Chapters 1415

1
Chapters 1415

• Options

2
Option Terminology

• Buy - Long
• Sell - Short
• Call
• Put
• Key Elements
• Exercise or Strike Price
• Premium or Price
• Maturity or Expiration

3
Market and Exercise Prices

• In the Money - exercise of the option would be
  profitable
• Call market pricegtexercise price
• Put exercise pricegtmarket price
• Out of the Money - exercise of the option would
  not be profitable
• Call market pricegtexercise price
• Put exercise pricegtmarket price
• At the Money - exercise price and asset price are
  equal

4
American vs. European Options

• American - the option can be exercised at any
  time before expiration or maturity
• European - the option can only be exercised on
  the expiration or maturity date

5
Different Types of Options

• Stock Options
• Index Options
• Futures Options
• Foreign Currency Options
• Interest Rate Options

6
Payoffs on Call Holders at Expiration

• Notation
• Stock Price ST Exercise Price X
• Payoff to Call Holder
• (ST - X) if ST gtX
• 0 if ST lt X
• Profit to Call Holder
• Payoff - Purchase Price

7
Payoffs on Call Writers at Expiration

• Payoff to Call Writer
• - (ST - X) if ST gtX
• 0 if ST lt X
• Profit to Call Writer
• Payoff Premium

8
Profit Profiles for Calls
Profit
Call Holder
0
Call Writer
Stock Price
9
Payoffs on Put Holders at Expiration

• Payoffs to Put Holder
• 0 if ST gt X
• (X - ST) if ST lt X
• Profit to Put Holder
• Payoff - Premium

10
Payoffs on Put Writers at Expiration

• Payoffs to Put Writer
• 0 if ST gt X
• -(X - ST) if ST lt X
• Profits to Put Writer
• Payoff Premium

11
Profit Profiles for Puts
Profits
Put Writer
0
Put Holder
Stock Price
12
Equity, Options Leveraged Equity - Text Example
Investment Strategy Investment Equity only Buy
stock _at_ 80 100 shares 8,000 Options only Buy
calls _at_ 10 800 options 8,000 Leveraged Buy
calls _at_ 10 100 options 1,000 equity Buy T-bills
_at_ 2 7,000 Yield
13
Equity, Options Leveraged Equity - Payoffs
Microsoft Stock Price 75 80 100 All
Stock 7,500 8,000 10,000 All
Options 0 0 16,000 Lev Equity
7,140 7,140 9,140
14
Equity, Options Leveraged Equity - Rates of
Return
Microsoft Stock Price 75 80 100 All
Stock -6.25 0 25 All Options -100
-100 100 Lev Equity -10.75
-10.75 14.25
15
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
16
Payoff of Long Call Short Put
Payoff
Long Call
Combined Leveraged Equity
Stock Price
Short Put
17
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

18
Put-Call Parity - Disequilibrium Example

• Stock Price 110 Call Price 17
• Put Price 5 Risk Free 10.25
• 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

19
Put-Call Parity Arbitrage
Immediate Cashflow in Six Months Position Cash
flow STlt105 STgt 105 Buy Stock -110 ST
ST Borrow X/(1r)T 100 100 -105 -105 Sell
Call 17 0 -(ST-105) Buy Put
-5 105-ST 0 Total 2 0 0
20
Option Strategies

• Protective Put
• Long Stock
• Long Put
• Covered Call
• Long Stock
• Short Call
• Straddle (Same Exercise Price)
• Long Call
• Long Put

21
Option Strategies

• Spreads - A combination of two or more call
  options or put options on the same asset with
  differing exercise prices or times to expiration
• Vertical or money spread
• Same maturity
• Different exercise price
• Horizontal or time spread
• Different maturity dates

22
Exotic Options

• Asian Options
• Barrier Options
• Lookback Options
• Currency-Translated Options
• Binary Options

23
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

24
Time Value of Options Call
Option value
Value of Call
Intrinsic Value
Time value
X
Stock Price
25
Factors Influencing Option Values Calls

• Factors Effect on value
• Stock price increases
• Exercise price decreases
• Volatility of stock price increases
• Time to expiration increases
• Interest rate increases
• Dividend Rate decreases

26
Binomial Option PricingText Example
200
100
50
Stock Price
27
Binomial Option PricingText Example
150
Alternative Portfolio Buy 1 share of stock at
100 Borrow 46.30 (8 Rate) Net outlay
53.70 Payoff Value of Stock 50 200 Repay
loan - 50 -50 Net Payoff 0
150
53.70
0
Payoff Structure is exactly 2 times the Call
28
Binomial Option PricingText Example
150
53.70
0
2C 53.70 C 26.85
29
Another View of Replication of Payoffs and Option
Values

• Alternative Portfolio - one share of stock and 2
  calls written (X 125)
• Portfolio is perfectly hedged
• Stock Value 50 200
• Call Obligation 0 -150
• Net payoff 50 50
• Hence 100 - 2C 46.30 or C 26.85

30
Black-Scholes Option Valuation

• Co Soe-dTN(d1) - Xe-rTN(d2)
• d1 ln(So/X) (r d s2/2)T / (s T1/2)
• d2 d1 - (s T1/2)
• where
• Co Current call option value.
• So Current stock price
• N(d) probability that a random draw from a
  normal dist. will be less than d.

31
Black-Scholes Option Valuation

• X Exercise price.
• d Annual dividend yield of underlying stock
• e 2.71828, the base of the nat. log.
• r Risk-free interest rate (annualizes
  continuously compounded with the same maturity as
  the option.
• T time to maturity of the option in years.
• ln Natural log function
• s Standard deviation of annualized cont.
  compounded rate of return on the stock

32
Call Option Example

• So 100 X 95
• r .10 T .25 (quarter)
• s .50 d 0
• d1 ln(100/95)(.10-0(.5 2/2))/(.5 .251/2)
• .43
• d2 .43 - ((.5)( .251/2)
• .18

33
Probabilities from Normal Dist.

• N (.43) .6664
• Table 17.2
• d N(d)
• .42 .6628
• .43 .6664 Interpolation
• .44 .6700

34
Probabilities from Normal Dist.

• N (.18) .5714
• Table 17.2
• d N(d)
• .16 .5636
• .18 .5714
• .20 .5793

35
Call Option Value

• Co Soe-dTN(d1) - Xe-rTN(d2)
• Co 100 X .6664 - 95 e- .10 X .25 X .5714
• Co 13.70
• Implied Volatility
• Using Black-Scholes and the actual price of the
  option, solve for volatility.
• Is the implied volatility consistent with the
  stock?

36
Put Option Value Black-Scholes

• PXe-rT 1-N(d2) - S0e-dT 1-N(d1)
• Using the sample data
• P 95e(-.10X.25)(1-.5714) - 100 (1-.6664)
• P 6.35

37
Put Option Valuation Using Put-Call Parity

• P C PV (X) - So
• C Xe-rT - So
• Using the example data
• C 13.70 X 95 S 100
• r .10 T .25
• P 13.70 95 e -.10 X .25 - 100
• P 6.35

38
Using the Black-Scholes Formula

• Hedging Hedge ratio or delta
• The number of stocks required to hedge against
  the price risk of holding one option
• Call N (d1)
• Put N (d1) 1
• Option Elasticity
• Percentage change in the options value given a
  1 change in the value of the underlying stock

39
Portfolio Insurance - Protecting Against Declines
in Stock Value

• Buying Puts - results in downside protection with
  unlimited upside potential
• Limitations
• Tracking errors if indexes are used for the puts
• Maturity of puts may be too short
• Hedge ratios or deltas change as stock values
  change