Simon Benninga. Financial Modelling 3-rd edition
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528 Chapter 19
The initial part of the payoff has slope
Y
/
X
1
, and the second increasing part
of the payoff pattern has slope
Y
/
X
2
. This payoff pattern is created by the
following formula:
Y
Y
X
XS
Y
X
T
Y
X
−−+
↑
1
1
2
0
1
max , max
Payoff of
written puts
()
(()SX
T
Y
X
−
↑
2
0
2
,
Payoff of
bought calls
To prove that this is indeed the payoff,
Y
Y
X
XS
Y
X
T
Y
X
−−+
↑
1
1
2
0
1
max , max
Payoff of
written puts
()
(()
()
SX
Y
Y
X
XS
Y
X
T
Y
X
T
−
=
−−=
↑
2
1
1
0
2
,
Payoff of
bought calls
11
1
12
2
2
2
2
SSX
YXSX
Y
Y
X
SX
Y
X
SXS
TT
T
TTT
<
≤<
+−= ≤
⎧
⎨
⎪
⎪
⎩
⎪
⎪
()
X
1
X
2
Y
529 The Black-Scholes Model
As an example of this kind of structured payoff security, Figure 19.3
shows the term sheet for a structured product issued by ABN-AMRO
bank. The payment on this “Airbag” security depends on the value of the
Stoxx50—an index of European stocks. Here are the details:
•
Issuance date: 24 March 2003
•
Terminal date: 24 March 2008
•
Cost: a1,020
•
Payment at terminal date:
Payment at maturity
1,000 1.33
Stoxx50
Stoxx50
Maturity
Initial
=
∗∗
⎛⎛
⎝
⎜
⎞
⎠
⎟
<
<
If Stoxx50 1,618.50
1,000 1,618.50 Stoxx50
Maturity
Matu
rrity
Maturity
Initial
2,158
1,000
Stoxx50
Stoxx50
If Stox
<
∗
⎛
⎝
⎜
⎞
⎠
⎟
xx50 2,158
Maturity
>
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
We recognize this security as one whose payoff has the form discussed
earlier:
Y
Y
X
XS
T
Y
X
↑
↑
−−
Bond payoff
written puts with
exercise pr
max ,
1
1
0
1
()
iice
purchased call
with exer
max ,
X
T
Y
X
Y
X
SX
1
2
2
2
0
+−
↑
()
ccise price X
2
where X
1
= 1,618.50, X
2
= 2,158, and Y = 1,000.
Here’s the spreadsheet for the payoff. Cell B7 shows the payoff defi ni-
tion given by the Airbag issuer, and cell B8 shows the payoff in the option
terms defi ned in the preceding formulas. The data table in cells A13:B29
shows that these two defi nitions are equivalent:
530 Chapter 19
Figure 19.3
The ABN-AMRO term sheet for its Euro Stoxx50 Airbag security.
531 The Black-Scholes Model
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5
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7
8
9
10
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13
14
15
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18
19
20
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23
24
25
26
27
28
29
30
ABC D
Y 1,000.00
X
1
1,618.50
X
2
2,158.00
S
T
2,373.80
Airbag payoff
By Airbag definition 1100.00 <-- =IF(B5<B3,B2*(B4/B3)*B5/B4,IF(B5>B4,B2*B5/B4,1000))
Option formula 1100.00 <-- =B2-B2/B3*MAX(B3-B5,0)+B2/B4*MAX(B5-B4,0)
S
T
Airbag
definition
Option
formula
<-- Data table headers hidden
0 0.00 0.00
100 61.79 61.79
500 308.93 308.93
750 463.39 463.39
1,000 617.86 617.86
1,250 772.32 772.32
1,618.5 1,000.00 1,000.00
1,750 1,000.00 1,000.00
2,000 1,000.00 1,000.00
2,158 1,000.00 1,000.00
2,500 1,158.48 1,158.48
2,750 1,274.33 1,274.33
3,000 1,390.18 1,390.18
3,250 1,506.02 1,506.02
3,500 1,621.87 1,621.87
3,750 1,737.72 1,737.72
Data table of payoffs
ABN-AMRO AIRBAG
0
200
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
0 1000 2000 3000 4000
Stoxx50 at Airbag expiration
Airbag payoff
Airbag definition
Option formula
To see how the Airbag is priced, we use the Black-Scholes model and
fi nd the Stoxx50 volatility implied by the Airbag price:
532 Chapter 19
When the Stoxx50 σ is 15.75 percent (cell B8), the Airbag’s price is
a1,020 (cell B14). The table shows the sensitivity of this price to the σ.
Notice that Airbag values are not very sensitive to σ. Doubling the sigma
from 10 percent to 20 percent increases the Airbag value by about a17.
This result occurs because of the offsetting values of the short put and
the long call in the Airbag.
We can do one more exercise on the Airbag. Using a two-dimensional
Data Table, we examine the Airbag’s price sensitivity to both time to
maturity T and the Stoxx50 volatility σ.
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6
7
8
9
10
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12
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14
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16
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18
19
20
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22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
CBA
Stoxx50 price today, S
0
2,158.0
X
1
1,618.50
X
2
2,158.0
1Y ,000.0
Risk-free interest rate for 5 years, r 7.00%
Time to maturity, T 5
Volatility of the Stoxx50, sigma 15.75%
Airbag components, value today
Bond paying X
1
at maturity
704.69 <-- =EXP(-B6*B7)*B5
Y/X
1
* written puts with exercise X
1
-4.69 <-- =-B5/B3*BSPut(B2,B3,B7,B6,B8)
Purchased call with exercise X
2
320.01 <-- =B5/B4*BSCall(B2,B4,B7,B6,B8)
Value of structured security today 1,020.00 <-- =SUM(B11:B13)
Table: Sensitivity of Airbag to Sigma 1,020.00 <-- =B14, data table header
0% 1,000.00
1% 1,000.00
3% 1,000.00
6% 1,000.16
9% 1,002.76
10% 1,004.57
11% 1,006.80
12% 1,009.34
13% 1,012.09
14% 1,014.95
15% 1,017.84
16% 1,020.70
17% 1,023.49
18% 1,026.16
19% 1,028.70
20% 1,031.11
21% 1,033.35
22% 1,035.45
23% 1,037.39
24% 1,039.19
25% 1,040.84
PRICING THE ABN-AIRBAG
Find the Implied Volatility
Airbag Pricing: Sensitivity to
σ
995
1,000
1,005
1,010
1,015
1,020
1,025
1,030
1,035
1,040
1,045
0 5 10 15 20 25
Stoxx50 volatility, (%)
σ
Airbag initial price
533 The Black-Scholes Model
The Airbag is a fairly stable security—its price varies no more than 10
percent for a wide variety of σs and for nearly all times to maturity.
19.7.3 A Reverse Convertible: Analyzing the UBS “Goals”
The Swiss bank UBS has issued a series of stock-linked securities called
Goals. All the Goals pay interest on the initial price; the fi nal repayment
depends on the market price of the underlying stock: If the stock price
is high, the Goals investor gets back her initial investment, and if the
stock price is low, the Goals investor is paid out in a package of shares
whose value is less than her initial investment.
An example of such a security is the Cisco-linked Goals issued by UBS
on 17 January 2001 (Figure 19.4). The main details of this security are as
follow:
The purchaser pays UBS $1,000 on 23 January 2001. In return, she gets
three payments of
$.
.
97 50
19 50
2
1=∗
⎛
⎝
⎜
⎞
⎠
⎟
percent
,000
on 23 July 2001, 23
January 2002, and 23 July 2002.
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
BACDEFG
H
Stoxx50 price today, S
0
2,158.0
X
1
1,618.50
X
2
2,158.0
Y 1,000.0
Risk-free interest rate for 5 years, r 7.00%
Time to maturity, T 5
Volatility of the Stoxx50, sigma 15.75%
Airbag components, value today
Bond paying X
1
at maturity
704.69 <-- =EXP(-B6*B7)*B5
Y/X
1
* written puts with exercise X
1
-4.69 <-- =-B5/B3*BSPut(B2,B3,B7,B6,B8)
Purchased call with exercise X
2
320.01 <-- =B5/B4*BSCall(B2,B4,B7,B6,B8)
Value of structured security today 1,020.00 <-- =SUM(B11:B13)
Time to maturity, T
1,020.00 543210.0001
5%
,
1,000.02
1,000.07 1,000.20 1,000.59 1,001.77 1,000.20
10%
1,004.57 1,006.22 1,008.40 1,011.13 1,013.78 1,000.40
15%
1,017.84
1,021.09 1,024.72 1,028.28 1,029.65 1,000.59
20% 1,031.11
1,035.21 1,039.61 1,043.69 1,044.54 1,000.79
V
olatility of the Stoxx50, sigma --> 25% 1,040.84 1,045.48 1,050.44 1,055.14 1,056.54 1,000.99
30% 1,047.16
1,052.22 1,057.69 1,063.09 1,065.58 1,001.19
35%
1,050.86
1,056.29 1,062.26 1,068.39 1,072.19 1,001.39
40% 1,052.66
1,058.44 1,064.88 1,071.75 1,076.95 1,001.59
45% 1,053.10
1,059.19 1,066.10 1,073.70 1,080.28 1,001.79
50% 1,052.55
1,058.94 1,066.29 1,074.59 1,082.53 1,001.99
ABN-AMRO AIRBAG SENSITIVITY TO TIME TO MATURITY AND SIGMA
Data table
header:
=B14
534 Chapter 19
THE SEC FILING FOR UBS AG $60,000,000 GOALS
- ----------------------------------------------------------------------------------------------------------------------------
Each note being offered has the terms described beginning on page S-14,
including the following:
- - Issuer: UBS AG
- - Issue: $60,000,000 USD principal amount
of GOALs due July 23, 2002 linked
to shares in the common stock of
Cisco Systems, Inc.
- - Coupon: 19.5% per annum, payable
semi-annually in arrears on each
January 23 and July 23 which shall
be composed of (1) an interest
coupon representing a rate of 5.2%
per annum and (2) a coupon
representing an option premium of
14.3% per annum
- - Initial price of $39.00 per share, subject to
underlying stock antidilution adjustments (strike price):
- - Key dates: Trade: January 17, 2001
Settlement: January 23, 2001
Determination: July 18, 2002
Maturity: July 23, 2002
- - Booking branch: The GOALs will be booked in UBS
AG, Jersey Branch
- - Proceeds at maturity are based on the closing price of Cisco Systems, Inc.
common stock three business days before maturity:
If the closing price of Cisco Systems, Inc. common stock is at or above the
initial price per share of $39.00, holders will receive a cash payment equal
to the principal amount of their GOALs.
If the closing price of Cisco Systems, Inc. is lower than the initial price
per share of $39.00, holders will receive 25.641 shares of Cisco Systems, Inc.
common stock for each $1,000 principal amount of their GOALs (the stock
redemption amount). Fractional shares will be paid in cash. The number of
shares received for each $1,000 invested will be calculated by dividing the
initial price per share of $39.00 into $1,000. The stock redemption amount and
the initial price per share of $39.00 (strike price) may change due to stock
splits or other corporate actions.
UBS AG
$60,000,000
19.5% GOALs DUE JULY 23, 2002
Figure 19.4
From the SEC fi ling for the UBS Cisco-linked Goals.
535 The Black-Scholes Model
On 23 July 2002, in addition to the $97.50 payment,
•
If the price of Cisco stock ≥ $39 per share, the purchaser of the security
gets $1,000.
•
If the price of Cisco stock < $39 per share, the purchaser gets
1
25 641
,000
39
= .
shares of Cisco.
The closing stock price of Cisco on 23 January 2001 was 42.625.
The continuously compounded risk-free interest rate at the time of the
Goals issuance was 5.2 percent annually (2.6 percent semiannually).
To analyze the Cisco-linked Goals, we start by noting that the cash
fl ow can be written as follows:
Time 0 1 2 3
−1,000
97.50 97.50 97.50
1,000 − 25.641
*
max(39 − S
T
, 0)
To prove that this defi nition is equivalent, we write a short spreadsheet
that compares the Goals payoff according to the UBS defi nition with its
payoff as we have described it:
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7
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12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
ABCDEFGHI
J
Cisco price, 23 July 2002, S
T
32
Payoff ratio 25.6410 <-- =1000/39
Terminal payoff
As described by UBS 820.51 <-- =IF(B2>39,1000,B2*B3)
In option terms 820.51 <-- =1000-B3*MAX(39-B2,0)
Data table: Comparing the payoff on Cisco-linked GOALS
Cisco stock price
on 23 July 2002, S
T
UBS
description
Alternative
option
description
820.51 820.51 <-- Data table headers, B5 and B6 respectively
00.000.00
10 256.41 256.41
15 384.62 384.62
20 512.82 512.82
22 564.10 564.10
24 615.38 615.38
26 666.67 666.67
30 769.23 769.23
32 820.51 820.51
34 871.79 871.79
38 974.36 974.36
39 1,000.00 1,000.00
40 1,000.00 1,000.00
42 1,000.00 1,000.00
44 1,000.00 1,000.00
46 1,000.00 1,000.00
47 1,000.00 1,000.00
50 1,000.00 1,000.00
EQUIVALENCE OF TWO WAYS OF WRITING THE PAYOFF
UBS Cisco-Lined GOALS Terminal Payoff
0
100
200
300
400
500
600
700
800
900
1000
01020304050
UBS
description
Alternative
option description
536 Chapter 19
The equivalence of these two defi nitions means that the purchaser of
the Cisco-linked Goals
•
Acquires a $1,000 bond paying 9.75 percent interest semiannually.
•
Writes UBS 25.641 puts with exercise X = 39 and with time to maturity
T = 1.5.
At the time the Goals were issued, the semiannual interest rate was
around 2.6 percent, far below 9.75 percent. Thus the bond component of
the Goals was worth much more than $1,000. However, the purchaser of
the Goals was giving UBS 25.641 puts. The value of this “gift” should be
accounted for in any analysis of the Goals.
We illustrate two ways to value the Goals. The fi rst method assumes
that the equilibrium net present value of any security should be zero.
Applying this method for the UBS security gives
−+
+
+
+
+
+
↑
1
97 50
1
97 50
1
1
1
05 10 15
,000
,097.50
Value=
.
()
.
() ()
...
rrr
$$205.11
at
Puts on Cisco ( ,
r
X
=
−∗ =
26
25 641 39
.%
.
TT ==15 0.)
In the next spreadsheet, we use this logic to price the puts embedded in
the UBS security, and we compare this price to the Black-Scholes
price.
Implicit put valuation ,000=−+
+
+
1
25 641
1
97 50
126
97 50
05
.
.
(.%)
.
(
.
1126
1
126
205 11
25 641
800
10 15
+
+
+
⎡
⎣
⎢
⎤
⎦
⎥
==
.%) ( .%)
$.
.
$.
..
,097.50
UBS is implicitly paying the Goals purchaser $8.00 per Cisco put.
However, as shown in the spreadsheet, the Black-Scholes price of such
a put is $11.71 if σ = 80 percent.
5
This result makes the Goals a bad
buy.
5. The Cisco-linked Goals was issued during the Nasdaq crash of the early 2000s. The
implied volatility of Cisco stock during this period varied between 80 and 120
percent.
537 The Black-Scholes Model
There is another way to look at the Goals. Consider a Goals purchaser
who buys the Goals and also buys 25.641 puts on Cisco with T = 1.5, X
= 39. This “engineered” combination of a Goals + 25.641 puts creates a
risk-free security:
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3
4
5
6
7
8
9
10
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12
13
14
15
16
17
18
19
20
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22
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24
25
26
27
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29
30
CBA
Annual risk-free rate 5.20%
Coupon rate 19.50%
Initial cost 1,000
Conversion ratio: number of shares of
Cisco received if share price is low
25.641 <-- =1000/39
Valuing the fixed payments at 5.20%
Fixed payments
Date Cash flow
23-Jan-01 (1,000.00)
23-Jul-01 97.50 <-- =$B$3*B4/2
23-Jan-02 97.50
23-Jul-02 1,097.50
PV of Goals bond component 205.11
<-- =XNPV(B2,B10:B13,A10:A13)
Value of 25.641 puts embedded in Goals 205.11 <-- =B14
Value per put 8.00
<-- =B16/25.641
This is what UBS is
paying
the Goals purchaser for the embedded puts.
Valuing the puts with Black-Scholes
S 42.625 Current stock price
X 39 Exercise price
r 5.20% Risk-free interest rate
T 1.5 Time to maturity of option (in years)
Sigma 80% Stock volatility
Put price 11.71
<-- =BSPut(B21,B22,B24,B23,B25)
Is the Goals a good buy? No <-- =IF(B17>B26,"Yes","No")
PRICING THE UBS GOALS IMPLICIT PUT
Technical note
: For didactic clarity, the computations use 5.2% as the interest rate for valuing
both the bond component of the Goals (rows 10-14) and for the option valuation. Given a 2.6%
semiannual discrete interest rate, it would be technically more correct to use an equivalent
continuously compounded interest rate of
LN((1.026)^2)
in the option computations. The reader
can confirm that the effect of this correction is negligible.