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CHAPTER 21: OPTION VALUATION
1.
The value of a put option also increases with the volatility of the stock.
We see this
from the putcall parity theorem as follows:
P = C – S
0
+ PV(X) + PV(Dividends)
Given a value for S and a riskfree interest rate, then, if C increases because of an
increase in volatility, P must also increase in order to maintain the equality of the
parity relationship.
2.
a.Put A must be written on the stock with the lower price.
Otherwise, given the
lower volatility of Stock A, Put A would sell for less than Put B.
b.
Put B must be written on the stock with the lower price.
This would explain
its higher price.
c.Call B must have the lower time to maturity.
Despite the higher price of Stock
B, Call B is cheaper than Call A.
This can be explained by a lower time to
maturity.
d.
Call B must be written on the stock with higher volatility.
This would explain
its higher price.
e.Call A must be written on the stock with higher volatility.
This would explain its
higher price.
3.
Exercise
Price
Hedge
Ratio
115
5/30 = 0.167
100
20/30 = 0.667
75
30/30 = 1.000
50
30/30 = 1.000
25
30/30 = 1.000
10
30/30 = 1.000
As the option becomes more in the money, the hedge ratio increases to a
maximum of 1.0.
211
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S
d
1
N(d
1
)
45
0.0268
0.4893
50
0.5000
0.6915
55
0.9766
0.8356
5.
a.uS
0
= 130
⇒
P
u
= 0
dS
0
= 80
⇒
P
d
= 30
The hedge ratio is:
5
3
80
130
30
0
dS
uS
P
P
H
0
0
d
u

=


=


=
b.
Riskless
Portfolio
S = 80
S = 130
Buy 3 shares
240
390
Buy 5 puts
150
0
Total
390
390
Present value = $390/1.10 = $354.545
c.The portfolio cost is: 3S + 5P = 300 + 5P
The value of the portfolio is: $354.545
Therefore: P = $54.545/5 = $10.91
6.
The hedge ratio for the call is:
5
2
80
130
0
20
dS
uS
C
C
H
0
0
d
u
=


=


=
Riskless
Portfolio
S = 80
S = 130
Buy 2 shares
160
260
Write 5 calls
0
100
Total
160
160
Present value = $160/1.10 = $145.455
The portfolio cost is: 2S – 5C = $200 – 5C
The value of the portfolio is: $145.455
Therefore: C = $54.545/5 = $10.91
Does P = C + PV(X) – S?
10.91 = 10.91 + 110/1.10 – 100 = 10.91
212
7.
d
1
= 0.3182
⇒
N(d
1
) = 0.6248
d
2
= –0.0354
⇒
N(d
2
) = 0.4859
Xe

r T
= 47.56
C = $8.13
8.
P = $5.69
This value is derived from our BlackScholes spreadsheet, but note that we could
have derived the value from putcall parity:
P = C + PV(X) – S
0
= $8.13 + $47.56

$50 = $5.69
9.
a.C falls to $5.5541
b.
C falls to $4.7911
c.C falls to $6.0778
d.
C rises to $11.5066
e.
C rises to $8.7187
10.
According to the BlackScholes model, the call option should be priced at:
[$55
×
N(d
1
)] – [50
×
N(d
2
)] = ($55
×
0.6) – ($50
×
0.5) = $8
Since the option actually sells for more than $8, implied volatility is greater than 0.30.
11.
A straddle is a call and a put.
The BlackScholes value would be:
C + P = S
0
N(d
1
)

Xe
–rT
N(d
2
) + Xe
–rT
[1

N(d
2
)]

S
0
[1

N(d
1
)]
= S
0
[2N(d
1
)

1] + Xe
–rT
[1

2N(d
2
)]
On the Excel spreadsheet (Spreadsheet 21.1), the valuation formula would be:
B5*(2*E4

1) + B6*EXP(

B4*B3)*(1

2*E5)
12.
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This note was uploaded on 01/24/2009 for the course BUSI 474 taught by Professor Ettinger during the Spring '09 term at A.T. Still University.
 Spring '09
 Ettinger
 Dividends, Interest, Interest Rate, Valuation, Volatility

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