338 THE HANDBOOK OF PORTFOLIO MATHEMATICS
54% disinvested. Fifty-four percent of 50 units is 27 units. Therefore, at the
present price level of the fund at this point in time, for the given interest
rate and volatility levels, the fund should be short 27 S&P units along with
its long position in cash stocks.
Because the delta needs to be recomputed on an ongoing basis, and
portfolio adjustments must be constantly monitored, the strategy is called
a dynamic hedging strategy.
One problem with using futures in the strategy is that the futures mar-
ket does not exactly track the cash market. Further, the portfolio you are
selling futures against may not exactly follow the cash index upon which
the futures market is traded. These tracking errors can add to the expense
of a portfolio insurance program. Furthermore, when the option being repli-
cated gets very near to expiration, and the portfolio value is near the strike
price, the gamma of the replicated option goes up astronomically. Gamma is
the instantaneous rate of change of the delta or hedge ratio. In other words,
gamma is the delta of the delta. If the delta is changing very rapidly (i.e., if
the replicated option has a high gamma), portfolio insurance becomes in-
creasingly more cumbersome to perform. There are numerous ways to work
around this problem, some of which are very sophisticated. One of the sim-
plest involves the concept of a perpetual option. For instance, you can
always assume that the option you are trying to replicate expires in, say,
three months. Each day you will move the replicated option’s expiration
date ahead by a day. Again, this high gamma usually becomes a problem
only when expiration draws near and the portfolio value and the replicated
option’s strike price are very close.
There is a very interesting relationship between optimalfand portfolio
insurance. When you enter a position, you can state thatfpercent of your
funds are invested. For example, consider a gambling game where your
optimalf is .5, biggest loss−1, and bankroll is $10,000. In such a case,
you would bet one dollar for every two dollars in your stake since−1, the
biggest loss, divided by−.5, the negative optimalf, is 2. Dividing $10,000 by
2 yields $5,000. You would, therefore, bet $5,000 on the next bet, which is
fpercent (50%) of your bankroll. Had we multiplied our bankroll of $10,000
byf(.5), we would have arrived at the same $5,000 result. Hence, we have
betfpercent of our bankroll.
Likewise, if our biggest loss were $250 and everything else the same,
we would be making one bet for every $500 in our bankroll since−$250/−.5
=$500. Dividing $10,000 by $500 means that we would make twenty bets.
Since the most we can lose on any one bet is $250, we have thus riskedf
percent, 50% of our stake in risking $5,000 ($250∗20).
Therefore, we can state that fequals the percentage of our funds at
risk, orf equals the hedge ratio. Remember, when discussing portfolios,