The following figure finally shows the full frequency distribution of the portfolio present
values. You can clearly see in Figure 18-4 the offsetting diversification effects of
combining a call with a put option:
In [ 39 ]: pvs = pv1 + pv2
plt.hist(pvs, bins= 50 , label=‘portfolio’);
plt.axvline(pvs.mean(), color=‘r’, ls=‘dashed’,
lw=1.5, label=‘mean = %4.2f’ % pvs.mean())
plt.xlim( 0 , 80 ); plt.ylim( 0 , 7000 )
plt.grid(); plt.legend()
Figure 18-4. Portfolio frequency distribution of present values
What impact does the correlation between the two risk factors have on the risk of the
portfolio, measured in the standard deviation of the present values? The statistics for the
portfolio with correlation are easily calculated as follows:
In [ 40 ]: # portfolio with correlation
pvs.std()
Out[40]: 16.736290069957963
Similarly, for the portfolio without correlation, we have:
In [ 41 ]: # portfolio without correlation
pv1 = 5 * portfolio.valuation_objects[‘eur_call_pos’].\
present_value(full=True)[ 1 ]
pv2 = 3 * portfolio.valuation_objects[‘am_put_pos’].\
present_value(full=True)[ 1 ]
(pv1 + pv2).std()
Out[41]: 21.71542409437863
Although the mean value stays constant (ignoring numerical deviations), correlation
obviously significantly decreases the portfolio risk when measured in this way. Again, this
is an insight that it is not really possible to gain when using alternative numerical methods
or valuation approaches.
