”’

from math import log, sqrt
from scipy import stats

S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma * 2 ) T / (sigma sqrt(T))
vega = S0 stats.norm.cdf(d1, 0.0, 1.0) * sqrt(T)
return vega

Implied volatility function

def bsm_call_imp_vol(S0, K, T, r, C0, sigma_est, it= 100 ):
”’ Implied volatility of European call option in BSM model.

            Parameters

            ==========

            S0  :   float

                            initial stock/index level

            K   :   float

                            strike  price

            T   :   float

                            maturity    date    (in year    fractions)

            r   :   float

                            constant    risk-free   short   rate

            sigma_est   :   float

                            estimate    of  impl.   volatility

            it  :   integer

                            number  of  iterations



            Returns

            =======

            simga_est   :   float

                            numerically estimated   implied volatility

            ”’

for i in range(it):
sigma_est -= ((bsm_call_value(S0, K, T, r, sigma_est) - C0)
/ bsm_vega(S0, K, T, r, sigma_est))
return sigma_est

These are only the basic functions needed to calculate implied volatilities. What we need

as well, of course, are the respective option quotes, in our case for European call options

on the VSTOXX index, and the code that generates the single implied volatilities. We will

see how to do this based on an interactive IPython session.

Let us start with the day from which the quotes are taken; i.e., our t = 0 reference day. This

is March 31, 2014. At this day, the closing value of the index was V 0 = 17.6639 (we

change from S to V to indicate that we are now working with the volatility index):

In  [ 1 ]:  V0  =   17.6639

For the risk-free short rate, we assume a value of r = 0.01 p.a.:

In  [ 2 ]:  r   =   0.01

All other input parameters are given by the options data (i.e., T and K) or have to be

calculated (i.e.,

imp

). The data is stored in a pandas DataFrame object (see Chapter 6) and

saved in a PyTables database file (see Chapter 7). We have to read it from disk into

memory:

In  [ 3 ]:  import pandas as pd

h5  =   pd.HDFStore(‘./source/vstoxx_data_31032014.h5’, ‘r’)

futures_data    =   h5[‘futures_data’]      #   VSTOXX  futures data

options_data    =   h5[‘options_data’]      #   VSTOXX  call    option  data

h5.close()

We need the futures data to select a subset of the VSTOXX options given their (forward)

moneyness. Eight futures on the VSTOXX are traded at any time. Their maturities are the