Options (2)
Class 20 Financial Management, 15.414 MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Today
Options
•
Option pricing
•
Applications: Currency risk and convertible bonds
Reading
•
Brealey and Myers, Chapter 20, 21
2
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Options
Gives the holder the right to either buy (call option) or sell (put
option) at a specified price.
Exercise, or strike, price Expiration or maturity date American vs. European option In-the-money, at-the-money, or out-of-the-money 3
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Option payoffs (strike = $50)
25
25
20
20
Buy a call
15
10
10
5
5
0
0
-5
30
40
Buy a put
15
50
60
70
Stock price
-5
5
30
40
-5
-10
-15
60
70
60
70
Stock price
5
Stock price
Stock price
0
50
0
30
40
Sell a call
50
60
70
-5
-10
-15
-20
-20
-25
-25
4
30
40
50
Sell a put
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Valuation Option pricing
How can we estimate the expected cashflows, and what is the
appropriate discount rate?
Two formulas
Put-call parity
Black-Scholes formula*
* Fischer Black and Myron Scholes
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Put-call parity
Relation between put and call prices
P + S = C + PV(X)
S = stock price
P = put price
C = call price
X = strike price
PV(X) = present value of $X = X / (1+r)t
r = riskfree rate
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Option strategies: Stock + put 70
25
65
20
60
Buy stock
55
15
50
Buy put
10
45
5
40
35
0
30
30
40
50
60
30
70
40
50
-5
Stock price
Stock price
70
65
60
55
50
45
Stock + put
40
35
30
30
40
50
Stock price
7
60
70
60
70
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Option strategies: Tbill + call 70
25
65
Buy Tbill with
FV = 50
60
55
20
15
50
Buy call
10
45
40
5
35
0
30
30
40
50
60
30
70
40
-5
Stock price
50
Stock price
70
65
60
55
50
45
Tbill + call
40
35
30
30
40
50
Stock price
8
60
70
60
70
MIT SLOAN SCHOOL OF MANAGEMENT
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Class 20
Example
On Thursday, Cisco call options with a strike price of $20 and an
expiration date in October sold for $0.30. The current price of Cisco
is $17.83. How much should put options with the same strike price
and expiration date sell for?
Put-call parity
P = C + PV(X) – S
C = $0.30, S = $17.83, X = $20.00 r = 1% annually → 0.15% over the life of the option Put option = 0.30 + 20 / 1.0015 – 17.83 = $2.44
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Black-Scholes
Price of a call option
C = S × N(d1) – X e-rT N(d2)
S = stock price
X = strike price
r = riskfree rate (annual, continuously compounded)
T = time-to-maturity of the option, in years
ln(S/X) + (r + σ2 /2) T
d1 =
σ T
d2 = d1 − σ T
N(⋅) = prob that a standard normal variable is less than d1 or d2
σ = annual standard deviation of the stock return
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Class 20
Cumulative Normal Distribution
0.5
N(-2) = 0.023
N(-1) = 0.159
N(0) = 0.500
N(1) = 0.841
N(2) = 0.977
0.4
0.3
0.2
0.1
0.0
-3.5 -3 -2.5 -2 -1.5 -1 -0.5
0
11
0.5
1
1.5
2
2.5
3
3.5
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Example
The CBOE trades Cisco call options. The options have a strike price
of $20 and expire in 2 months. If Cisco’s stock price is $17.83, how
much are the options worth? What happens if the stock goes up to
$19.00? 20.00?
Black-Scholes
S = 17.83, X = 20.00, r = 1.00, T = 2/12, σ2003 = 36.1% ln(S/X) + (r + σ2 /2)
T
d1 =
= -0.694
σ T
d2 = d1 − σ T = -0.842 Call price = S × N(d1) – X e-rT N(d2) = $0.35
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MIT SLOAN SCHOOL OF MANAGEMENT
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Class 20
Cisco stock price, 1993 – 2003
$90
80
70
60
50
40
30
20
10
0
Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug93
94
95
96
97
98
99
00
01
02
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Cisco returns, 1993 – 2003
40%
30%
20%
10%
-20%
-30%
-40%
14
Aug-02
Aug-01
Aug-00
Aug-99
Aug-98
Aug-97
Aug-96
Aug-95
Aug-94
-10%
Aug-93
0%
MIT SLOAN SCHOOL OF MANAGEMENT
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Class 20
Cisco option prices $6
Payoff (intrinsic value)
Option price
5
Today's price (2 months)
4
3
2
1
0
15
16
17
18
19
20
21
Stock price
15
22
23
24
25
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Option pricing
Factors affecting option prices
Stock price (S)
Exercise price (X)
Time-to-maturity (T)
Stock volatility (σ)
Interest rate (r)
Dividends (D)
Call option
Put option +
–
+
+
–
+
+
+
+
–
–
+
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MIT SLOAN SCHOOL OF MANAGEMENT
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Class 20
Example 2
Call option with X = $25, r = 3%
Time to expire
T = 0.25
T = 0.50
Stock price
Std. deviation
$18
25
32
30%
30
30
Call option
$0.02
1.58
7.26
18
25
32
50
50
50
0.25
2.57
7.75
18
25
32
30
30
30
0.14
2.29
7.68
18
25
32
50
50
50
0.76
3.67
8.68
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Option pricing 18
0 months
16
1 month
Option price
14
3 months
12
6 months
10
8
6
4
2
0
9
13
17
21
25
29
Stock price
18
33
37
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Using Black-Scholes
Applications
Hedging currency risk
Pricing convertible debt
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15.414
Class 20
Currency risk
Your company, headquartered in the U.S., supplies auto parts to
Jaguar PLC in Britain. You have just signed a contract worth ₤18.2
million to deliver parts next year. Payment is certain and occurs at
the end of the year.
The $ / ₤ exchange rate is currently s$/₤ = 1.4794.
How do fluctuations in exchange rates affect $ revenues? How can
you hedge this risk?
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
s$/₤, Jan 1990 – Sept 2001 2.1
Volatility
Full sample: 9.32%
After 1992: 8.34%
After 2000: 8.33%
After 2001: 7.95%
1.95
1.8
1.65
1.5
1.35
1.2
J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
$ revenues as a function of s$/₤
$ 32
30
28
$26.9 million
26
24
22
20
1.30
1.34
1.38
1.42
1.46
1.50
Exchange rate
22
1.54
1.58
1.62
1.66
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Currency risk
Forwards
1-year forward exchange rate = 1.4513
Lock in revenues of 18.2 × 1.4513 = $26.4 million
Put options*
S = 1.4794, σ = 8.3%, T = 1, r = -1.8%*
Strike price
Min. revenue
Option price
1.35
1.40
1.45
$24.6 M
$25.5 M
$26.4 M
$0.012
$0.026
$0.047
Total cost (×18.2 M)
$221,859
$470,112
$862,771
*Black-Scholes is only an approximation for currencies; r = rUK – rUS
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
$ revenues as a function of s$/₤
$31
30
with put option
29
28
27
with forward contract
26
25
24
23
22
1.30
1.34
1.38
1.42
1.46
1.50
Exchange rate
24
1.54
1.58
1.62
1.66
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Convertible bonds Your firm is thinking about issuing 10-year convertible bonds. In the
past, the firm has issued straight (non-convertible) debt, which
currently has a yield of 8.2%.
The new bonds have a face value of $1,000 and will be convertible
into 20 shares of stocks. How much are the bonds worth if they pay
the same interest rate as straight debt?
Today’s stock price is $32. The firm does not pay dividends, and
you estimate that the standard deviation of returns is 35% annually.
Long-term interest rates are 6%.
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Payoff of convertible bonds
$ 1,500
Convertible into 20 shares
1,400
1,300
Convert if stock price > $50
(20 × 50 = 1,000)
1,200
1,100
1,000
900
800
700
600
500
30
34
38
42
46
50
54
Stock price
26
58
62
66
70
MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Convertible bonds
Suppose the bonds have a coupon rate of 8.2%. How much
would they be worth?
Cashflows*
Year
1
2
3
4
Cash
$82
$82
$82
$82
…
10
$1,082
Value if straight debt: $1,000
Value if convertible debt: $1,000 + value of call option
* Annual payments, for simplicity
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414
Class 20
Convertible bonds
Call option
X = $50, S = $32, σ = 35%, r = 6%, T = 10
Black-Scholes value = $10.31
Convertible bond
Option value per bond = 20 × 10.31 = $206.2
Total bond value = 1,000 + 206.2 = $1,206.2
Yield = 5.47%*
*Yield = IRR ignoring option value
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