Transcript Options : A Primer

Options : A Primer
By A.V. Vedpuriswar
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.
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 contract’s
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.
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.

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 < 0, a call option is out of the money.
If S = X, a call option is said to be at the money.
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 < 0, a put option is out of the money.
If S = X, a put option is said to be at the money.
Intrinsic value
 An option’s 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
=
Max[0, S – X]
 Similarly, the intrinsic value of a put is (X – S) or 0, whichever
is greater. That is
P
= Max[0, X – S]
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.
intrinsic value.
So it does not have any
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.
Problem
I own a call option on the S&P 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
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.
Problem
Repeat the earlier problem if it is a European Put.
Solution
Present value of strike price = 65/(1+.05)0.33 .
So pay off
=
63.96
=
63.96 – 63
= .96
Problem
A $35 call on a stock trading at $38 is priced at $5. What is the
time value?
Solution
Intrinsic value
=
38 – 35
Total value
Time value
=
5–3
=
3
=
5
=
2
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
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
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:
45
4.5
44
43
2.5
42
41
40
0.5
-3+ (40-38) = -1
36
-1.5
42
38
37
44
-1
39
40 -3
-3
-3
38
35
-3.5
-5.5
1
2
Stock price
3
Pay Off
4
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
42
-3 – 3 + (42 - 40) = - 4
44
-3 – 3 + (44 – 40) = -2
=-6
Suppose you have bought a $40 call and a $40 put each with
premium of 3. What is the pay off if stock price = 36, 38, 40, 42,
44?
Solution:
50
4.5
45
40
2.5
35
0.5
30
25
20
-3+ (40-38)-3 = - 2
-2
36
40
38
-3-3+ (33-40) = -2
15
-4
10
44 -2
42
-4
-3+ (40-38)-3 = - 4
-3-3+ (42-40) = - 4
-3.5
-5.5
-6
5
-1.5
-3-3=-6
0
-7.5
1
2
3
Stock price
4
Pay Off
5
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.
Solution
a)
Spot price < 1.40;
Options will not be exercised.
Pay off = - .32 + .28 = - .04
b)
1.40 < spot price < 1.60;
exercised
$1.40 call option will be
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
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
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
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.
Solution
Buy Put A and Sell Put B
Certain cash flows = 10 – 4 = 6
S < 113 , Both options are exercised.
Pay off = (113 – S) – (118 – S) + 6
=
1
113 < S < 118 , only Put B is exercised
Pay off = 6 – (118 – S)
= S – 112
S > 118, neither option is exercised
Pay off
= 6
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 < 30 No option is 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
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)
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
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
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 holder’s 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.
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.
 Let’s consider a LIBOR-based interest rate put option with
the same features as the call that we just discussed.
Assume the option has
amount of $1,000,000.
a strike rate of 5% and notional
 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.

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.
Caps
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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.
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.

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.
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 asset’s value. It
would be cheaper to simply buy the underlying asset.
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.
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/(1+RFR)T
 At expiration time, t = T, this portfolio will pay cT – ST + X.
That is, we will collect cT = Max[0, 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
 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 / (1+RFR)T ≥ 0
Which allows us to conduct that: c0 ≥ S0 – X/(1+RFR)T

 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.
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 option’s 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 doesn’t 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/(1+RFR)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.
 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.

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.
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.
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.

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
Portfolio 2
:
=
50.28
=
0.72
Protective put
Buy stock, Buy put
Investment = 40 + 11
=
51
So profit from arbitrage = 51 – 50.28
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
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
Options trading in India

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
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 S&P CNX Nifty index.
S&P 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.

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


.
S&P 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-themoney 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 S&P 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.




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 BlackScholes model of calculation of options premiums.