1. Options Markets: Introduction
CHAPTER 20
Options Markets: Introduction

2. Options
Derivatives are securities that get their value from the price of other securities.
Derivatives are contingent claims because their payoffs depend on the value of other securities.
Options are traded both on organized exchanges and OTC.

3. Option Terminology Types Options Clearing Corporation Key Elements
Exercise or Strike Price
Premium or Price
Maturity or Expiration
Types
American Options - the option can be exercised at any time before expiration
European Options – can only be exercised at expiration
Options Clearing Corporation
Guarantees contract performance

4. The Option Contract: Calls
A call option gives its holder the right to buy an asset:
At the exercise or strike price
On or before the expiration date
This is a “long” strategy.
Exercise the option to buy the underlying asset if market value > strike.

5. The Option Contract: Puts
A put option gives its holder the right to sell an asset:
At the exercise or strike price
On or before the expiration date
This is a “short” strategy
Exercise the option to sell the underlying asset if market value < strike.

6. Market and Exercise Price Relationships
In the Money - exercise of the option would be profitable
Call: exercise price < market price
Put: exercise price > market price
Out of the Money - exercise of the option would not be profitable
Call: market price < exercise price.
Put: market price > exercise price.
At the Money - exercise price and asset price are equal

7. Example 20.1 Profit and Loss on a Call
A January 2010 call on IBM with an exercise price of $130 was selling on December 2, 2009, for $2.18.
The option expires on the third Friday of the month, or January 15, 2010.
If IBM remains below $130, the call will expire worthless.

8. Example 20.1 Profit and Loss on a Call
Suppose IBM sells for $132 on the expiration date.
Option value = stock price-exercise price
$132- $130= $2
Profit = Final value – Original investment
$ $2.18 = -$0.18
Option will be exercised to offset loss of premium.
Call will not be strictly profitable unless IBM’s price exceeds $ (strike + premium) by expiration.

9. Example 20.2 Profit and Loss on a Put
Consider a January 2010 put on IBM with an exercise price of $130, selling on December 2, 2009, for $4.79.
Option holder can sell a share of IBM for $130 at any time until January 15.
If IBM goes above $130, the put is worthless.

10. Example 20.2 Profit and Loss on a Put
Suppose IBM’s price at expiration is $123.
Value at expiration = exercise price – stock price:
$130 - $123 = $7
Investor’s profit:
$ $4.79 = $2.21
Holding period return = 46.1% over 44 days!

11. Different Types of Options
Stock Options
Index Options
Futures Options
Foreign Currency Options
Interest Rate Options

12. Profit Potential Investing in Stocks
Investing in Options

13. Payoffs and Profits at Expiration - Calls
Notation
Stock Price = ST Exercise Price = X
Payoff to Call Holder
(ST - X) if ST >X
0 if ST < X
Profit to Call Holder
Payoff - Purchase Price

14. Payoffs and Profits at Expiration - Calls
Payoff to Call Writer
- (ST - X) if ST >X
0 if ST < X
Profit to Call Writer
Payoff + Premium

15. Payoff and Profit to Call Options
Investing in Options

16. Payoffs and Profits at Expiration - Puts
Payoffs to Put Holder
0 if ST > X
(X - ST) if ST < X
Profit to Put Holder
Payoff - Premium

17. Payoffs and Profits at Expiration – Puts
Payoffs to Put Writer
0 if ST > X
-(X - ST) if ST < X
Profits to Put Writer
Payoff + Premium

18. Profit Potential of a Put Option
Investing in Options

19. Option versus Stock Investments
Could a call option strategy be preferable to a direct stock purchase?
Suppose you think a stock, currently selling for $100, will appreciate.
A 6-month call costs $10 (contract size is 100 shares).
You have $10,000 to invest.

20. Option versus Stock Investments
Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100.
Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling for $10. (This would require 10 contracts, each for 100 shares.)
Strategy C: Purchase 100 call options for $1,000. Invest your remaining $9,000 in 6-month T-bills, to earn 3% interest. The bills will be worth $9,270 at expiration.

21. Option versus Stock Investment
Investment Strategy Investment
Equity only Buy shares $10,000
Options only Buy options $10,000
Leveraged Buy options $1,000
equity Buy 3% $9,000
Yield

22. Strategy Payoffs

23. Protective Put
Puts can be used as insurance against stock price declines.
Protective puts lock in a minimum portfolio value.
The cost of the insurance is the put premium.
Options can be used for risk management, not just for speculation.

24. Table 20.1 Value of Protective Put Portfolio at Option Expiration
Buy a Put to protect a stock investment.

25. Covered Calls Purchase stock and write calls against it.
Call writer gives up any stock value above X in return for the initial premium.
If you planned to sell the stock when the price rises above X anyway, the call imposes “sell discipline.”

26. Table 20.2 Value of a Covered Call Position at Expiration

27. Straddle
Long straddle: Buy call and put with same exercise price and maturity.
The straddle is a bet on volatility.
To make a profit, the change in stock price must exceed the cost of both options.
You need a strong change in stock price in either direction.
The writer of a straddle is betting the stock price will not change much.

28. Table 20.3 Value of a Straddle Position at Option Expiration

29. Figure 20.9 Value of a Straddle at Expiration

31. Spreads
A spread is a combination of two or more calls (or two or more puts) on the same stock with differing exercise prices or times to maturity.
Some options are bought, whereas others are sold, or written.
A bullish spread is a way to profit from stock price increases.

32. Put-Call Parity
The call-plus-bond portfolio (on left) must cost the same as the stock-plus-put portfolio (on right):
If the prices are not equal arbitrage will be possible.
To exploit the arbitrage you buy the cheap portfolio and sell the other.

33. Put Call Parity - Disequilibrium Example
Stock Price = Call Price = 17
Put Price = Risk Free = 5%
Maturity = 1 yr X = 105
117 > 115
Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

34. CHAPTER 21
Option Valuation

35. 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

36. Table 21.1 Determinants of Call Option Values

37. Restrictions on Option Value: Call
Call value cannot be negative. The option payoff is zero at worst, and highly positive at best.
Call value cannot exceed the stock value.
Value of the call must be greater than the value of levered equity.
Lower bound = adjusted intrinsic value:
C > S0 - PV (X) - PV (D)
(D=dividend)

38. Early Exercise: Calls
The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires.
The value of American and European calls is therefore identical.
The call gains value as the stock price rises. Since the price can rise infinitely, the call is “worth more alive than dead.”

39. Early Exercise: Puts
American puts are worth more than European puts, all else equal.
The possibility of early exercise has value because:
The value of the stock cannot fall below zero.
Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money.

40. Determining Option Prices
Three stages of discussion
1) Assume risk neutral investors & equal probability of stock values between two extremes (not in text)
2) Relax assumption of risk neutral investors & add uncertainty (appended by info from outside the text)
3) assume stock values are normally distributed

41. Pricing of options (risk neutrality & uniform probability distributions)
Valuing a Call
Stock pays no div and expires in 1 year
Two possible future values of the stock with equal probability
Probability distribution is rectangular shaped
With risk neutral investors use a discount rate = RF
Example
Stock price 50 or 10
RF = 10%
Exercise price = 30
Present Value of stock

42. Value of call
this equation represents the probability (second part) * average value if in the money
Probability distribution for the call value is rectangular

43. Value of put

44. Relationships between option values and stock values
shift of stock price distribution to right
Now the stock value ranges from 20 to 60 (an increase in average stock price
E(VS ) = An increase
E(VC ) = An increase
E(VP ) = A decrease

45. Effect of changes in variance on option values
expansion of the range of possible stock prices
But same expected return of stock
This will isolate the effect of changes in variance
New range 10 to 70
From the text example
E(VS ) = An increase
E(VC ) = An increase
E(VP ) = An increase

46. Conclusions about prices of options
1) as stock price inc
Value of call increases & put decreases
2) as volatility of stock price increases
increases the value of calls and puts
3) given change in variance, $ price change of out of money options greater than in the money
4) options more volatile in percent terms than stocks
5) out of money more volatile than in the money

47. Binomial Option Pricing: Text Example
120 10
100 C 90
Call Option Value
X = 110
Stock Price

48. Two state call option pricing call X = 110 Price stock = 100
Two rates of return on the stock are possible
-10% & +20%
The price can then be $90 or 120$
RF = 10%
Hedge an investment
buy one and sell other
Say buy stock and sell options
How many options to sell
spread for options
if Stock = call = 10
Stock = = 0
calc end value of portfolio
own 1 share and sold 3 calls
If stock goes to 90, have 1 share worth 90
Owners of options will not exercise
Value of portfolio = 90
If stock goes 120, have 1 share worth 120
But owners of the calls want to exercise
They give you 110 each = 330
But you must buy the stock at 120
Must spend and extra $30
Net value of portfolio – (360 – 330)
= 90
Portfolio = 90 regardless of stock price
Implies a RF return

49. Implies a RF return if RF = 10%
Solve for option price that would give a RF return
Could do the same thing but reverse the transaction buy calls & short the stock
(Note no investment on your part)
If options sell for anything other than 6.06 then get an arbitrage situation
Market conditions will force the option price at 6.06
Binomial Option Model

50. Two state put option pricing for a single period
since puts and stock price move opposite a hedge here would include a sale of both or a purchase of both
Use formula for # of calls to buy for the equivalent # for puts
again eventually solve for put option price
example need 2 stocks for 3 puts
If stock goes to 120
Have two shares worth 240 & 3 worthless puts
Total value = 240
If stock goes to 90
Have two shares of stock worth 180
Three puts worth 20 each = 60
Use formula from above to solve for the value of the puts that will give Rp = RF = 10%

51. Expanding to Consider Three Intervals
Assume that we can break the year into three intervals.
For each interval the stock could increase by 20% or decrease by 10%.
Assume the stock is initially selling at $100.

52. Expanding to Consider Three Intervals

53. Black-Scholes Option Valuation
Co = SoN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)
d2 = d1 - (T1/2)
where
Co = Current call option value
So = Current stock price
N(d) = probability that a random draw from a normal distribution will be less than d

54. Black-Scholes Option Valuation
X = Exercise price
e = , the base of the natural log
r = Risk-free interest rate (annualized, continuously compounded with the same maturity as the option)
T = time to maturity of the option in years
ln = Natural log function
Standard deviation of the stock

55. Example 21.1 Black-Scholes Valuation
So = 100 X = r = .10 T = .25
= .50 (50% per year)
Thus:
Black and Scholes

56. Call Option Value Implied Volatility
Implied volatility is volatility for the stock implied by the option price.
Using Black-Scholes and the actual price of the option, solve for volatility.
Is the implied volatility consistent with the stock?

57. Black-Scholes Model with Dividends
The Black Scholes call option formula applies to stocks that do not pay dividends.
What if dividends ARE paid?
One approach is to replace the stock price with a dividend adjusted stock price
Replace S0 with S0 - PV (Dividends)

58. Example 21.3 Black-Scholes Put Valuation
P = Xe-rT [1-N(d2)] - S0 [1-N(d1)]
Using Example 21.2 data:
S = 100, r = .10, X = 95, σ = .5, T = .25
We compute:
$95e-10x.25( )-$100( ) = $6.35

59. Put Option Valuation: Using Put-Call Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
P = e -.10 X
P = $6.35

60. 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 option’s value given a 1% change in the value of the underlying stock

61. Portfolio Insurance
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

62. Hedging and Delta The appropriate hedge will depend on the delta.
Delta is the change in the value of the option relative to the change in the value of the stock, or the slope of the option pricing curve.
Change in the value of the option
Change of the value of the stock
Delta =

63. Example 21.6 Speculating on Mispriced Options
Implied volatility = 33%
Investor’s estimate of true volatility = 35%
Option maturity = 60 days
Put price P = $4.495
Exercise price and stock price = $90
Risk-free rate = 4%
Delta = -.453

64. Table 21.3 Profit on a Hedged Put Portfolio

65. Example 21.6 Conclusions
As the stock price changes, so do the deltas used to calculate the hedge ratio.
Gamma = sensitivity of the delta to the stock price.
Gamma is similar to bond convexity.
The hedge ratio will change with market conditions.
Rebalancing is necessary.

66. Delta Neutral
When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral.
The portfolio does not change value when the stock price fluctuates.

67. Table 21.4 Profits on Delta-Neutral Options Portfolio

68. Empirical Evidence on Option Pricing
The Black-Scholes formula performs worst for options on stocks with high dividend payouts.
The implied volatility of all options on a given stock with the same expiration date should be equal.
Empirical test show that implied volatility actually falls as exercise price increases.
This may be due to fears of a market crash.

69. CHAPTER 22
Futures Markets

70. Futures and Forwards
Forward – a deferred-delivery sale of an asset with the sales price agreed on now.
Futures - similar to forward but feature formalized and standardized contracts.
Key difference in futures
Standardized contracts create liquidity
Marked to market
Exchange mitigates credit risk

71. Basics of Futures Contracts
A futures contract is the obligation to make or take delivery of the underlying asset at a predetermined price.
Futures price – the price for the underlying asset is determined today, but settlement is on a future date.
The futures contract specifies the quantity and quality of the underlying asset and how it will be delivered.

72. Basics of Futures Contracts
Long – a commitment to purchase the commodity on the delivery date.
Short – a commitment to sell the commodity on the delivery date.
Futures are traded on margin.
At the time the contract is entered into, no money changes hands.

73. Basics of Futures Contracts
Profit to long = Spot price at maturity - Original futures price
Profit to short = Original futures price - Spot price at maturity
The futures contract is a zero-sum game, which means gains and losses net out to zero.

74. Figure 22.2 Profits to Buyers and Sellers of Futures and Option Contracts

75. Figure 22.2 Conclusions
Profit is zero when the ultimate spot price, PT equals the initial futures price, F0 .
Unlike a call option, the payoff to the long position can be negative because the futures trader cannot walk away from the contract if it is not profitable.

76. Existing Contracts
Futures contracts are traded on a wide variety of assets in four main categories:
Agricultural commodities
Metals and minerals
Foreign currencies
Financial futures

77. Trading Mechanics
Electronic trading has mostly displaced floor trading.
CBOT and CME merged in 2007 to form CME Group.
The exchange acts as a clearing house and counterparty to both sides of the trade.
The net position of the clearing house is zero.

78. Trading Mechanics
Open interest is the number of contracts outstanding.
If you are currently long, you simply instruct your broker to enter the short side of a contract to close out your position.
Most futures contracts are closed out by reversing trades.
Only 1-3% of contracts result in actual delivery of the underlying commodity.

79. Margin and Marking to Market
Marking to Market - each day the profits or losses from the new futures price are paid over or subtracted from the account
Convergence of Price - as maturity approaches the spot and futures price converge

80. Margin and Trading Arrangements
Initial Margin - funds or interest-earning securities deposited to provide capital to absorb losses
Maintenance margin - an established value below which a trader’s margin may not fall
Margin call - when the maintenance margin is reached, broker will ask for additional margin funds

81. Trading Strategies Speculators Hedgers
seek to profit from price movement
short - believe price will fall
long - believe price will rise
seek protection from price movement
long hedge - protecting against a rise in purchase price
short hedge - protecting against a fall in selling price

82. Basis and Basis Risk
Basis - the difference between the futures price and the spot price, FT – PT
The convergence property says FT – PT= 0 at maturity.

83. Basis and Basis Risk
Before maturity, FT may differ substantially from the current spot price.
Basis Risk - variability in the basis means that gains and losses on the contract and the asset may not perfectly offset if liquidated before maturity.

84. Futures Pricing
Spot-futures parity theorem - two ways to acquire an asset for some date in the future:
Purchase it now and store it
Take a long position in futures
These two strategies must have the same market determined costs

85. Spot-Futures Parity Theorem
With a perfect hedge, the futures payoff is certain -- there is no risk.
A perfect hedge should earn the riskless rate of return.
This relationship can be used to develop the futures pricing relationship.

86. Hedge Example: Section 22.4
Investor holds $1000 in a mutual fund indexed to the S&P 500.
Assume dividends of $20 will be paid on the index fund at the end of the year.
A futures contract with delivery in one year is available for $1,010.
The investor hedges by selling or shorting one contract .

87. Hedge Example Outcomes
Value of ST , ,030
Payoff on Short
(1,010 - ST)
Dividend Income
Total 1, , ,030

88. Rate of Return for the Hedge

89. The Spot-Futures Parity Theorem
Rearranging terms

90. Arbitrage Possibilities
If spot-futures parity is not observed, then arbitrage is possible.
If the futures price is too high, short the futures and acquire the stock by borrowing the money at the risk free rate.
If the futures price is too low, go long futures, short the stock and invest the proceeds at the risk free rate.

91. Spread Pricing: Parity for Spreads

92. Spreads
If the risk-free rate is greater than the dividend yield (rf > d), then the futures price will be higher on longer maturity contracts.
If rf < d, longer maturity futures prices will be lower.
For futures contracts on commodities that pay no dividend, d=0, F must increase as time to maturity increases.

93. Futures Prices vs. Expected Spot Prices
Expectations
Futures prices equals the future spot
Normal Backwardation
Futures price is bid down and will rise toward to a point where futures price = future spot price
Contango
Demand for the contract drives up the price, so over time the futures price will fall to the future spot price.

94. Figure 22.7 Futures Price Over Time, Special Case