Options : A Primer

1
Options A Primer

• By A.V. Vedpuriswar

2
Introduction

• An option contract gives its owner the right,
  but not the legal obligation, to conduct a
  transaction involving an underlying asset at a
  predetermined future date (the exercise date) and
  at a predetermined price (the exercise or strike
  price).
• An option gives the option buyer the right to
  decide whether or not the trade will eventually
  take place.
• The seller of the option has the obligation to
  perform if the buyer exercises the option.
• To acquire these rights, owner of the option must
  pay a price called the option premium to the
  seller of the option.

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Types of options

• American options may be exercised at any time up
  to an including the contract's expiration date.
• European options can be exercised only on the
  contracts expiration date.
• If two options are identical (maturity,
  underlying stock, strike price, etc.), the value
  of the American option will equal or exceed the
  value of the European option.
• The owner of a call option has the right to
  purchase the underlying asset at a specific price
  for a specified time period.
• The owner of a put option has the right to sell
  the underlying asset at a specific price for a
  specified time period.

4
In the money, Out of the money

• If immediate exercise of the option would
  generate a positive payoff, it is in the money.
• If immediate exercise would result in a loss
  (negative payoff), it is out of the money.
• When the current asset price equals the exercise
  price, exercise will generate neither a gain nor
  loss, and the option is at the money.

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In the money call options

• If S X 0, a call option is in the money.
• S X is the amount of the payoff a call holder
  would receive from immediate exercise, buying a
  share for X and selling it in the market for a
  great price S.
• If S X
• If S X, a call option is said to be at the
  money.

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In the money put options

• If X S 0, a put option is in the money.
• X S is the amount of the payoff from immediate
  exercise, buying a share for S and exercising the
  put to receive X for the share.
• If X S
• If S X, a put option is said to be at the
  money.

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Intrinsic value

• An options intrinsic value is the amount by
  which the option is in-the-money.
• It is the amount that the option owner would
  receive if the option were exercised.
• An option has zero intrinsic value if it is at
  the money or out of the money, regardless of
  whether it is a call or a put option.
• The intrinsic value of a call option is the
  greater of (S-X) or 0. That is C Max0, S
  X
• Similarly, the intrinsic value of a put is (X
  S) or 0, whichever is greater. That is
• P Max0, X S

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Problem
A call option has an exercise price of 40 and the
underlying stock is trading at 37. What is the
intrinsic value?
Solution If we exercise the option, loss 3 The
stock is 3 out of the money. So it does not have
any intrinsic value.
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Problem
A put option has an exercise price of 40 and the
underlying stock is trading at 37. What is the
intrinsic value?
Solution I can buy from the market at 37 and sell
to the option writer for 40. So the intrinsic
value is 3.
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Problem
I own a call option on the SP 500 with an
exercise price of 900. During expiration, the
index was trading at 912. If the multiplier is
250, what is the profit I make?
Solution Notionally I can buy at 900 and sell at
912. Profit (912 900) (250) 3000
11
Problem
Calculate the lowest possible price for an
American put option with a strike price of 65, if
the stock is trading at 63 and the risk free rate
is 5. The expiration of the option is after 4
months.
Solution The minimum price 65 63 2.
Otherwise risk free profits can be made by
arbitraging.
12
Problem
Repeat the earlier problem if it is a European
Put.
Solution Present value of strike price
65/(1.05)0.33 . 63.96 So pay off 63.96
63 .96
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Problem
A 35 call on a stock trading at 38 is priced at
5. What is the time value?
Solution Intrinsic value 38 35 3 Total
value 5 Time value 5 3 2
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Problem
A call option with exercise price 40, has a
premium of 3. What is the pay off if the stock
price 38, 40, 42, 44?
Solution Stock Price Pay
off 38 -3 40 -3 42 -3 (42 40) -
1 44 -3 (44 40) 1
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Problem
A put option with exercise price 40 has a premium
of 3. What is the pay off if the stock price
38, 40, 42, 44?
Solution Stock Price Pay off 38 -3
(40 38) - 1 40 -3 42 -3 44 -3
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A put option with exercise price 40 has a premium
of 3. What is the pay off if the stock price
38, 40, 42, 44?
Solution
-3 (40-38) -1
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Problem
Suppose you have bought a 40 call and a 40 put
each with premium of 3. What is the pay off is
the stock price 36, 38, 40, 42, 44?
Solution Stock Price Pay off 36 -3
(40 36) - 3 - 2 38 -3 (40 38)
3 - 4 40 -3 3 - 6 42 -3
3 (42 - 40) - 4 44 -3 3 (44 40)
-2
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Suppose you have bought a 40 call and a 40 put
each with premium of 3. What is the pay off is
stock price 36, 38, 40, 42, 44?
Solution
-3 (40-38)-3 - 2
-3-3 (33-40) -2
-3 (40-38)-3 - 4
-3-3 (42-40) - 4
- 3 - 3 - 6
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Problem
A trader adopts a combination of the following
strategies a) Purchase of call option Strike
price 1.40/Euro Premium 0.32 b) Sale
of call option Strike price
1.60/Euro Premium 0.28 Determine the pay
off.
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Solution
a) Spot price exercised. Pay off - .32 .28 - .04 b)
1.40 will be exercised Pay off - .04 S 1.40 S
1.44 C) Spot price 1.60 Both options will
be exercised Pay off - .04 S 1.40
(S-1.60) - .04 S 1.40 S 1.60 .16
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Problem
A trader buys the following options
simultaneously construct the pay off table. Put
option Strike price 1.71 premium 0.10 Call
option Strike price 1.75 premium 0.05
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Solution
Spot price 1.71, only put option is
exercised Pay off - 0.10 0.05 1.71
S 1.56 S 1.71 spot price 1.75 no option
is exercised pay off - 0 .15 Spot price
1.75 , only call option is exercised pay off
- 0.15 S 1.75 S 1.90
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Problem
A stock trades at 108 and there are two European
options currently available. Strike
Price Premium Put A 113 4 Put
B 118 10 Explain how
arbitraging can take place.
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Solution
Buy Put A and Sell Put B Certain cash flows 10
4 6 S off (113 S) (118 S) 6 1 113 118 , only Put B is exercised Pay off 6 (118
S) S 112 S 118, neither option is
exercised Pay off 6
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Problem
The following call options are trading Option
Strike Price Premium Put A 113
4 Put B 118 10 Explain how
arbitraging can take place.
Solution Sell B, Buy A S exercised , profit 10 - 4 6 30 S 35 only
A is exercised , profit 6 (S-30) S - 24 S
35 both options are exercised , profit
6(S-30)- (S-35) 11
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Problem
Suppose you bought a put on a stock selling for
60 with a strike price of 55, for a 5 premium.
What is the maximum gain possible?
Solution Maximum gain - 5 (55-0)
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Problem
I write a covered call on a 40 stock with an
exercise price of 50 for a premium of 2. what
will be my maximum gain?
Solution Covered call means writing a call and
buying the stock. Premium received 2 Cash paid
for buying stock 40 Maximum gain will be when
the option is not exercised and the stock price
reaches 50. Then stock can be sold for 50 40
10 So Maximum gain 10 2 12
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Problem
What will be the maximum loss in the previous
problem?
Solution If stock price falls to zero, pay off
2 0 2 Cash paid for buying stock
40 Maximum loss 2 40 - 38
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Specialised options

• Bond options are most often based on Treasury
  bonds because of their active trading.
• Index options settle in cash, nothing is
  delivered, and the payoff is made directly to the
  option holders account.
• Options on futures sometimes called futures
  options, give the holder the right to buy or sell
  a specified futures contract on or before a given
  date at a given futures rice, the strike price.
• Call options on futures contracts give the
  holder the right to enter into the long side of a
  future contract at a given futures price.
• Put options on futures contracts give the
  holder the option to take on a short futures
  position at a future price equal to the strike
  price.

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Interest rate options

• Interest rate options are similar to stock
  options except that the exercise price is an
  interest rate and the underlying asset is a
  reference a rate such as LIBOR.
• Interest rate options are also similar to FRAs .
• They are settled in cash, in an amount that is
  based on a notional amount and the spread between
  the strike a rate and the reference rate.
• Most interest options are European options.

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• Consider a long position in a LIBOR-based
  interest rate call option with a notional amount
  of 1,000,000 and a strike rate of 5.
• If at expiration, LIBOR is greater than 5, the
  option can be exercised and the owner will
  receive 1,000,000 x (LIBOR 5).
• If LIBOR is less than , the option expires
  worthless and the owner receives nothing.

32

• Lets consider a LIBOR-based interest rate put
  option with the same features as the call that we
  just discussed.
• Assume the option has a strike rate of 5 and
  notional amount of 1,000,000.
• If at expiration, LIBOR falls below 5 the
  option writer (short) must pay the put holder an
  amount equal to 1,000,000 x (5 - LIBOR).
• If at expiration, LIBOR is greater than 5, the
  option expires worthless and the put writer makes
  no payments.

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Problem
I have bought a call option on 90 day LIBOR with
a notional principal of 2 million and a strike
rate of 4. At the expiration of the option, if
LIBOR is 5, what is the compensation I will
receive?
Solution (2,000,000) (.05 - .04) (90/360)
5000 This compensation will be received 90
days after expiration.
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Caps

• An interest rate cap is a series of interest
  rate call options, having expiration dates that
  correspond to the reset dates on a floating-rate
  loan.
• Caps are often used to protect a floating-rate
  borrower from an increase in interest rates.
• Caps place a maximum (upper limit) on the
  interest payments on a floating-rate loan.
• A cap may be structured to cover a certain number
  of periods or for the entire life of a loan.
• The cap will make a payment at any future
  interest payment due date whenever the reference
  rate exceeds the cap rate.

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Floors

• An interest rate floor is a series of interest
  rate put options, having expiration dates that
  correspond to the reset dates on a floating-rate
  loan.
• Floors are often used to protect a floating-rate
  lender from a decline in interest rates.
• Floors place a minimum (lower limit) on the
  interest payments that are received from a
  floating-rate loan.

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Collars

• An interest rate collar combines a cap and a
  floor.
• A borrower with a floating-rate loan may buy a
  cap for protection against rates above the cap
  and sell a floor in order to defray some of the
  cost of the cap.

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Call Option value

• Lower bound. Theoretically, no option will sell
  for less than its intrinsic value and no option
  can take on a negative value.
• This means that the lower bound for any option is
  zero for both American and European options.
• Upper bound. The maximum value of either an
  American or a European call option at any time t
  is the time-t share price of the underlying
  stock.
• This makes sense because no one would pay a price
  for the right to buy an asset that exceeded the
  assets value. It would be cheaper to simply buy
  the underlying asset.

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Put Option value bounds

• Upper bound for put options. The price for an
  American put option cannot be more than its
  strike price.
• This is the exercise value in the event the
  underlying stock price goes to zero.
• However, since European puts cannot be exercised
  prior to expiration, the maximum value is the
  present value of the exercise price discounted at
  the risk-free rate.
• Even if the stock price goes to zero, and is
  expected to stay at zero, the intrinsic value, X,
  will not be received until the expiration date.

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Valuing call options

• For a European call option, construct the
  following portfolio
• A long at-the money European call option with
  exercise price X, expiring at time t T
• A long discount bond priced to yield the
  risk-free rate that pays X at option expiration.
• A short position in one share of the underlying
  stock priced at S0 X
• The current value of this portfolio is c0 S0
  X/(1RFR)T

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• At expiration time, t T, this portfolio will
  pay cT ST X.
• That is, we will collect cT Max0, ST X) on
  the call option, pay ST to cover our short stock
  position, and collect X from the maturing bond.
• If ST X, the call is in-the-money, and the
  portfolio will have a zero payoff because the
  call pays ST X, the bond pays X, and we pay
  ST to cover our short position.
• That is, the time t T payoff is ST X X
  ST 0.
• If X ST the call is out-of-the-money, and the
  portfolio has a positive payoff equal to X ST
  because the call value, cT is zero, we collect X
  on the bond, a pay - ST to cover the short
  position.
• So, the time t T payoff is 0 X ST X - ST

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• Note that no matter whether the option expires
  in-the-money, at-the-money, or out-of-the-money,
  the portfolio value will be equal to or greater
  than zero. We will never have to make a
  payment.
• To prevent arbitrage, any portfolio that has no
  possibility of a negative payoff cannot have a
  negative value. Thus, we can state the value of
  the portfolio at time t 0 as
• c0 S0 X / (1RFR)T 0
• Which allows us to conduct that c0 S0
  X/(1RFR)T

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• Given two puts that are identical in all
  respects except exercise price, the one with the
  higher exercise price will have at least as much
  value as the one with the lower exercise price.
• This is because the underlying stock can be sold
  at a higher price.
• Similarly, given two calls that are identical in
  every respect except exercise price, the one with
  the lower exercise price will have at least as
  much value as the one with the higher exercise
  price.
• This is because be underlying stock can be
  purchased at a lower price.

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Option value and time to expiration

• For American options and in most cases for
  European options, the longer the time to
  expiration, the greater the time value and, other
  things equal, the greater the options premium
  (price).
• For far out-of-the-money options, the extra time
  may have no effect, but we can say the
  longer-term option will be no less valuable that
  the shorter-term option.

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• The case that doesnt fit this pattern is the
  European put.
• The minimum value of an in-the-money European put
  at any time t prior to expiration is X/(1RFR)T-t
  St.
• While longer time to expiration increases option
  value through increased volatility, it decreases
  the present value of any option payoff at
  expiration.
• For this reason, we cannot state positively that
  the value of a longer European put will greater
  than the value of a shorter-term put.

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• If volatility is high and the discount rate low,
  the extra time value will be the dominant factor
  and the longer-term put will be more valuable.
• Low volatility and high interest rates have the
  opposite effect and the value of a longer-term
  in-the-money put option can be less than the
  value of a shorter-term put option.

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Put Call Parity

• Our derivation of put-call parity is based on the
  payoffs of two portfolio combinations, a
  fiduciary call and a protective put.

47
Fiduciary call

• A fiduciary call is a combination of a
  pure-discount, riskless bond that pays X at
  maturity and a call with exercise price X.
• The payoff for a fiduciary call at expiration is
  X when the call is out of the money, and X (S
  X) S when the call is in the money.

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Protective put

• A protective put is a share of stock together
  with a put option on the stock.
• The expiration date payoff for a protective put
  is (X-S) S X when the put is in the money,
  and S when the put is out of the money.
• When the put is the money, the call is out of
  the money, both portfolios pay X at expiration.
• Similarly, when the put is out of the money and
  the call is in the money, both portfolios pay S
  at expiration.

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• If exercised, an American call will pay St X,
  which is less than its minimum value of St
  X/(1RFR)T-t. Thus, there is no reason for early
  exercise of an American call option on stocks
  with no dividends.
• For American call options on dividend-paying
  stocks, the argument presented above against
  early exercise does not necessarily apply.
• Keeping in mind that options are not typically
  adjusted for dividends, it may be advantageous to
  exercise an American call prior to the stock's
  ex-dividend date, particularly if the dividend is
  expected to significantly decrease the price of
  the stock.
• For American put options, early exercise may be
  warranted if the company that issued the
  underlying stock is in bankruptcy so that its
  stock price is zero.
• It is better to get X now than at expiration.
• Similarly, a very low stock price might also
  make an American put worth more dead than alive.

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Volatility and option value

• Greater volatility in the value of an asset or
  interest rate underlying an option contract
  increases the values of both puts and calls (and
  caps and floors).
• The reason is that options are one-sided."
• Since an options value falls no lower than zero
  when it expires out of the money, the increased
  upside potential (with no greater downside risk)
  from increased volatility, increases the options
  value.

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Problem
A stock is selling at 40, 3 month 50 put is
selling for 11, a 3 month 50 is selling 1.
The risk free rate is 6. How much can be made
on arbitrage.
Solution Portfolio 1 Fiduciary call Buy Call,
Invest in Bond Investment 1 50/(1.06).25
50.28 Portfolio 2 Protective put Buy
stock, Buy put Investment 40 11 51 So
profit from arbitrage 51 50.28 0.72
52
Problem
The current stock price is 52 and the risk free
rate is i5. A 3month 50 put is quoting at
1.50. Estimate the price for a 3 month 50 call.
Solution Fiduciary call C 50 /
(1.05).25 Protective put 52 1.5 To prevent
arbitrage, we write C 50/(1.05).25 52
1.5 Or C 53.5 40.39 4.11
53
Problem
The current stock price is 53 and the risk free
rate is 5. A 3 month European 50 call is
quoting 3. What is the price of a 3 month 50
put?
Solution To prevent arbitrage, we write C
50/(1.05).25 53 P Or P 53 3 -
49.39 0.61
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Options trading in India

• NSE introduced trading in index options on June
  4, 2001.
• The options contracts are European style and cash
  settled and are based on the popular market
  benchmark SP CNX Nifty index.
• SP CNX Nifty options contracts have 3
  consecutive monthly contracts, additionally 3
  quarterly months of the cycle March / June /
  September / December and 5 following semi-annual
  months of the cycle June / December would be
  available, so that at any point in time there
  would be options contracts with at least 3 year
  tenure available.

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• On expiry of the near month contract, new
  contracts (monthly/quarterly/ half yearly
  contracts as applicable) are introduced at new
  strike prices for both call and put options, on
  the trading day following the expiry of the near
  month contract.
• SP CNX Nifty options contracts expire on the
  last Thursday of the expiry month.
• If the last Thursday is a trading holiday, the
  contracts expire on the previous trading day.

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• New contracts with new strike prices for existing
  expiration date are introduced for trading on the
  next working day based on the previous day's
  index close values, as and when required.
• In order to decide upon the at-the-money strike
  price, the index closing value is rounded off to
  the nearest applicable strike interval.
• The in-the-money strike price and the
  out-of-the-money strike price are based on the
  at-the-money strike price.
• The value of the option contracts on Nifty may
  not be less than Rs. 2 lakhs at the time of
  introduction.

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• The permitted lot size for futures contracts
  options contracts shall be the same for a given
  underlying or such lot size as may be stipulated
  by the Exchange from time to time.
• The price step in respect of SP CNX Nifty
  options contracts is Re.0.05.
• Base price of the options contracts, on
  introduction of new contracts, would be the
  theoretical value of the options contract arrived
  at based on Black-Scholes model of calculation of
  options premiums.

58

• The base price of the contracts on subsequent
  trading days, will be the daily close price of
  the options contracts. The closing price shall be
  calculated as follows
• If the contract is traded in the last half an
  hour, the closing price shall be the last half an
  hour weighted average price.
• If the contract is not traded in the last half an
  hour, but traded during any time of the day, then
  the closing price will be the last traded price
  (LTP) of the contract.
• If the contract is not traded for the day, the
  base price of the contract for the next trading
  day is arrived at based on Black-Scholes model of
  calculation of options premiums.