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Title: 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
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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
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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.
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