Today

1
Today

• Options
• Option pricing
• Applications Currency risk and convertible bonds
  
• Reading
• Brealey, Myers, and Allen Chapter 20, 21

2
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
Option payoffs (strike 50)
4
Valuation

• Option pricing
• How risky is an option? 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

5
Option Values

• Intrinsic value - profit that could be made if
  the option was immediately exercised (?????????)
• Call stock price - exercise price
• Put exercise price - stock price
• Time value - the difference between the option
  price and the intrinsic value

6
Time Value of Options Call
Option value
Value of Call
Intrinsic Value
Time value
X
Stock Price
7
Factors Influencing Option Values

• Factor
• Stock price
• Exercise price
• Volatility of stock price
• Time to expiration
• Interest rate
• Dividend Rate

8
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 / (1r)t r riskfree rate
9
Option strategies Stock put
10
Put-Call Parity Relationship

• Long a call and write a put simultaneously.
• Call and put are with the same exercise price and
  maturity date.

11
Put-Call Parity Relationship
ST lt X ST gt X Payoff for Call Owned
0 ST - X Payoff for Put Written -( X -ST)
0 Total Payoff ST - X ST X PV of (ST-X)
S0 - X / (1 rf)T
12
Payoff of Long Call Short Put
Payoff
Long Call
Combined Leveraged Equity
Stock Price
Short Put
13
Arbitrage Put Call Parity

• Since the payoff on a combination of a long call
  and a short put are equivalent to leveraged
  equity, the prices must be equal.
• C - P S0 - X / (1 rf)T
• If the prices are not equal, arbitrage will be
  possible

14
Put Call Parity - Disequilibrium Example

• Stock Price 110, Call Price 17, Put Price
  5
• Risk Free 5 per 6 month (10.25 effective
  annual yield)
• Maturity .5 yr X 105
• C - P gt S0 - X / (1 rf)T
• 17- 5 gt 110 - (105/1.05)
• 12 gt 10
• Since the leveraged equity is less expensive,
  acquire the low cost alternative and sell the
  high cost alternative

15
Put-Call Parity Arbitrage
Immediate Cashflow in Six Months Position Cash
flow STlt105 STgt 105 Buy Stock -110 ST
ST Borrow 100 X/(1r)T 100 100 -105 -105 S
ell Call 17 0 -(ST-105) Buy Put
-5 105-ST 0 Total 2 0 0
16
Example
17
Black-Scholes
18
Cumulative Normal Distribution
19
Example
20
Option pricing
21
Using Black-Scholes

• Applications
• Hedging currency risk
• Pricing convertible debt

22
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?
23
S/, Jan 1990 Sept 2001
24
revenues as a function of S/
25
Currency risk
26
revenues as a function of S/
27
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? Todays 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.
28
Payoff of convertible bonds
29
Convertible bonds
30
Convertible bonds
Call option X 50, S 32, s 35, r 6,
T 10 Black-Scholes value 10.31
Convertible bond