Before deciding about what kind of portfolio insurance will be
used some points have to be clarified:
Example 3.5
We proceed from the assumption that the investor has decided to
invest in stock. Depending on the type of return of the object we
distinguish two cases (for negative returns, as storage costs of
real values for example, the approach can be applied analogously):
- i) continuous dividend yield
- ii) ex ante known discrete yields with a time 0
discounted total value of
The data of the example is shown in Table
2.7. The
volatility can be interpreted as a measure of variability of the
stock price. The notion of volatility is an essential part of
option pricing and will be treated extensively later. Placing our
considerations at the beginning

of the investment the time
to maturity is

For both strategies the goal is
to determine the number

of stocks and/or (European) options
and their delivery price
Table 2.7:
Data of Example 2.5
Data of Example 2.5: |
|
Current point of time  |
0 |
Available capital  |
100000 EUR |
Target floor  |
95000 EUR |
Investment horizon  |
2 years |
Current stock price  |
100 EUR |
Continuously compounded annual interest rate  |
0.10 |
Annual stock volatility  |
0.30 |
Dividends during time to maturity |
|
Case i): continuous dividends  |
0.02 |
Fall ii): dividends with present value  |
5 EUR |
|
Case i)
The stock pays a continuous dividend at rate

= 2% p.a. which
he reinvests immediately. At maturity

the position in the
stock grew from

stocks to

with

Thus, for strategy 1 he has to buy in

the same number of put
options. Since the amount he wants to invest in

is

it
must hold
 |
(3.21) |
The investor chooses the put options delivery price

such that
his capital after two years does not drop below the floor

he
is aiming at. That is, exercising the puts in time

(if

) must give the floor

which gives the second condition
 |
(3.22) |
Substituting equation (
2.22) into equation
(
2.21) gives
 |
(3.23) |
Thanks to the Black-Scholes pricing formula for European options
that will be derived later in Section
6.2
the put price is expressed as a function of the parameter

The
delivery price which he is looking for can be computed by solving
equation (
2.23) numerically, for example by means of the
Newton-Raphson method. In this case,

is equal to

. To
be sure that the capital value does not drop below the floor

EUR he buys

stocks and

puts with delivery price

The price of
the put option given by the Black-Scholes formula is

EUR/put.
SFEexerput.xpl
Following the corresponding strategy 2 he invests
EUR in bonds at time 0 which gives compounded to
time
exactly the floor
EUR. For the
remaining capital of
EUR he buys
calls with delivery price
which have a price of
EUR/call according to the Black-Scholes formula. From
the put-call parity follows the equivalence of both strategies,
i.e. both portfolios consisting of stocks and puts respectively
zero bonds and calls have at each time
the same value:
 |
(3.24) |
where

and

due to
equation (
2.22). Table
2.8 shows the risk
decreasing effect of the insurance.
Table:
The effect of a portfolio insurance in case i) on
portfolio value and return.
SFEoptman.xpl
|
Non-insured |
Insured |
Insured |
|
portfolio |
portfolio |
portfolio in % |
Stock price  |
Value |
Return |
Value |
Return |
of the non- |
EUR![$ ]$](sfehtmlimg319.gif) |
EUR![$ ]$](sfehtmlimg319.gif) |
%
p.a. |
EUR![$ ]$](sfehtmlimg319.gif) |
%
p.a. |
insured portfolio |
70 |
72857 |
-27 |
95000 |
-5 |
130 |
80 |
83265 |
-17 |
95000 |
-5 |
114 |
90 |
93673 |
-6 |
95000 |
-5 |
101 |
100 |
104081 |
+4 |
95400 |
-5 |
92 |
110 |
114489 |
+15 |
104940 |
+5 |
92 |
120 |
124897 |
+25 |
114480 |
+14 |
92 |
130 |
135305 |
+35 |
124020 |
+24 |
92 |
140 |
145714 |
+46 |
133560 |
+34 |
92 |
|
Fig.:
The effect of a portfolio insurance: While the straight line
represents the value of the insured portfolio as a function of
the stock price, the dotted line represents the value of the
non-insured portfolio as a function of the stock price.
SFEoptman.xpl
|
Case ii)
Until maturity the stock pays dividends with a time 0 discounted
total value
EUR which are after distribution
immediately invested in bonds. At time
the dividend yield has
a compounded value of
EUR where
denotes the time to maturity. Reasoning as in case i) and
taking the dividend
into account he buys
stocks
respectively
puts, and obtains the following equations
 |
(3.25) |
and
 |
(3.26) |
The substraction of the cash dividend

from the stock price

in the option price formula cannot be justified until we
introduced the binomial model in Chapter
7.
Briefly, in a perfect market the stock price decreases
instantaneously by the amount of the distributed dividend.
Otherwise, an arbitrage opportunity arises. Substituting equation
(
2.26) into equation (
2.25) gives:
 |
(3.27) |
Solving the equations analogously as in case i) the number

of
stocks and puts and the delivery price

for strategy 1 are
obtained:

und
For strategy 2 he buys

calls at a price of

EUR/call with a delivery price

He invests

in bonds. For case ii) the effect of the
portfolio insurance for both strategies is shown in Table
2.9 taking into account the time

compounded total
dividend.
Table 2.9:
The effect of a portfolio insurance in case ii) on
portfolio value and return.
|
Non-insured |
Insured |
Insured |
|
portfolio |
portfolio |
portfolio in % |
Stock price  |
Value |
Return |
Value |
Return |
of the non- |
EUR![$ ]$](sfehtmlimg319.gif) |
EUR![$ ]$](sfehtmlimg319.gif) |
%
p.a. |
EUR![$ ]$](sfehtmlimg319.gif) |
% p.a. |
insured portfolio |
70 |
76107 |
-24 |
94996 |
-5 |
125 |
80 |
86107 |
-14 |
94996 |
-5 |
110 |
90 |
96107 |
-4 |
94996 |
-5 |
99 |
96.42 |
102527 |
+3 |
94996 |
-5 |
93 |
100 |
106107 |
+6 |
98313 |
-2 |
93 |
110 |
116107 |
+16 |
107579 |
+8 |
93 |
120 |
126107 |
+26 |
116844 |
+17 |
93 |
130 |
136107 |
+36 |
126110 |
+26 |
93 |
140 |
146107 |
+46 |
135375 |
+35 |
93 |
|
The example shows how a portfolio insurance can be carried out by
means of options in principle. In practice, the following problems
frequently occur: