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
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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
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Option payoffs (strike = $50) 25 25
20 20
1515
1010
55
00
-5-5 30 40 50 60 70 30 40 50 60 70
Buy a call Buy a put
Stock price Stock price
-5
0
5
30 40 50 60 70 -5
0
5
30 40 50 60 70
Sell a call Sell a put
-25
-20
-15
-10
Stock price
-25
-20
-15
-10
Stock price
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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
30
35
40
45
50
55
60
65
70
30 40 50 60 70 -5
0
5
10
15
20
25
30 40 50 60 70
30 35
40 45
50 55 60 65
70
30 40 50 60 70
Buy stock Buy put
Stock + put
Stock price Stock price
Stock price
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Option strategies: Tbill + call
30 35 40 45 50 55 60 65 70
30 40 50 60 70 -5
0
5
10
15
20
25
30 40 50 60 70
30 35
40 45
50 55 60 65
70
30 40 50 60 70
Buy Tbill with FV = 50
Buy call
Tbill + call
Stock price Stock price
Stock price
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 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 T /2) 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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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Cumulative Normal Distribution
0.0
0.1
0.2
0.3
0.4
0.5 N(-2) = 0.023 N(-1) = 0.159 N(0) = 0.500 N(1) = 0.841 N(2) = 0.977
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
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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 T /2)d1 = = -0.694σ 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
15.414 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- Aug-93 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%
0%
-10%
-20%
-30%
-40%
Aug
-93
Aug
-94
Aug
-95
Aug
-96
Aug
-97
Aug
-98
Aug
-99
Aug
-00
Aug
-01
Aug
-02
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Cisco option prices O
ptio
n pr
ice
$ 6
5
4
3
2
1
0 15 16 17 18 19 20 21 22 23 24 25
Payoff (intrinsic value)
Today's price (2 months)
Stock price
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Option pricing
Factors affecting option prices
Call option Put option Stock price (S) Exercise price (X) Time-to-maturity (T) Stock volatility (σ) Interest rate (r) Dividends (D)
+ – – + + + + + + – – +
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Example 2
Call option with X = $25, r = 3%
Time to expire Stock price Std. deviation Call option
T = 0.25
$18 25 32
18 25 32
30% 30 30
50 50 50
$0.02 1.58 7.26
0.25 2.57 7.75
T = 0.50
18 25 32
18 25 32
30 30 30
50 50 50
0.14 2.29 7.68
0.76 3.67 8.68
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MIT SLOAN SCHOOL OF MANAGEMENT
15.414 Class 20
Option pricing O
ptio
n pr
ice
16
14
12
10
8
6
4
2
0
0 months
1 month
3 months
6 months
9 13 17 21 25 29 33 37 Stock price
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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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MIT SLOAN SCHOOL OF MANAGEMENT
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
1.2
1.35
1.5
1.65
1.8
1.95
2.1 Volatility Full sample: 9.32% After 1992: 8.34% After 2000: 8.33% After 2001: 7.95%
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
24
22
20 1.30 1.34 1.38 1.42 1.46 1.50 1.54 1.58 1.62 1.66
$26.9 million
Exchange rate
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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 Total cost (×18.2 M) 1.35 $24.6 M $0.012 $221,859 1.40 $25.5 M $0.026 $470,112 1.45 $26.4 M $0.047 $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
29
28
27
26
25
24
23
22 1.30 1.34 1.38 1.42 1.46 1.50 1.54 1.58 1.62 1.66
with forward contract
with put option
Exchange rate
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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
$
500
600
700
800
900
1,000
1,100
1,200
1,300
1,400
1,500 Convertible into 20 shares
Convert if stock price > $50 (20 × 50 = 1,000)
30 34 38 42 46 50 54 58 62 66 70Stock price
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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 … 10
Cash $82 $82 $82 $82 $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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