9.220
Chapter 22 - Options
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Options
 If you have an option, then you have the right to
do something. I.e., you can make a decision or
take some action.



The option owner has a choice to make.
Usually the choice can be made over time after more
information is known.
Having an option is valuable; options never have a
negative value, because, at worst, the option owner
can discard the option and not take any action.
2
A European Call Option
gives the holder the right to buy the underlying asset for a
prescribed price (exercise/strike price), on a prescribed date
(expiry date).
A European Put Option
gives the holder the right to sell the underlying asset for a
prescribed price (exercise/strike price), on a prescribed date
(expiry date).
American Options
exercise is permitted at any time during the life of the option
(call or put).
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Underlying Asset (S)
The specific asset on which an option contract is based (e.g. stock,
bond, real-estate, etc.).
For traded Stock Options: one call (put) option contract represents
the right to buy (sell) 100 shares of the underlying stock.
Strike/Exercise Price (E)
The specified asset price at which the asset can be bought (sold) by
the holder of a call (put) if s/he exercised his/her right.
Expiration Date (T)
The last day an option exists.
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Option Writer


The option writer is the person who created (or wrote) the
option and then sold it to someone else. As the writer did not
previously own the option, the act of writing and selling the
option is equivalent to short-selling the option.
The option writer has sold an option or right to the option
owner.

Thus the writer has taken on an obligation to act on the
instruction of the option owner as specified by the option
contract.

When selling the option to the buyer, the writer receives as
compensation the “option premium”= the price the buyer
pays for the option.
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Writer:
Seller of an option (takes a short position in the option).
Holder:
Buyer of the option (takes a long position in the option).
Elements of an option contract:
 type (put or call)
 style (American or European)
 underlying asset (stock/bond/etc…)
 unit of trade
 exercise price
 expiration date
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I’m “in the money”
 An option is said to be “in the money” if
exercising it would produce a positive payoff.
 An “at the money” option would generate a zero
payoff if exercised.
 An “out of the money” option would generate a
negative payoff if exercised. (Thus an out of the
money option would never be exercised.)
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Option Payoffs at Expiration
 It is useful to examine the payoffs of
options when they are about to expire. At
this point in time, the owner is forced to
make the decision to exercise or to
abandon the option. The option owner will
exercise if the option is in the money when
it is about to expire.
 Consider Call and Put Options…
8
Holding A European Call Option Contract- An Example
European style IBM corp. September 100 call:
entitles the buyer (holder) to purchase 100 shares of IBM common stock at
$100 per share (E), at the options expiration date in September (T).
At the options expiration date (T):
For the Call Option Holder
If ST > E=$100:
Exercise the call option - pay $100 for an IBM stock with a market
value of ST (e.g. ST=$105).
Payoff at T: ST - E = $105-$100=$5 > 0.
If ST  E=$100:
Can buy IBM stocks in the market for ST (e.g. ST=$90). Holder will
not choose to exercise (option expires worthless).
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Payoff at T: $0.
Holding a European Call
Conclusion:
A call option holder will never lose at T (expiration), since his/her payoff is
never negative:
If ST  E=$100
Call option value at T
If ST > E=$100
$0
ST - E = ST - $100
Payoff
at T
450
0
ST
E=$100
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For the Call Option Writer (Short Seller)
If ST > E = $100: Holder will exercise. Writer will deliver an IBM stock
with a market value of ST ($105) to the holder, in return
for E dollars ($100).
Payoff at T: E-ST = $100-$105= - $5 < 0.
If ST  E = $100: Holder will not exercise.
Payoff at T: $0.
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For the Call Option Writer (Short Seller)
Conclusion:
A call option writer will never gain at T (expiration), since his/her payoff is
never positive:
If ST  E=$100
Call option value at T
If ST > E=$100
$0
E - ST = $100 - ST
Payoff
at T
0
ST
E=$100
450
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Holding A European Put Option Contract- An Example
European style IBM corp. September 100 put
entitles the buyer (holder) to sell 100 shares of IBM corp. common stock at
$100 per share (E), at the option’s expiration date in September (T).
At the options expiration date (T):
For the Put Option Holder
If ST < E=$100:
Exercise the put option - receive $100 for an IBM stock with a
market value of ST (e.g. ST=$90).
Payoff at T: E - ST = $100-$90=$10 > 0.
If ST  E=$100:
Can sell IBM stocks in the market for ST (e.g. ST=$105). Holder
will not choose to exercise (option expires worthless).
Payoff at T: $0.
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Holding A European Put
Conclusion:
A put option holder will never lose at T, since his/her payoff is never
negative:
If ST  E=$100
Put option value at T
If ST < E=$100
$0
E - ST = 100 - ST
Payoff
at T
$100
450
0
ST
E=$100
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For the Put Option Writer (Short Seller)
If ST<E = $100: Holder will exercise. Writer will pay E dollars
($100) in return for an IBM stock (worth ST = $90).
Payoff at T: ST - E = $90-$100 = -$10< 0.
If ST  E = $100 : Holder will not exercise.
Payoff at T: $0.
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For the Put Option Writer (Short Seller)
Conclusion:
A put option writer will never gain at T, since his/her payoff is never
positive:
If ST  E=$100
Put option value at T
If ST < E=$100
$0
ST - E = ST - 100
Payoff
at T
0
450
E=$100
ST
- $100
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Combinations of Options
You purchase a BCE stock, and simultaneously write (short
sell) the July $85 European call option.
Your payoff diagram at expiration in July (T):
Payoff
Buy Stock
at T
Payoff
Short Sell Call
Payoff
at T
at T
$85
450
0
Combination
ST
0
ST
E=$85
450
0
ST
E=$85
Same as Short Sell a Put
and Buy A T-bill
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The Put-Call Parity Relationship*
You purchase the BCE stock, the July $85 put option, and
short sell the July $85 call option (both options are
European). Your payoff at expiration in July (T):
If ST = $100
If ST = $80
Stock (S)
Put (P)
Call(C)
__
Total (Certain) Payoff:
To calculate the PV of the certain payoff ($85=E) today, we
use the risk-free rate:
S  P  C  Ee
*Only for European Options
Tr f
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The Put-Call Parity Portfolio
You purchase the BCE stock, the July $85 European put
option, and short sell the July $85 European call option
Your payoff at expiration in July (T):
Payoff
Buy Stock
at T
Short Sell Call
$85
450
0
Buy Put
Combination
E=$85
450
ST
E=$85
ST
E=$85
450
ST
E=$85
A Certain
Payoff of
E=$85
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Using Put-Call-Parity (PCP) to Replicate Securities
A synthetic security
 Definition - A portfolio of other securities which will pay the
same future cash flows as the security being replicated.
 Since payoffs at expiration (cash flows) are the same for the synthetic
security and the original security under all states of the world, their
current prices must be identical.
 Otherwise, if one is currently cheaper than the other, an arbitrage
opportunity will exist: buy (long) the cheaper security today for
the lower price, and simultaneously short sell the expensive
security for the higher price. This results in a positive initial
cash flow.
 This positive cash flow is an arbitrage profit (“free lunch”), since
at expiration, the cash flows from both positions will offset each other,
and the total cash flow at expiration will be zero.
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How Do We Replicate Securities?
 The Put-Call-Parity (PCP) Relationship:
Trf
Ee
 S  P C
This is a risk free T-bill that pays E dollars in T years
 Recall that the PCP portfolio was created by:
Long one stock (+S), Long one put (+P), and Short one call (-C)
 We saw that this is equivalent to:
Long a T-bill (+Ee-Trf)
 Thus, we replicated a long position in a T-bill with: long stock, long put
and short call.
 For security replication purposes, use PCP with the following rule:
Long is “+”
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Short is “-”
A Synthetic Stock


We first rearrange the PCP equation to isolate S:
Tr
S  Ee f  C  P
According to the above replication rule:
Long one stock (+S) =
Long a T-bill (+Ee-Trf) & Long one call (+C) & Short one put (-P),

The payoff (cash flow) at maturity:
ST < E
E
ST > E
E
Long Call
0
ST - E
Short Put
-(E - ST)
$0
+ST
+ST
Long T-bill
Total Replicated Payoff:

Conclusion - holding the replicated portfolio is the same as holding the
stock
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A Synthetic Call

We first rearrange the PCP equation to isolate C:
Trf

C  S  P  Ee
According to the above replication rule:
Long one call (+C) =
Long one stock (+S) & Long one put (+P) & Short a T-bill (-Ee-Trf)

The payoff (cash flow) at maturity:
ST < E
ST > E
Long Stock
Long Put
Short T-bill
Total Replicated Payoff:
Conclusion - holding the replicated portfolio is the same as holding a
call
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A Synthetic Put

We first rearrange the PCP equation to isolate P:
Trf

P  Ee
C S
According to the above replication rule:
Long one put (+P) =
Long a T-bill (+Ee-Trf) & Long one call (+C) & Short one stock (-S)

The payoff (cash flow) at maturity:
ST < E
ST > E
Long T-bill
Long Call
Short Stock
Total Replicated Payoff:

Conclusion - holding the replicated portfolio is the same as holding a
put
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Bounding The Value of An American Call
The value of an American call can never be:
 below the difference b/w the stock price (S) and the exercise price (E).

If C < S - E: investors will pocket an arbitrage profit.

Example: S = $100, E = $90, C = $8
=>
C = 8 < 10 = S – E

Arbitrage Strategy:
Buy the call for $8, and exercise it immediately by paying the exercise price
($90)to get the stock (worth $100). This results in an immediate arbitrage profit
(i.e. “free lunch”) of: -8-90+100= $2
Excess demand will force C to rise to $10

As long as there is time to expiration, we will have C > $10 = S - E
 Above the value of the underlying stock (S)

If it is, buy the stock directly
Boundary Conditions
value of the
American call
will be here
Payoff
at t
450
0
St
E=$90
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Determinants of American Option Pricing
For an American Call:
C = C (S, E, T, , r)
(+) (-) (+) (+) (+)
S - The higher the share price now, the higher the profit from exercising.
Thus the higher the option price will be.
E - The higher the exercise price now, the more it needs to be paid on
exercise. Thus, the lower the option value will be.
T - The more time there is to expiration, the higher the chance that the
stock price will be higher at T, and the higher the option value will be.
 - The larger the volatility, the more probable a profitable outcome, thus
the higher C is.
r - The higher the interest rate, the P.V. of the future exercise price
decreases. The call price will increase.
For an American Put:
P = P(S, E, T, , r)
(-) (+) (+) (+) (-)
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Determinants of American Option Pricing
Determinants of
Relation to
Relation to
Option Pricing
Call Option
Put Option
Stock price
Positive
Negative
Strike price
Negative
Positive
Risk-free rate
Positive
Negative
Volatility of the stock
Positive
Positive
Time to expiration date
Positive
Positive
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