Part

1

• Part V
• Essentials of Options

2
A Quick Recap

• Options are by design different from forward and
  futures contracts.
• The buyer of the options contract is called the
  Holder or the Long, and he has a right.
• The seller of the contract is called the Writer
  or the Short and he has an obligation.

3
Recap (Cont)

• Call Options give the holder the right to buy the
  underlying asset at a pre-specified price.
• Put Options give the holder the right to sell the
  underlying asset at a pre-specified price.
• All option contracts have a specified expiration
  date after which they become null and void.

4
Recap (Cont)

• Options contract which can be exercised only at
  the time of expiration are called European
  options.
• Contracts which can be exercised at any time,
  upto and including the time of expiration, are
  called American options.
• Most exchange traded options are American.

5
Associated Terms

• The following terms are important in the context
  of options.
• Option Price or Premium
• Strike Price or Exercise Price
• Expiration Date or Exercise Date or Strike Date
  or Maturity Date

6
Price or Premium

• This is the cost of acquisition of the option.
• It is payable by the buyer to the writer at the
  outset.
• Thus unlike in the case of a forward or a futures
  contract, the long has to pay the short to get
  into an options contract.

7
Price or Premium (Cont)

• The difference is because in the case of a
  forward/futures contract, both the parties have
  an equivalent obligation.
• In the case of an options contract however, one
  party is acquiring a right from the other.
• And, no one will give away a right for free.

8
Strike Price or Exercise Price

• This is the price payable per unit of the
  underlying asset, if a call option is exercised
  by the holder.
• It is the price receivable per unit of the
  underlying asset, if a put option is exercised by
  the holder.

9
Exercise Price (Cont)

• Thus when the buyer of an options contract pays
  the option premium, he merely acquires the right
  to transact.
• If he subsequently decides to go through with the
  transaction, he must pay to acquire the
  underlying asset in the case of call options.

10
Exercise Price (Cont)

• Or else he must be paid when he delivers the
  underlying asset in the case of put options.

11
Expiration Date

• This is the point in time after which the
  contract becomes null and void.
• It is the only point in time at which a European
  option can be exercised.
• It is the last point in time at which an American
  option can be exercised

12
Example of a Call Option

• Consider European calls on Reliance expiring on
  the last Thursday of September.
• Let the exercise price be Rs 400.
• Let the option premium be Rs 15.
• Option premia are always quoted on a per share
  basis.

13
Example (Cont)

• The contract size, which is the number of shares
  of stock underlying the contract is 100 shares in
  the U.S., irrespective of the company on whose
  shares the contract is written.
• In India the contract size varies from company to
  company.

14
Example (Cont)

• In the case of Reliance, the contract size is 600
  shares.
• Thus the buyer has to pay 15 x 600
• Rs 9000 to the writer at the outset.
• This is a sunk cost and cannot be recovered.
• In exchange the buyer acquires the right to buy
  600 shares at the time of expiration at a price
  of Rs 400 per share.

15
Example (Cont)

• What will happen at expiration?
• If the stock price is greater than Rs
  400, then the option will be exercised.
• This is because it is worth paying Rs 400 for an
  asset that is selling for more than Rs 400.
• Otherwise the option will simply be allowed to
  expire worthless.

16
Example (Cont)

• For instance, why pay Rs 400 for an asset that is
  selling at say Rs 395.
• Remember that since an option is a right, the
  holder cannot be forced to exercise.
• Notice that the spot price at expiration need not
  be greater than the sum of the exercise price and
  the premium, in order to trigger off exercise.

17
Example (Cont)

• That is, the terminal stock price need not exceed
  Rs 400 Rs 15 Rs 415, before the holder opts
  to exercise.
• This is because sunk costs are irrelevant while
  taking investment decisions.

18
The Irrelevance of Sunk Costs

• Assume that the terminal stock price is
• Rs 405.
• If the option is exercised the profit is
• ? 600(405 400) 9000 (6000)
• If the option is not exercised
• ? (9000)
• Obviously it is better to lose Rs 6000.

19
The Case of Puts

• If the options had been puts instead of calls,
  then the holder would exercise only if the spot
  price at expiration were to be less than Rs 400.
• Obviously, it is attractive to sell the stock for
  Rs 400, when the prevailing market price is less
  than Rs 400.

20
Puts (Cont)

• Otherwise it is best to allow the options to
  expire worthless.
• For instance if the spot price is Rs 405, why
  should the option holder deliver under the
  contract for Rs 400.

21
Profit Bounds

• For a call holder the maximum profit is
  unlimited, since theoretically, there is no upper
  bound on the price of the asset.
• Thus if the call is exercised
• p (ST X) C, which has no upper bound.
• ST is the stock price, X is the exercise price
  and C is the premium.

22
Profit Bounds (Cont)

• If the call is not exercised
• p -C
• For a call writer the maximum profit is the
  option premium.
• This is because the best thing that can happen
  from his standpoint is that the holder does not
  exercise, and he consequently gets to retain the
  entire premium.

23
Profit Bounds (Cont)

• Thus if the call is not exercised
• p C.
• If the call were to be exercised the writer has
  to deliver a share, whose price is theoretically
  unbounded, at the exercise price. That is
• ? C (ST X)

24
Profit Bounds (Cont)

• Thus the maximum possible loss for a call writer
  is infinite.

25
Puts and Profits

• In the case of a put holder the profit is given
  by
• (X ST) P
• The maximum possible value is X P.
• This is because the lowest possible stock price
  is 0, since stocks have limited liability.
• The maximum possible loss is once again equal to
  the premium paid p -P

26
Puts and profits (Cont)

• For a put writer the maximum possible profit is
  the premium.
• This is because the best thing that can happen to
  him is that the option is not exercised.
• His loss if the put is exercised is
• p P (X ST) which has a lower bound of
• (P X) -(X-P)

27
Zero Sum Games

• Thus both calls and puts are Zero Sum Games.
• One mans profit is always another mans loss.

28
Payoffs and Profits

• Symbolically the payoff from an option for a call
  holder is
• Max0, ST X
• The profit is Max0, ST X C
• The payoff for a call writer is
• -Max0, ST X Min0, X ST
• The profit is Min0, X ST C

29
Payoffs and Profits (Cont)

• The payoff for a put holder is
• Max0, X ST
• The profit is
• Max0, X ST P
• The payoff for a put writer is
• Min0, ST - X
• The profit is Min0, ST - X P

30
Exchange Trade OTC Options

• Exchange traded options were introduced for the
  first time by the Chicago Board Options Exchange
  (CBOE) in 1973.
• Until then options were only traded Over the
  Counter.

31
Exchange Traded vs. OTC (Cont)

• OTC options are customized, in the sense that the
  exercise price, the expiration date, and the
  contract size are negotiated between the buyer
  and the seller.
• Exchange traded options are however standardized
  like futures contracts.
• That is the allowable exercise prices and
  expiration dates are specified by the exchange.

32
Exchange Traded vs. OTC (Cont)

• Individual buyers and sellers can incorporate any
  of the allowable exercise prices and expiration
  dates into their agreements, but cannot design
  their own contracts.
• The contract size too is specified by the
  exchange.

33
Exchange Traded Options (Cont)

• The advantage of standardization is that volumes
  tend to be high and transactions costs tend to be
  low.
• Secondly because of high volumes, these markets
  tend to be liquid.
• Besides standardized option contracts can be
  offset by taking counter-positions, without
  necessarily involving the original counter-party.

34
Counter-Positions

• Taking a counter-position means that if you have
  originally bought a call/put, you now sell an
  identical call/put.
• By identical we mean that the offsetting contract
  should be on the same asset, and have the same
  exercise price and time to expiration.

35
Counter-Positions (Cont)

• Similarly if you have sold a call/put, you would
  now have to buy an identical call/put in order to
  offset.

36
Illustration

• Aditi had bought an options contract on Reliance
  from Rakesh a week ago.
• The contract terms have specified an exercise
  price of Rs 350 and the contract is scheduled to
  expire at the end of June.
• Now assume that Aditi wants to get out of her
  position.

37
Illustration (Cont)

• All she has to do, is to find a person on the
  floor of the exchange who would like to go long
  in a contract on Reliance expiring in June, with
  an exercise price of 350.
• This person need not be Rakesh, the individual
  with whom she initially traded.

38
Standardization Offsetting

• Offsetting is easy when the contracts are
  standardized.
• In the case of customized contracts, there is an
  infinite number of exercise prices and expiration
  dates that can be specified, as a consequence of
  which the odds of finding a third party who is
  willing to transact as per the original contract
  are severely reduced.

39
Credit Risk

• In the case of exchange traded options, credit
  risk is minimized because there is a
  clearinghouse which becomes the effective buyer
  for every seller and the effective seller for
  every buyer.
• However, unlike in the case of a futures
  contract, the clearinghouse has to guarantee only
  the performance of the writer.

40
Credit Risk (Cont)

• This is because a performance guarantee is
  required only when a party has an obligation and
  not when he has a right.
• And remember both call and put holders have
  rights, as a consequence of which there is no
  fear of non-performance.

41
OTC Markets

• The OTC market is dominated by institutional
  investors.
• Contracts are entered into privately by large
  corporations, financial institutions, and
  sometimes even governments.
• When buying an OTC option you have to be either
  familiar with the creditworthiness of the writer
  or else seek a guarantee.

42
OTC Markets (Cont)

• Nevertheless OTC markets always carry an element
  of credit risk.
• They do offer certain advantages however.
• Firstly terms and conditions like expiration
  dates and exercise prices can be tailored to the
  specific needs of the two parties.

43
OTC Markets (Cont)

• Often the contract may be on an asset on which an
  exchange traded contract is not available.
• Since the market is private, neither the public
  nor other investors need to know about the
  transaction taking place.
• However, seeking privacy need not mean that an
  illegal activity is taking place.

44
OTC Markets (Cont)

• The OTC market is unregulated.
• Consequently government approval is not required
  to design new types of contracts.

45
FLEX Options

• Their disadvantages not withstanding, customized
  contracts have an appeal particularly for
  institutional investors.
• For many institutions, exchange designed
  contracts are often inadequate and they desire
  their freedom to create their own contracts.

46
FLEX Options (Cont)

• Traditionally, an institution in need of a
  tailor-made contract has had to seek out another
  like minded institution like a commercial bank
  who is seeking to write an option with similar
  features.
• Of late, in response to competition the exchanges
  have been making an effort to grab a slice of the
  growing OTC market.

47
FLEX Options (Cont)

• To do this, they have created products known as
  FLEX options for stock indices and E-FLEX options
  for equity shares, where FLEX stands for FLexible
  EXchange.
• In order to trade in these options, an investor
  has to submit what is called a Request for Quote
  or RFQ.

48
RFQs

• The RFQ will contain the details of the contract
  sought by the investor, namely whether it is a
  call or a put, the exercise price, the time to
  maturity, and whether they want a European or an
  American style contract.
• The RFQ is then acted upon by market makers who
  submit quotes for the premium.

49
FLEX Options

• Both FLEX and E-FLEX options are cleared by the
  clearinghouse.

50
Major U.S. Equity Options Exchanges Contract
Volumes in Millions in 2001
Exchange Nickname Equities Options Volume Index Options Volume Quotation Symbol
Chicago Board CBOE 254 52 CO
American Amex 204 1 A
Pacific P-Coast 103 - P
Philadelphia Philly 96 5 X
Int. Securities ISE 65 - I
51
Other Global Options Exchanges
NAME LOCATION
BMF Sao Paulo
Paris Bourse Paris
EUREX Frankfurt
LIFFE London
Tokyo Stock Exchange Tokyo
52
Underlying Assets

• Equities The CBOE itself trades options on about
  1400 stocks.
• Indices Examples include DJIA, SP 100, and the
  SP 500
• Interest Rates
• Foreign Exchange

53
Moneyness

• Let us denote the current stock price by St and
  the exercise price by X.
• If St gt X, the call option is said to be in the
  money.
• Example St 110 X 100
• If St lt X the call option is said to be out of
  the money.
• Example St 90 X 100

54
Moneyness (Cont)

• If St X the call option is said to be at the
  money.
• Example St 100 X 100
• For put options, if St gt X, the option is said to
  be out of the money.
• Example St 110 X 100
• If St lt X, the put option is said to be in the
  money.

55
Moneyness (Cont)

• If St X, the put option is said to be at the
  money.
• If St is very close to X, both call and put
  options are said to be near the money.
• Obviously, an option, whether a call or a put
  will exercised only if it is in the money.

56
Expiration Dates

• Stock options contracts in the U.S expire on the
  Saturday following the third Friday of the
  expiration month.
• That is, if the first day of the month is a
  Saturday, then the contracts will expire on the
  fourth Saturday, else they will expire on the
  third Saturday.
• The last day of trading is the third Friday.

57
Expiration Dates (Cont)

• In India stock and index options expire on the
  last Thursday of the expiration month.
• If the last Thursday happens to be a market
  holiday, then the contracts will expire on the
  previous business day.

58
Available Expiration Months

• The methods used in the U.S. and in India are
  different from each other.
• In the U.S, a company on whose shares options are
  allowed for trading, is assigned at the outset to
  either a January, February, or a March cycle.
• The January cycle comprises of
• January, April, July, and October

59
Expiration Months (Cont)

• The February cycle comprises of
• February, May, August, and November.
• The March cycle comprises of
• March, June, September, and December.
• At any point in time, the available months for a
  stock will be the current month, the following
  month, and the next two months of the cycle to
  which it has been assigned.

60
Illustration

• Assume that today is 1 September 2002 and that
  XYZ corporation is assigned to a February cycle.
• Contracts will therefore be available for
  September 2002, October 2002, November 2002, and
  February 2003.

61
Illustration (Cont)

• September is the current month, October the
  following month, and November and February are
  the next two months from the February cycle.
• Once the September contracts expire, the
  available months will be
• October 2002, November 2002, February 2003, and
  May 2003.

62
LEAPS

• In addition both the CBOE and the Amex offer long
  term options with upto two years to maturity
  called Long Term Equity Anticipation Securities
  or LEAPS.

63
INDIA

• SEBI guidelines permit contracts with upto 12
  months to maturity.
• But currently we only have contracts with a
  maximum of three months to expiration.
• So we have contracts for the current month, and
  the following two months.

64
INDIA (Cont)

• For instance, on 1 September 2002 we will have
  the following contracts
• September 2002, October 2002, and November 2002.

65
Exercise Prices

• The exchange has to specify the allowable
  exercise prices.
• There will always be an at the money or near the
  money contract since these are of the maximum
  possible interest from the standpoints of both
  the longs as well as the shorts.

66
Exercise Prices (Cont)

• Consequently at the money contracts have the
  maximum trading volume.
• In addition there will be a number of in the
  money and out of the money contracts available at
  any point in time.
• The exchange in India guarantees that a minimum
  of 7 exercise prices will be provided for
  contracts with a given expiration date.

67
Exercise Prices (Cont)

• Three of these contracts will be in the money,
  three out of the money, and one at or near the
  money.
• The intervals between exercise prices would
  depend on the price of the underlying stock, and
  would be determined as per the following schedule.

68
Exercise Price Intervals
Stock Price Strike Price Interval
S lt Rs 50 Rs 2.50
Rs 50 lt S lt Rs 250 Rs 5
Rs 250 lt S lt Rs 500 Rs 10
Rs 500 lt S lt Rs 1000 Rs 20
Rs 1000 lt S lt Rs 2500 Rs 30
S gt Rs 2500 Rs 50
69
Illustration

• Assume that the January contracts on Reliance
  have just expired and that April contracts are
  being introduced.
• Let the prevailing share price be Rs 647.
• Since 647 is in between 250 and 500, the
  applicable strike price interval is Rs 20.

70
Illustration (Cont)

• In order to determine the at the money exercise
  price, the stock price will be rounded off to the
  nearest multiple of the strike price interval,
  which in this case is Rs 640.
• Thus contracts with an exercise price of 640,
  which represent near the money options, will be
  allowed for trading.

71
Illustration (Cont)

• The strike prices for the three in the money
  contracts and the three out of the money
  contracts will then be determined with reference
  to the at the money exercise price, in accordance
  with the prescribed strike price interval.

72
Illustration (Cont)

• Thus contracts with exercise prices of 580, 600,
  620, 660, 680, and 700 will be allowed for
  trading.
• Now assume that at the end of the day, the stock
  price is 695.
• The next morning, using the same logic as above,
  the exercise price for a near the money contract
  will be set at 700.

73
Illustration (Cont)

• With reference to this exercise price, four in
  the money contracts are already available.
• Thus three new exercise prices which correspond
  to three out of the money calls, namely 72, 740,
  and 760 will be allowed for trading.

74
Illustration (Cont)

• No matter how volatile the stock price may be
  during the day, new exercise prices will not be
  introduced during the course of trading on any
  day.
• Fresh exercise prices will be introduced as
  applicable only on the following morning.

75
The U.S. System

• The exercise prices in the U.S are determined as
  per the following schedule.

76
The U.S. System (Cont)
Stock Price Strike Price Interval
S lt 25 2.50
25 lt S lt 200 5
S gt 200 10
77
The U.S. System (Cont)

• For instance if a stock has a price of say
  21.5, and contracts with a new expiration month
  are being introduced, then to start with two
  exercise prices, namely, 22.50 and 20 will be
  allowed.
• If the price moves to 24 then automatically an
  exercise price of 25 will be permitted.

78
The U.S. System (Cont)

• Index options have exercise prices in intervals
  of 5.
• These rules are however flexible and can be
  modified by the exchange if in its opinion, such
  changes are necessary to attract larger trading
  volumes.

79
Exercise Prices

• So at any given point in time contracts with many
  different exercise prices will be trading for
  each of the expiration months.
• The number of different exercise prices that are
  observable at a point in time, would depend on
  the movement in the price of the underlying stock
  from the inception of trading in contracts for
  that expiration month.

80
Option Class

• All contracts on a given stock which are of the
  same type, that is calls or puts, are said to
  constitute an Option Class.
• For instance all the calls that are available on
  IBM at a point in time, irrespective of their
  strike price or the expiration date, would be
  said to constitute an Option Class.

81
Option Series

• All the contracts in a given class, that is, the
  Call Class or the Put Class, and which have the
  same exercise price and the same expiration date,
  are said to constitute an Options Series.
• Thus all call options contracts on XYZ stock with
  X 75 and expiring in June 2003 would constitute
  an Options series.

82
Exercising Options

• When an investor decides to exercise he has to
  inform his brokerage firm which will notify the
  clearing firm through whom the order was
  originally cleared.
• Of course the brokerage firm itself may be
  empowered to clear in certain cases.

83
Exercising (Cont)

• The clearing firm will then place an exercise
  order with the Options Clearing Corporation which
  is the major clearinghouse for options in the
  U.S.
• In India, clearing is undertaken by the National
  Securities Clearing Corporation (NSCCL).

84
Exercising (Cont)

• The clearinghouse will then randomly select a
  clearing firm through which someone has written
  the same option.
• The clearing firm thus chosen will then choose a
  particular writer who has written the option in
  question.
• Such a writer is said to be assigned.

85
Exercising (Cont)

• The procedure adopted by a clearing firm for the
  purpose of assigning, has to be established and
  made known to its customers in advance.
• In general, in the case of call options, the
  writer will deliver the stock, and will receive
  the exercise price from the holder.

86
Exercising (Cont)

• For puts, the holder will deliver the stock and
  will received the exercise price from the writer.

87
Cash Settlement

• Cash settlement is used for Index options
  globally.
• It has also been specified as the method of
  settlement to be adopted in India, till our
  markets achieve the desired level of maturity.
• Indices obviously cannot be delivered.

88
Cash Settlement (Cont)

• For an index represents a portfolio of stocks,
  which can be large (500 in the case of the SP
  500), weighted in particular proportions.
• So delivery of an index is technically feasible
  but practically difficult.

89
Cash Settlement (Cont)

• So if an index option is exercised the holder
  will receive the difference between the current
  index value and the exercise price, in the case
  of call options.
• In the case of puts, the holder will receive the
  difference between the exercise price and the
  current index level, from the writer.

90
Cash Settlement (Cont)

• In India this procedure is currently being
  followed for stock options as well.
• For instance assume that the stock price of
  Reliance is Rs 350, and that an investor is
  holding a call option with an exercise price of
  Rs 320.
• If he decides to exercise, he will receive
• 600 x (350 320) Rs 18,000.

91
Cash Settlement (Cont)

• No shares will change hands.
• Similarly if the current stock price is Rs 350
  and an investor is holding put options with an
  exercise price of Rs 375 on Reliance, he will
  receive
• 600 x (375 350) Rs 15,000
• were he to decide to exercise.

92
Arbitrage Free Conditions

• The relationships and conditions that we are
  going to demonstrate must be satisfied if
  arbitrage is to be ruled out.
• Violation of any of these conditions would
  tantamount to the presence of an arbitrage
  opportunity.

93
Non-Negative Premia

• The option price or premium cannot be negative.
• What would a negative premium imply?
• It would mean that the writer is prepared to pay
  the holder to buy the option.
• If so, the holder can acquire the option, pocket
  the payment, and simply forget about the contract.

94
Non-Negative Premia (Cont)

• The reason he can afford to be nonchalant is
  because he need not worry about the possibility
  of a subsequent cash outflow.
• This is because an option is a right and not an
  obligation and consequently the holder cannot be
  forced to exercise under adverse circumstances.

95
Properties of American Options

• We will use the symbol CA,t to denote the price
  of an American call option at time t.
• The stock price at that time will be denoted by
  St and the exercise price of the option by X.
• We can state that
• CA,t ? Max0,(St X)

96
American Options (Cont)

• Proof
• If (St X) lt 0, then all we can say is that
• CA,t ? 0, since the option premium cannot be
  negative.
• However, if (St X) gt 0, then
• CA,t ? St X
• To prove this let us assume the converse.

97
American Options (Cont)

• Assume that CA,t lt St X gt 0
• If so, an investor can buy an option and
  immediately exercise it.
• He will make a profit given by
• ? St X CA,t
• which is clearly an arbitrage profit, because it
  is costless and risk-less.

98
American Options (Cont)

• Similarly, if we denote the premium for an
  American put option at time t by PA,t then it can
  be demonstrated that
• PA,t ? Max0,(X St)
• Once again PA,t gt 0 if (X St) lt 0, because a
  put option cannot have a negative premium.
• However if (X St) gt 0, then the put premium
  must be greater than or equal to this.

99
American Options (Cont)

• Otherwise, an arbitrageur will simply buy the put
  option and immediately exercise it.

100
The Put-Call Parity Theorem

• What we are now going to demonstrate is a
  condition that is valid for European options on
  non-dividend paying stocks.
• By a non-dividend paying stock we mean a stock
  that will not pay a dividend during the life of
  the option.
• Analogous relationships can be derived for
  European options paying one or more dividends.

101
Put-Call Parity

• The relationship for American options is slightly
  different and will not be covered here.
• Consider the strategy depicted below and the
  corresponding cash flows.

102
(No Transcript)
103
Analysis

• Let us analyze the above table carefully.
• Such tables are very common in the course of
  study of options.
• The first column indicates the transactions that
  form components of the overall strategy.
• The second column indicates the cash flow
  associated with each transaction.

104
Analysis (Cont)

• All inflows will be positive and outflows will be
  negative.
• The third and fourth columns represent the
  situation at the time of expiration of the
  option.
• The key variable of interest is the stock price
  at expiration and its level with respect to the
  exercise price.

105
Analysis (Cont)

• There are therefore two possible situations.
• The stock price can either be less than the
  exercise price, or else it can be greater than
  it.
• In our case the overall cash flow at expiration
  is zero, irrespective of the level of the
  terminal stock price.

106
Analysis (Cont)

• Consequently to rule out arbitrage, the initial
  cash flow must be non-positive.
• Thus to rule out arbitrage we require that

107
Analysis (Cont)

• However if the LHS of the above equation were to
  be less than zero, then we can reverse the above
  strategy and make arbitrage profits as shown
  below.

108
(No Transcript)
109
Analysis (Cont)

• Hence to preclude arbitrage in either case, it
  must be true that

Or in other words
110
Analysis (Cont)

• This is the Put-Call parity relationship.
• It states that the difference between the price
  of a European call and a European put with the
  same exercise price and expiration date, will be
  equal to the difference between the stock price
  and the present value of the exercise price for
  non-dividend paying stocks.

111
Intrinsic Value Time Value

• The intrinsic value of an option is equal to the
  amount by which it is in the money, if it is in
  the money, else it is equal to zero.
• Therefore the intrinsic value of a call is
• Max0, (St X)
• While that of a put is
• Max0, (X St)

112
I.V T.V (Cont)

• The difference between an options premium and
  its intrinsic value is called the time value of
  the option, also known as the speculative value
  of the option.
• From our earlier analysis, it is obvious that
  both American calls and puts must be worth at
  least their intrinsic values.

113
I.V T.V (Cont)

• Thus American options will always have a
  non-negative time value.
• What about European options? From put-call parity
  we know that

114
I.V T.V (Cont)

• Look at the RHS.
• The value of the put option will always be
  greater than or equal to zero.
• The difference between the exercise price and its
  present value will also be non-negative.
• Thus if the option is in the money, its time
  value will be non-negative.

115
I.V and T.V (Cont)

• What if the option is out of the money?
• If so the entire premium is the time value by
  definition, which has to be non-negative since
  option premia cannot be negative.
• Thus European calls on non-dividend paying stocks
  will always have a non-negative time value.

116
I.V T.V (Cont)

• What about European Puts? From put-call parity

117
I.V T.V (Cont)

• Once again, if the put is out of the money, the
  entire premium is due to the time value, which
  consequently has to be non-negative.
• What if the option is in the money?
• If so, the intrinsic value will be positive.
• The call premium will be non-negative.
• The difference between the present value of X and
  X, will be non-positive.

118
I.V T.V (Cont)

• So whether or not the time value is negative or
  not would depend on which item in the expression,
  the call premium or the difference between the
  present value of X and X, is larger.
• Obviously, the lower the value of the call
  premium, the lower will be the time value.

119
I.V T.V (Cont)

• For a given exercise price, the lower the stock
  price the lower will be the call premium.
• Thus the more out of the money the call is, the
  lower will be the time value of the European put.

120
I.V T.V (Cont)

• Thus certain deep in the money European put
  options can have a negative time value.

121
Determining Option Values

• Pricing futures contracts was relatively easy.
• All that we had to do was to derive a pricing
  relationship that would preclude both cash and
  carry as well as reverse cash and carry arbitrage.

122
Option Values (Cont)

• This was feasible because a futures contract
  entails an obligation on the part of both the
  parties.
• Options however are more complex from a valuation
  standpoint.
• This is because the holder has a right and not an
  obligation.

123
Option Values (Cont)

• The attractiveness of the right in the case of a
  European option would depend on the holders
  perception of his being able to exercise the
  option at maturity, and the corresponding payoff.

124
Option Values (Cont)

• American options are considerably more complex
  because at every instant the decision has to be
  taken as to whether or not to exercise.
• Similarly from a writers standpoint, what is
  important is the possibility of the holder not
  exercising and of his consequently being able to
  retain the premium.

125
Option Values (Cont)

• Thus in the case of options, valuation entails
  the postulation of a process for the evolution of
  the stock price through time.
• Corresponding to every hypothesis about the price
  process, we will get a theoretical option premium.

126
Option Values (Cont)

• In certain cases, we will be able to derive
  precise mathematical formulae for the option
  price, or what we call closed-form solutions.
• In other cases we will have to make do with
  numerical approximations.

127
Variables Influencing the Option Premium

• The current stock price The is obviously a major
  factor in determining the option value.
• Everything else remaining the same, the higher
  the prevailing stock price, the greater will be
  the value of a call option and the lower will be
  the value of a put option.

128
Variables (Cont)

• The Exercise Price The higher the exercise
  price, for given values of other variables, the
  lower will be the value of the call option and
  the higher will be the value of the put option.
• Dividends The payment of a dividend will lead to
  a decline in the value of the stock price.

129
Variables (Cont)

• Thus dividends which are paid during the life of
  the option, will lead to a reduction in call
  values and an increase in put values.
• Exchange traded options are not payout protected
  from the standpoint of cash dividends. What this
  means is that the terms of the option contract
  will not be amended if the stock were to pay a
  dividend.

130
Variables (Cont)

• Volatility Modern financial theory is based on
  the assumption that all investors are risk
  averse.

131
Variables (Cont)

• Consequently, an increase in the volatility, as
  measured by variance of the rate of return of the
  asset, will be perceived negatively, and will
  lead to a greater risk premium being demanded,
  which will be manifested by a lower price.

132
Variables (Cont)

• Volatility however has a positive impact on the
  option price.
• Since the holder is protected on one side, his
  maximum loss is limited to the premium.

133
Variables (Cont)

• Thus an increase in volatility will be perceived
  positively, although it signals a greater
  probability of both high as well as low stock
  prices.
• This argument is valid for both call and put
  options.

134
Variables (Cont)

• Time to Maturity American calls and puts, and
  European calls on non-dividend paying stocks will
  always have a non-negative time value, whereas
  European puts on non-dividend paying stocks may
  have either a positive or a negative time value
  depending on the extent to which the option is in
  the money.

135
Variables (Cont)

• At expiration however, an option must have a zero
  time value.
• That is, at expiration, the option premium must
  be equal to its intrinsic value.
• We will prove this result by assuming the
  converse.
• That is, assume that the call option premium is
  greater than ST X gt 0.

136
Variables (Cont)

• If so, the arbitrageur will sell the call.
• Of course it will be exercised.
• But his cash flow, which is
• C (ST X) is by assumption guaranteed to be
  positive.
• This is a clear arbitrage profit.

137
Variables (Cont)

• What if the option is out of the money and C gt 0.
• If so an investor can sell the call and not worry
  about exercise, thereby assuring himself of an
  arbitrage profit.
• Thus all calls, whether American or European,
  must sell for their intrinsic values at
  expiration.

138
Variables (Cont)

• Thus, in general, keeping other variables
  constant, the value of an option will decline
  with time.
• We use the words in general, because American
  calls and puts and European calls have a
  non-negative time value prior to expiration,
  which must decline to approach zero at expiration.

139
Variables (Cont)

• However, in the case of certain deep in the money
  European puts, the time value may be negative
  before expiration, in which case it will increase
  so as to approach zero at expiration.
• In such cases the value of the option will
  increase with the passage of time.

140
Variables (Cont)

• It is for this reason, that options are called
  Wasting Assets.
• That is, in most cases, their values decline with
  the passage of time.
• The risk-less rate of interest To assess the
  impact of the risk-less rate, consider a person
  who is contemplating the purchase of a stock.

141
Variables (Cont)

• Before proceeding further we will demonstrate
  that the price of a call option can never exceed
  the prevailing stock price.

142
(No Transcript)
143
Strategy

• This table is slightly different from the ones
  that we have studied earlier.
• Although we still have two columns for the
  possible stock price at expiration, we have an
  additional column where we have considered the
  possibility of early exercise of the option.

144
Strategy (Cont)

• This is because we are seeking to prove a result
  for both European as well as American options.
• And in the case of American options, a result is
  valid only if we also take into account the
  possibility of the option being exercised early.

145
Strategy (Cont)

• As we can see,, once the strategy is put in
  place, there is no further possibility of a
  negative cash flow.
• Therefore to preclude arbitrage the initial cash
  flow must be less than zero.
• That is
• Ct St lt 0 or Ct lt St

146
Strategy (Cont)

• Thus a call option can never be worth more than
  the price of the stock on which it is written.
• Intuitively, it would be irrational to pay more
  than the stock price for the option, which merely
  gives the right to acquire the stock subsequently
  by paying an additional amount equal to the
  exercise price.

147
Strategy (Cont)

• Now let us consider the case of the investor who
  has an amount equal to the current stock price
  with him.
• One option, instead of buying the stock, would be
  to buy a call and invest the difference at the
  risk-less rate of interest.
• The higher the risk-less rate the more attractive
  will this option be.

148
Strategy (Cont)

• Thus the higher the risk-less rate, the greater
  will be the option premium.
• What about puts?
• Consider the case of a person who owns a stock
  and is contemplating its sale.
• One option would be to buy a put which will
  ensure a subsequent minimum selling price of X.

149
Strategy (Cont)

• Or else he could sell the asset immediately in
  the spot market and invest the proceeds at the
  risk-less rate of interest.
• The higher the risk less rate, everything else
  being the same, the more attractive will be the
  second option.
• Consequently the higher the risk-less rate, the
  lower will be the put value.

150
Margining

• Options like futures are highly levered
  instruments.
• Consequently they cannot be traded on the margin,
  that is, by borrowing a part of the amount
  required to pay the premium.
• Thus an option buyer must pay up the full premium
  upfront.

151
Margining (Cont)

• Options however impose a performance obligation
  on the writer, if the buyer were to exercise.
• Consequently writers are required to post
  performance guarantees or collateral.

152
Margining on U.S. Exchanges Other Than Futures
Exchanges

• The method adopted on such exchanges is called
  contract value margining.
• In the case of calls, it depends on whether the
  position is covered or naked.
• A naked call position is one where the writer has
  written a call without having the stock in his
  possession.

153
Margining (Cont)

• On the other hand, writers of covered calls
  already own the stock at the time of writing the
  option.

154
Margining for Naked Calls Puts

• In the U.S, the writer must deposit the premium
  plus 20 of the value of the stock for call
  options.
• If the call happens to be out of the money, the
  requirement is reduced by the amount by which the
  call is out of the money.

155
Margining (Cont)

• However the margin must at all times be at least
  equal to the premium plus 10 of the value of the
  stock.
• Thus the formula can be expressed as
• MaxC .10S, C .20S (Max0,X-S)

156
Illustration

• Assume an investor writes a call option with an
  exercise price of 100.
• Let the prevailing stock price be 102, and
  assume that the premium is 6.50.
• Thus the required margin is
• 100x.20x102 100x6.50 2,690.

157
Illustration (Cont)

• If however the stock price were to be 90 and
  the corresponding premium 2.50, then the
  required margin would be
• 100xMax2.50 .10x90, 2.50 .20x90-(Max0,100-90
  )
• 100xMax11.50, 10.50 1,150

158
Puts and Margins

• The formula for puts is similar.
• It can be expressed as
• MaxP.10S,P.20S-(Max0,S-X)

159
Illustration

• Consider the case of an investor who writes a put
  option with X 100, when the stock price is
  102.
• Let the premium be 2.50.

160
Illustration (Cont)

• The margin will be
• 100xMax2.50.10x100,2.50.20x100-(Max0,102-100
  )
• 100xMax12.50,20.50
• 2,050

161
Index Options

• In the case of index options the margin
  requirements are slightly less, since indices are
  generally less volatile than individual stocks.
• Consequently instead of a 20 margin, a 15
  requirement is imposed for such options.

162
Covered Calls

• Naked options are extremely risky for the
  writers.
• In the case of call options, they have to acquire
  the stock at the prevailing market price, which
  has no upper bound, if they are called upon to
  deliver.

163
Covered Calls (Cont)

• In the case of puts, they have to arrange to pay
  the exercise price for a stock which may have a
  substantially lower market value.
• Consequently many brokers will not allow
  investors to a write naked options.
• Such privileges are usually restricted to wealthy
  investors who can afford large losses.

164
Covered Calls (Cont)

• In the case of a covered call the investor owns
  the stock on which he has written a call.
• Since the stock is in his possession, there is no
  fear of nonperformance at the time of delivery.
• Hence no margin is required for the short call
  position.

165
Covered Calls (Cont)

• However, most investors who write covered calls,
  tend to buy the stock on the margin.
• That is, they borrow a part of the funds required
  for the purchase of the stock from the broker.
• The sale of a call option has implications for
  the amount of money that they can borrow.

166
Covered Calls (Cont)

• In the U.S. the minimum margin for a stock
  purchase is 50.
• That is, at least 50 of the cost of acquisition
  of the share must be provided by the investor.
• If the investor were to write a covered call
  which is at or out of the money, then the option
  premium can be used to reduce the margin
  requirement for the stock.

167
Illustration

• Consider a stock that is trading at 100.
• Assume that call options with an exercise price
  of 100 are quoting at 6.50.
• If the investor were to acquire the stock on the
  margin, he would have to invest at least
• 100x.50x100 5000 of his own funds.

168
Illustration (Cont)

• However, if he were to write a covered call, he
  need put up only
• 5000 100x6.50 4,350
• He would have to deposit greater margin, however,
  if the call were to be in the money.

169
Illustration (Cont)

• For instance, if the stock price were to be
  102, and the corresponding option premium were to
  be 8, then the minimum amount that he would be
  required to deposit would be
• 100x102 100x8.00 100x.50x100
• 4,400.

170
Portfolio Based Margining

• In a portfolio based approach, all futures and
  options positions in the same underlying asset or
  commodity, that have been established by an
  investor, are regarded as a portfolio.
• This portfolio is then treated as a single entity
  for the purpose of margining.

171
SPAN

• The most widely used portfolio margining system
  is SPAN or Standard Portfolio Analysis of Risk.
• SPAN is used by LIFFE in London for all its
  products, and is used by other major exchanges
  like MATIF in France, and the Singapore Exchange.

172
SPAN (Cont)

• SPAN is also the method of choice in major
  futures exchanges in the U.S like the CME and the
  CBOT.
• However stock and index options traded on the
  CBOE and other exchanges in the U.S., are
  margined using a contract value approach.

173
SPAN (Cont)

• In India, the exchanges have adopted SPAN.
• SPAN was developed by the CME in 1988.
• Exchanges which have adopted this system, use it
  at both the clearing member level, as well as at
  the level of the client.

174
SPAN (Cont)

• SPAN focuses on the overall risk of a portfolio
  of options and futures contracts on an underlying
  asset or commodity.
• The objective is to determine the maximum loss
  that a portfolio may reasonably be expected to
  suffer from one day to the next.

175
SPAN (Cont)

• SPAN is very easy to use in practice, because of
  the way that it has been structured.
• The complex aspects of margin calculations, like
  option valuation, are done at the level of the
  exchanges and clearinghouses which use SPAN.

176
SPAN (Cont)

• The results of such computations are referred to
  as Risk Arrays.
• The clearinghouse concerned will then package
  these risk arrays and other required data inputs
  into a file called a SPAN risk parameter file.
• This file is then transmitted to the various
  users once a day, or at times more often.

177
SPAN (Cont)

• This aspect is called the SPAN front-end.
• A user of SPAN like a clearing member has to
  simply use these risk parameter files in
  conjunction with his portfolio details, in order
  to calculate the required margin.
• This step requires only simple arithmetic
  operations, and is called the SPAN back-end.

178
SPAN (Cont)

• This two-part process makes SPAN very easy to
  use, and is mainly responsible for its growing
  popularity.

179
SPAN (Cont)

• One one hand, since the valuation and
  re-valuation of the underlying derivative assets,
  under changing market conditions, is done by the
  clearinghouse, it ensures that all end users
  compute their margin requirements in accordance
  with the specifications of the clearinghouse.

180
SPAN (Cont)

• On the other hand, the ease of operation of the
  back-end facilitates its use by investors.
• In practice, SPAN can operate on virtually any
  kind of platform.