As we have already conjectured in the beginning, options can be used in
order to provide an insurance towards adverse movements of the market. If
one manages a well diversified portfolio that mimics the index [with beta
equal to one and dividend yield equal to the market dividend yield], then
the procedure is straightforward. However, sometimes the traded options
required are either not available or not sufficiently liquid. In such cases
one can create options synthetically. There are two ways in achieving this.
Using the put-call parity relationship. This stems from the fact that
![$\displaystyle S_{T}-K=\max \left[ S_{T}-K,0\right] -\max \left[ K-S_{T},0\right]$](DOLimg1243.gif)
,
suggesting that under risk neutrality
![$\displaystyle \mathbf{E}_{Q}\left[ S_{T}-K\right] =\mathbf{E}_{Q}\max \left[ S_{T}-K,0\right] -\mathbf{E}_{Q}\max \left[ K-S_{T},0\right]$](DOLimg1244.gif)
.
In addition, under risk neutrality the stock price dynamics [with
the dividend yield] are given by the following SDE
![$\displaystyle \frac{dS\left( t\right) }{S\left( t\right) }=\left[ r\left( t\right) -\delta
\left( t\right) \right] dt+\sigma \left( t\right) dB\left( t\right)$](DOLimg1245.gif)
,
giving that
. We also
have that
The above expressions imply the put-call parity relationship
This relationship allows one to replicate a call [or put] by using the
corresponding put [or call] and the money market account.
Another approach would be to construct an option completely synthetically,
using the delta hedging methodology and the money market alone. In this
context one wants to create a position which will match the delta of any
given option. A put option on the portfolio, for example, has delta equal to
where now
is the strike price [for the insurance] and
is the volatility of the portfolio [which is beta time the market
volatility, although in this case
]. Remember that the BS
methodology suggests that a portfolio composed by
![$\displaystyle \left.
\begin{tabular}{l}
$-1$ derivative [with price $P$] \\
$+\Delta _{P}$ shares of the underlying
\end{tabular}\right\} \Rightarrow$](DOLimg1257.gif)
gives the risk free rate of return
Apparently, in order to replicate a derivative, one has to take the inverse
position to the one that hedges it. Therefore, by employing a dynamic
strategy where a fraction
of the original portfolio is always
invested in riskless assets, the fund manager ensures that the overall
position will match the given delta.
The above are illustrated in the following
Example 45 (Portfolio insurance I)
Say that a fund manager has a well-diversified portfolio, which mirrors the
FTSE250 index [

]. The value of the portfolio is

, whereas the value of the index is

pts. The
manager wants to insure that the portfolio she is holding will not decline
by more than

over a six month period. Say that the risk free rate is

p.a., and the dividend yield for both
FTSE250 and the
portfolio is

p.a. The volatility of the
index is

p.a. Since

and the dividend
yields are the same, if the portfolio looses

of its value, it is
implied that the
FTSE250 index has lost

of its value too.
Therefore the fund manager wants to insure against the index falling below

pts. The strategy that he should employ would be buying six
month put options with strike price

. Since today the
value of the portfolio is

times the index, she
should buy

such options.
Now we turn into examining what the price of such an insurance would be.
Example 46 (cont. The value of the insurance I)
Suppose that the BS formula is correct. The value of a put option with the
above characteristics would be given by
implying that

. Since the fund manager wants to buy

of these contracts, the value of the insurance is

.
Put options can be replicated using calls and the put call parity. In this
setting
Example 47 (cont. Using calls I)
The fund manager can also create a synthetic put, using call option and the
put call parity
This strategy would include
This strategy would have the same value as the one involved put options.
A delta hedging strategy implies that in fact the put option can be
approximated by borrowing or lending part of the portfolio at the risk free
rate of return. This would be very handy if for example options with strike
price
did not exist.
Example 48 (cont. Using

-hedging I)
The delta of a put option on a dividend paying asset such as the index is
![$\displaystyle \Delta _{P}=e^{-\delta T}\left[ \mathcal{N}\left( d_{1}\right) -1\right]$](DOLimg1278.gif)
,
which in this case is

. This implies that the put
option is approximated by selling one third of the portfolio and invest this
amount [

] in risk free securities.
When the portfolio's beta and/or the dividend yield do not match the market,
the procedure is slightly more complicated. One has to utilize the CAPM in order to match the insurance required for the portfolio with the
equivalent market moves. In addition, it can be shown that the optimal
hedging ratio that we first encountered when we discussed the futures
contracts still holds: if the value of the portfolio is
times the
index, then
put contracts have to be bought. One
can restate the above in the following way: one has to by
times the contracts she would buy if beta was equal to one. Now suppose that
the initial setting of the above example is slightly changed to illustrate
these differences.
Example 49 (Portfolio insurance II)
Now suppose that the beta of the portfolio is

, and that
the portfolio's dividend yield is

p.a. Since

p.a. or

over six months, if the value of the portfolio
declines by

over the next six months, the
total return will be

, or

p.a.
CAPM will now give us the corresponding market total return
 |
 |
, or |
|
 |
 |
p.a., or over six months. |
|
This is the
total return. Since the
FTSE250 dividend yield
is

p.a., or

over six months, we expect the index
to have fallen by

over the six month period, down to

pts. This means that in order to insure against a

drop of the
portfolio value, the fund manager needs to use six month put options with
exercise price

. Today the value of the portfolio is

times the index, therefore the optimal hedge ratio
dictates that she should buy

such options.
We will now see that the cost of the insurance has increased significantly
Example 50 (cont. The value of the insurance II)
The value of a put option with the new characteristics would now be

. Since the fund manager wants to buy now

of these
contracts, the value of the insurance is

.
The replicating procedure remains the same
Example 51 (cont. Using calls II)
The strategy creating a synthetic put, using a call option and the put call
parity is the following
This strategy would have the same value as the one involved put options.
If the fund manager wants to consider a delta hedging strategy, she must
calculate the delta of the portfolio. This is not as straightforward as in
the previous case.8.2
Example 52 (cont. Using

-hedging II)
The insurance can be seen as a put option on the
portfolio instead of
the
index, with strike price

. In addition, the
CAPM
implies that the volatility of the well diversified portfolio is

p.a. In this context the delta of the
portfolio would be
![$\displaystyle \Delta _{\Pi }=e^{-\delta T}\left[ \mathcal{N}\left( d_{1}\right) -1\right]$](DOLimg1306.gif)
,
where now
giving

. The put option can be approximated by
selling

of the portfolio and by investing this amount [

] in risk free assets.
Kyriakos
2003-03-17