Computational Finance

1
Computational Finance

• Lecture 4
• Options Theory
• Part I Basics

2
Options

• Options give the holders a right to buy or sell
  the underlying asset by a certain date for a
  certain price.
• The difference of forward contracts and options
• Forward contracts the obligation
• Options the right

3
Call and Put Options

• Call options buy
• Put options sell
• The price in the contract is known as the
  exercise price or strike price the date is known
  as the expiration date or maturity.

4
Option Positions

• There are two sides to every option contract
• Long position buying the option
• Short position selling (or writing) the
  option
• Four types of option positions
• Long a call Short a call
• Long a put Short a put

5
European and American Options

• European-style options only can be exercised on
  the date of the maturity of the contract.
• American-style options can be exercised at any
  time up to the maturity of the contract.

6
Example

• As an example, consider an option on Intels
  stock. Suppose that the strike price is 20 per
  share and the maturity is May 13, 2009.
• If this is a European call option, the long
  position is entitled a right to buy Intel shares
  at the price of 20 per share on May 13, 2009.
• If this is an American call option, the long
  position has a right to buy Intel shares at 20
  per share any time before May 13, 2009.

7
Payoffs of Long Position in Call Options

• Suppose that Intel stock price turns out to be
  25 per share on May 13, 2009.
• The long position buys shares at the price of 20
  per share by exercising the option. He/She buys
  shares at lower price than the spot price. The
  gain he/she realizes is 25-20 5 per share.

8
Payoffs of Long Position in Call Options

• Suppose that Intel stock price turns out to be
  15 per share on May 13, 2009.
• The contract charges a higher price than the spot
  market. Of course, the holder would not like to
  exercise it.
• Options are rights. The holders are not required
  to exercise them if they do not want to. The
  contract will be left to mature without
  exercising.

9
Payoffs of Long Position in Call Options

• In general, suppose that the strike price is ,
  and the underlying asset price at the maturity is
  . Then, the payoff of the long position of the
  option should be

10
Diagram of Payoffs for Longing a Call

• Payoff Call
  Options
• 
  K Stock Price

11
Payoffs for Shorting a Call

• The writer of a call option has liability to
  satisfy the requirement of the long position if
  he/she asks to exercise options.
• In the previous example,
• If 25 per share, the option is exercised.
  The writer loses 5 per share.
• If 15 per share, the option is not
  exercised.

12
Diagram of Payoffs for Short Positions

• Payoff Call Options
• 
  K Stock Price

13
Options Premium (Option Price)

• The long position of an option always receive
  non-negative payoffs in the future while the
  writer always has non-positive payoffs.
• The long position must give a compensation to the
  writer of an options. The compensation is known
  as the options premiums or options prices.

14
Uses of Options

• Like other derivatives, options provide us two
  major uses
• Risk hedge
• Consider an investor who in Feb 13 owns 1,000
  Microsoft shares. The current stock price is 20.
  The investor is concerned that the share price
  might decline further in the next 3 months. The
  investor could buy 3-month put options on
  Microsoft with a strike price of 19, which
  grants a right to sell the shares at 19.

15
Uses of Options

• Options also can be used as a tool of speculation
• Consider an investor who is interested in
  Microsoft shares. He believes that there will be
  a strong surge in the price. Investing in
  Microsoft options is an attractive approach. Say,
  buy a call option with the strike price of 20.
  When the share price increases as he predicts, he
  make a profit by the difference of the spot price
  and the strike price. Meanwhile, he do not need
  to own stocks physically and the price of options
  is cheap.

16
Intrinsic Value of Options

• The values of an option contract comes from the
  potential positive payoffs. For example, on the
  expiration date the price of a call option is
• We apply this value at any time and call it as
  the intrinsic value of a call option.
• A call option is in the money, at the money and
  out of the money, depending on , ,
  

17
Time Value of Options

• Even out-of-the-money options have values at
  earlier times, since they provide the potential
  for future exercise.

18
Time Value of Options

• Suppose that you hold a European call option on a
  stock. It expires in 1 year. Todays stock price
  is 50 per share, and the strike price is 65 per
  share. Then, the intrinsic value is
• But the option has value today because there is a
  chance that the stock price might increase to
  over 65 within 1 year.

19
Other Factors Affecting Option PricesVolatility

• Example Call option with strike price 30. Two
  stocks, A and B. A is more volatile and B is more
  placid.
• A
• Price at maturity 10 20 30
  40 50
• Payoffs 0 0 0
  10 20
• B
• Price at maturity 20 25 30
  35 40
• Payoffs 0 0 0
  5 10

20
Other Factors Affecting Option PricesVolatility

• The values of options should increase with
  volatility. The payoff of options are related
  with extreme events of underlying. So, when the
  underlying asset becomes more volatile, the
  options should be more valued.

21
Other Factors Affecting Option PricesRisk Free
Interest Rates

• Risk free interest rates
• The interest rates in the economy affect the
  stock market. Usually stock prices tend to fall
  when interest rates rise.
• On the other hand, the present values of
  exercise prices decrease when interest rates
  rise.

22
Options Strategies

• It is common to invest in combination of options
  in order to implement special hedging or
  speculative strategies.
• Bull/Bear spreads
• Straddles
• Butterfly spreads

23
Options Strategies

• Bull spread
• A bull spread is designed to profit from a
  moderate rise in the price of the underlying
  security.
• For instance, we construct it by buying a call
  option with a low exercise price, and selling
  another call option with a higher exercise price.
  

24
Options Strategies

• Bull spread
• Suppose that I buy a call option with strike
  price 100 and write another with strike price
  120. Both of them have the same maturity and on
  the same underlying.

25
Diagram of Payoffs for Bull Spreads

• Payoff
• 20
• 
  100 120 Stock Price

26
Options Strategies

• Bear spread
• A bear spread is used when the options trader
  is moderately bearish on the underlying asset.

27
Options Strategies

• Bear spread
• Suppose that I write a put option with strike
  price 100 and buy another with strike price
  120. Both of them have the same maturity.

28
Diagram of Payoffs for Bull Spreads

• Payoff
• 20
• 
  100 120 Stock Price

29
Options Strategies

• Straddles
• A straddle consists of two options a call and
  a put. Both of them have the same maturity and
  the same strike price.
• For example, you buy a call and a put with
  strike price 100.

30
Diagram of Payoffs for Straddles

• Payoff
• Such straddle makes
• a profit when the
• underlying moves a
• long way from the strike
  
• price.
• 
  100 Stock Price

31
Options Strategies

• Short Straddles
• Option traders also are able to construct
  straddles to make profit for placid market
  movements. For instance, sell both a call and a
  put with strike price 100.

32
Diagram of Payoffs for Straddles

• Payoff
• Such straddle makes
• a profit when the
• underlying moves little.
• 
  100 Stock Price

33
Options Strategies

• Butterfly spread
• Buying a call with strike of 90, writing two
  calls stuck at 100 and buying a 110 call.

34
Diagram of Payoffs for Butterfly Spreads

• Payoff
• 90
  100 110 Stock Price

35
Risky Option Strategies

• No matter what kinds of option combination you
  are taking, you just stake a possible market
  movement. It could be very risky when operating
  improperly.

36
Barings Bank Collapse and Option Trading

• In 1995, Barings Bank with more than 230 years
  operating history collapsed. The culprit was a
  derivative trader, Nick Leeson, who incurred a
  loss of US1.4 billion.
• Nick Leeson initially caused a huge loss in
  futures trading. To recoup it, he placed a short
  straddle on the Nikkei Index on Jan. 1995.

37
Barings Bank Collapse and Option Trading

• Essentially he bet that the Japanese market would
  not move very significantly.
• However, an earthquake hit the Kobe city early in
  the morning on January 17 and sent Asia markets
  into tumbling. It failed Nicks dream to recover
  his losses and broke the back of Barings bank.
  

38
Barings Bank Collapse and Option Trading

• Of course, we can blame everything on Nick
  Leesons bad luck.
• But Barings bank also should take some
  responsibility. For instance, Nick could do large
  amount of speculative trading without any
  authorization. The bank's own deficient internal
  auditing and risk management practices also
  should be blamed.

39
Put-Call Parity

• For European options there is a simple
  theoretical relationship between the prices of
  corresponding puts and calls put-call parity.
• We can derive this relationship by the
  combination of options.

40
Put-Call Parity

• Imagine you buy one European call option with
  strike price and maturity date . Meanwhile,
  you write a European put on the same underlying
  stock with the same strike price and the same
  maturity date.
• What is the payoff for you at the maturity?

41
Put-Call Parity

• The payoff is
• It is the same as the payoff of a forward
  contract with the delivery price
• Recall the value of such forward contract should
  be
• Put-call parity

42
American Options and Early Exercises

• American options owners are entitled the right to
  exercise the options at any time up to the
  maturities of the options.
• The main point of interest with American options
  is of course deciding when to exercise.
• Bermuda options options are allowed to be
  exercised on specified dates before the maturity.