489: 0.142177,
46490: 0.07219}
Calibration Procedure
Calibration of an option pricing model is, in general, a convex optimization problem. The
most widely used function used for the calibration — i.e., the minimization — is the
mean-squared error (MSE) for the model option values given the market quotes of the
options. Assume there are N relevant options, and also model and market quotes. The
problem of calibrating a financial model to the market quotes based on the MSE is then
given in Equation 19-1. There, and are the market price and the model price of the
