Ralph Vince - Portfolio Mathematics

340 THE HANDBOOK OF PORTFOLIO MATHEMATICS

upon an expiration date, volatility level, and other input parameters to the
particular option model you intend to use. These inputs will give you the
option’s delta at any given point in time. Once the delta is known, you can
determine what your active equity should be. Since the delta for the account,
the variableHin Equation (10.06a), must equal the delta for the call option
being replicated:

H=

(n

∑

i= 1

fi

)

∗

active$

total equity

Therefore:

H

∑n

i= 1

fi

=

active$

total equity

ifH<

∑n

i= 1

fi (10.07)

Otherwise:

H=

active$

total equity

= 1

Since active$/total equity is equal to the percentage of active equity, we
can state that the percentage of funds we should have in active equity, of
the total account equity, is equal to the delta on the call option divided by
the sum of thefvalues of the components. However, you will note that if
His greater than the sum of thesefvalues, then it is suggesting that you
allocate greater than 100% of an account’s equity as active. Since this is not
possible, there is an upper limit of 100% of the account’s equity that can be
used as active equity.
Portfolio insurance is great in theory, but poor in practice. As witnessed
in the 1987 stock market crash, the problem with portfolio insurance is
that, when prices plunge, there isn’t any liquidity at any price. This does not
concern us here, however, since we are looking at the relationship between
active and inactive equity, and how this is mathematically similar to portfolio
insurance.
The problem with implementing portfolio insurance as a realloca-
tion technique, as detailed here, is that reallocation is taking place con-
stantly. This detracts from the fact that a dynamic fractionalfstrategy
will asymptotically dominate a static fractionalfstrategy. As a result,
trying to steer performance by way of portfolio insurance as a dynamic
fractional f reallocation strategy probably isn’t such a good idea. How-
ever, anytime you use fractional f, static or dynamic, you are employing
a form of portfolio insurance.