1. Options
Chapter 2.5
Chapter 15

2. Learning Objectives
Understand key terms related to options and options markets
Compute payoffs and profits to option holders and writers
Calculate potential profits from various options strategies
Describe the put-call parity relationship

3. Derivative
A derivative is a security who’s value is dependent on another assets
Base Asset Ex: commodity prices, bond and stock prices, or market index values
Derivatives are contingent claims
Their payoffs depend on the value of another securities.
Options are a specific type of derivative
The holder has the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at a price agreed upon today.

4. Options: Calls and Puts
Call: The owner of a call has the right, but not the obligation, to BUY an asset in the future at the strike or exercise price
Value decreases as the strike price increases
Put: The owner of a put has the right, but not the obligation, to SELL an asset in the future at the strike or exercise price
Value increases with the strike price
Value of both calls and puts increases with time to expiration, WHY?

5. Rights and Obligations
Buyer:
Gets all the Rights
Seller:
Gets all the Obligations
Calls
Right to Buy the asset
Is obliged to Sell the asset
Puts
Right to Sell the asset
Is obliged to Buy the asset

6. Futures Contracts
An agreement made today regarding the delivery of an asset (or in some cases, its cash value) at a specified delivery or maturity date for an agreed-upon price (futures price) to be paid at contract maturity
Long position: Take delivery at maturity
Short position: Make delivery at maturity

7. Comparison Option Futures Contract
Right, but not obligation, to buy or sell; option is exercised only when it is profitable
Options must be purchased
Both sides have an obligation
Long position must buy at the futures price
Short position must sell at futures price
Futures contracts are entered into without cost

8. The Option Contract
The purchase price of the option is called the premium.
Stock options cover 100 shares & premium is on a per share basis
Sellers (or Writers) of options receive premium.
If the Holder (or Buyer) exercises the option, the Writer must deliver (call) or take delivery (put) of the underlying asset.

9. Options Terminology Exercising the Option
Using the option
Strike Price or Exercise Price
The price specified by the option
Spot Price
The market price
Expiration Date
The option’s maturity date
European & American options
Europeans can only be exercised at expiration.
Americans can be exercised at any time up to expiration. BUT NEVER ARE

10. Options Terminology
In-the-Money: Exercising the option results in a profit
Call: exercise price < market price
Put: exercise price > market price
At-the-Money: Exercising the option results in 0 profit
exercise price = market price
Out-of-the-Money: Exercising the option results in a loss
Call: market price < exercise price
Put: market price > exercise price

11. Call Payoff (Intrinsic Value)
Call Payoffs
Holder of the option
Pays the premium at time=0
Has the right to exercise the option at time=T
Notation
Stock Price at T = ST
T is maturity, t is any other time
Exercise Price = X
Don’t Exercise
Exercise
Call Payoff (Intrinsic Value)
ST - X

12. Call Holder Payoff and Profit
Value of the Call at T: CT = Max [ST – X, 0]
What is the value of the Call at t?
ST >X
ST < X CT
ST – X
Profit
ST - X - Ct
-Ct

13. Payoff and Profit to Call Option at Expiration
What is the strike price?
What is the premium?

14. What is the value of a Call at t?
At maturity CT = Max [ST – X, 0]
Would you be willing to sell for St-X at t?
Hint: What could happened to the stock price?
Time value is the premium a rational investor will pay above an options intrinsic value
This base on the likelihood that the stock price will move making the option more valuable
Time Value = Option Price – Intrinsic Value

15. Call Writer Payoff and Profit
ST >X
ST < X
Exercise Cost
-(ST – X)
Profit
-(ST - X) + Ct Ct

16. Payoff and Profit to Call Writers at Expiration

17. Calls: A Zero Sum Game Call Option If ST < X If ST > X Decision
No exercise
Exercise
Option Payoff (holder)
ST – X
Option Profit (holder) -C
(ST – X) – C
Option Payoff (writer)
- (ST – X)
Option Profit (writer) +C
C - (ST – X)

18. Profit and Loss on a Call
A February 2013 call on IBM with an exercise price of $195 was selling on January 18, 2013, for $3.65.
The option expires on the third Friday of the month, or February 15, 2013.
If the price of IBM on Feb 15, 2013 is $194, what is the call worth?

19. Profit and Loss on a Call (cont.)
Suppose IBM sells for $197 at expiration
Remember: strike = $195, premium = $3.65
What is the value of the option?
Call Intrinsic value = stock price-exercise price
Will the option be exercised?
What is the Profit/Loss on this investment?
Profit = Final value – Original investment
What must the price of IBM be for the option to break-even?

20. Call Option Problem
At time=0 you buy a call option on IBM for $ The option gives you the right to buy 100 shares of IBM stock at time=T at $65
What is the payoff to you if ST = $70?
What is the payoff for the writer if ST = $70?
What is the payoff to you if ST = $60?
What is the payoff for the writer if ST = $60?

21. Puts Gives holder the right (but not the obligation) to sell an asset:
At the exercise or strike price
On or before the expiration date
Exercise the option to sell the underlying asset if market value < strike.

22. Put Payoff (Intrinsic Value)
Puts Payoffs
Holder of the option
Pays the premium at time=0
Has the right to exercise the option at time=T
Notation
Stock Price at T = ST
Exercise Price = X
Don’t Exercise
Exercise
Put Payoff (Intrinsic Value)
X - ST

23. Put Holder Payoff and Profit
Value of the Call at T: PT = Max [X - ST , 0]
ST <X
ST > X
Payoff
X - ST
Profit
X - ST - Pt
-Pt

24. Payoff and Profit to Put Option at Expiration

25. Put Writer Payoff and Profit
ST <X
ST > X
Exercise Cost
-(X - ST)
Profit
-(X - ST) + Pt Pt

26. Puts: Another Zero Sum Game
Put Option
ST < X
If ST > X
Decision
Exercise
No Exercise
Option Payoff (holder)
X – ST
Option Profit (holder)
(X-ST) - P -P
Option Payoff (writer)
- (X-ST)
Option Profit (writer)
+P - (X-ST) +P

27. Profit and Loss on a Put
Consider a February 2013 put on IBM with an exercise price of $195, selling on January 18, for $5.00.
Option holder can sell a share of IBM for $195 at any time until February 15.
If IBM sells for $196, what is the put worth?

28. Profit and Loss on a Put Suppose IBM’s price at expiration is $188.
What is the value of the option?
Put value = Exercise price- Stock price
Will the option be exercised?
What is the Profit/Loss on this investment?
Profit = Final value – Original investment
What is the HPR?

29. Put Option Problem
At time=0 you buy a put option on ITT stock for $ The option gives you the right to sell 100 shares of ITT stock at time=T at $50
What is the payoff to you if ST = $55?
What is the payoff to the put seller if ST = $55?
What is the payoff to you if ST = $45?
What is the payoff to the put seller if ST = $45?

30. 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.

31. Option versus Stock Investment
Investment Strategy Investment
Equity only Buy $100 (100 shares) $10,000
Options only Buy $10 (1,000 options) $10,000
Options + Buy $10 (100 options) $1,000
T-Bills Buy 3% Yield $9,000

32. Strategy Payoffs

33. Option Versus Stock Investment

34. Strategy Conclusions
The all-option portfolio, B, responds more than proportionately to changes in stock value; it is levered.
Portfolio C, T-bills plus calls, shows the insurance value of options.
C ‘s T-bill position cannot be worth less than $9270.
Some return potential is sacrificed to limit downside risk.

35. Combining Options
Puts and calls can serve as the building blocks for more complex option contracts
Can be used to manage risk
This is financial engineering
Allows you to tailor the risk-return profiles to meet your client’s desires
Ex: Protective Puts: Underlying asset and put are combined to guarantee a minimum valuation
Put is insurance against stock price declines.

36. Protective Put at Expiration

37. 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.”

38. Covered Call Position at Expiration

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

40. Straddle Position at Expiration

41. Spread
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 (written)
A bullish spread is a way to profit from moderate stock price increases
EX. Buy a call with a strike of $20 and sell a call with a strike price of $25

42. Bullish Spread Position at Expiration

43. Put Call Parity P –C = PV (X) - St
This is the relation between a put and call with the same exercise price (E) and maturity
It comes from replicating portfolios:
The payoffs from buying a call and selling a put is the same as the payoffs from buying the stock and borrowing the PV of the exercise price
P –C = PV (X) - St

44. Payoff-Pattern of Long Call–Short Put Position

45. Portfolios Portfolio 1: Buy a call and Write a put
Portfolio 2: Buy the stock but Borrow PV (X)
Levered Equity positions
If the payoffs are the same the price must be the same
-C+P = -S0 + Xe-rT
C-P = S0 – Xe-rT

46. Proof by Counter Example
Assume that:
Stock Price = Call Price = 14
Put Price = Risk Free = 5%
Maturity = 6 months Strike Price = 105
Portfolio 1 costs
= -9
Portfolio 2 costs:
e-5 = -7.59
Two different costs for the sample payoffs → ABRITRAGE

47. Arbitrage Strategy Payoff

48. Put Call Parity Example
What is the value of a put with an exercise price of $51, if the stock is currently trading at $49. The price of the corresponding call option is $4.65. According to put-call parity, if the effective annual risk-free rate of interest is 4% and there are three months until expiration, what should be the value of the put?
FYI: It doesn’t matter if compounding is monthly or quarterly