Chapter 22 Options

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• 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.

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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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• WriterSeller 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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Im 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

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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 gt E100 Exercise
the call option - pay 100 for an IBM stock with
a market value of ST (e.g. ST105). Payoff
at T ST - E 105-1005 gt 0. If ST ?
E100 Can buy IBM stocks in the market for ST
(e.g. ST90). Holder will not choose to
exercise (option expires worthless).
Payoff at T 0.
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Holding a European Call
Conclusion A call option holder will never lose
at T (expiration), since his/her payoff is never
negative If ST ? E100 If ST gt
E100 Call option value at T 0 ST - E
ST - 100
Payoff at T
450
0
ST
E100
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For the Call Option Writer (Short Seller)
If ST gt 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 lt 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 ? E100 If ST gt E100
Call option value at T 0 E - ST 100 -
ST
Payoff at T
0
ST
E100
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 options expiration date in
September (T). At the options expiration date
(T) For the Put Option Holder If ST lt E100
Exercise the put option - receive 100 for an
IBM stock with a market value of ST (e.g.
ST90). Payoff at T E - ST 100-9010 gt
0. If ST ? E100 Can sell IBM stocks in the
market for ST (e.g. ST105). 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 ? E100 If ST lt
E100 Put option value at T 0 E - ST
100 - ST
Payoff at T
100
450
0
ST
E100
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For the Put Option Writer (Short Seller)
If STltE 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 -10lt 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 ? E100 If ST lt
E100 Put option value at T 0 ST - E
ST - 100
Payoff at T
0
ST
E100
450
- 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)
Buy Stock
Short Sell Call
Combination
Payoff at T
Payoff at T
Payoff at T
85
450
0
0
0
ST
ST
ST
E85
E85
450
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 (85E)
  today, we use the risk-free rate

Only for European Options
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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)
Buy Stock
Short Sell Call
Combination
Buy Put
Payoff at T
E85
85
450
450
0
ST
ST
ST
E85
E85
E85
450
A Certain Payoff of E85
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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
• 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
• Short is -

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A Synthetic Stock

• We first rearrange the PCP equation to isolate
  S
• 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 lt E ST gt E
• Long T-bill E E
• Long Call 0 ST - E
• Short Put -(E - ST) 0
• Total Replicated Payoff ST ST
• 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
• 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 lt E ST gt 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
• 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 lt E ST gt 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 lt S - E investors will pocket an arbitrage
  profit.
• Example S 100, E 90, C 8 gt C 8 lt
  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
  -8-90100 2
• Excess demand will force C to rise to 10
• As long as there is time to expiration, we will
  have C gt 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
E90
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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