Class 20 Financial Management, 15 · 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)

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


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


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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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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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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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

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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

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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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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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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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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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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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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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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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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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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

17

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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

18

41


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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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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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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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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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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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%.

25


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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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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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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Documents

Class 20 Financial Management, 15 · 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)