The Geometry of Leverage Space Portfolios 337
expiration date for this hypothetical option which you are going to create.
Let’s say that the expiration date you choose is the date on which this quarter
ends.
Now, you will figure the delta (the instantaneous rate of change in the
price of a call option relative to the change in price of the underlying instru-
ment) for this 100 call option with the chosen expiration date. Suppose the
delta is .5. This means that you should be 50% invested in the given stock.
Thus, you would have only 50 shares of stock rather than the 100 shares
you would have if you were not practicing portfolio insurance. As the value
of the stock increases, so, too, will the delta, and likewise the number of
shares you hold. The upside limit is a delta at 1, where you would be 100%
invested. In our example, at a delta of 1, you would have 100 shares.
As the stock decreases, so, too, does the delta, and likewise the size of
the position in the stock decreases. The downside limit is at a delta of 0,
where you wouldn’t have any position in the stock.
Operationally, stock fund managers have usednoninvasive methodsof
dynamic hedging. Such a technique involves not having to trade the cash
portfolio. Rather, the portfolio as a whole is adjusted to what the current
delta should be, as dictated by the model by using stock index futures, and,
sometimes, put options. One benefit of a technique using futures is that
futures have low transactions cost.
Selling short futures against the portfolio is equivalent to selling off part
of the portfolio and putting it into cash. As the portfolio falls, more futures
are sold, and as it rises, these short positions are covered. The loss to the
portfolio, as it goes up and the short futures positions are covered, is what
accounts for the portfolio insurance cost, the cost of the replicated put op-
tions. Dynamic hedging, though, has the benefit of allowing us to closely
estimate this cost at the outset. To managers trying to implement such a
strategy, it allows the portfolio to remain untouched, while the appropriate
asset allocation shifts are performed through futures trades. This noninva-
sive technique of using futures permits the separation of asset allocation
and active portfolio management.
To someone implementing portfolio insurance, you must continuously
adjust the portfolio to the appropriate delta. This means that, say, each
day, you must input into the option pricing model the current portfolio
value, time until expiration, interest rate levels, and portfolio volatility, to
determine the delta of the put option you are trying to replicate. Adding
this delta (which is a number between 0 and−1) to 1 will give you the
corresponding call’s delta. This is the hedge ratio, the percentage that you
should be investing in the fund.
Suppose your hedge ratio for the present moment is .46. Let’s say that
the size of the fund you are managing is the equivalent of 50 S&P futures
units. Since you want to be only 46% invested, it means you want to be