Math 476 567 Actuarial Risk Theory

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Math 476 / 567 Actuarial Risk Theory

• Fall 2009
• University of Illinois at Urbana-Champaign
• Professor Rick Gorvett
• Options and Put-Call Parity
• August 27, 2009

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A Sampling ofOptionsand Other Derivatives
through History
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Ancient Greece

• There is the anecdote of Thales the Milesian
  and his financial device He was reproached for
  his poverty, which was supposed to show that
  philosophy was of no use. According to the story,
  he knew by his skill in the stars while it was
  yet winter that there would be a great harvest of
  olives in the coming year so, having a little
  money, he gave deposits for the use of all the
  olive-presses in Chios and Miletus, which he
  hired at a low price because no one bid against
  him. When the harvest-time came, and many were
  wanted all at once and of a sudden, he let them
  out at any rate which he pleased, and made a
  quantity of money. Thus he showed the world that
  philosophers can easily be rich if they like, but
  that their ambition is of another sort
• - Aristotle, Politics, Book One, Part XI

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Phoenician Shipping

• Merchants and ship-owners used options to hedge
  their ships and cargoes

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Mesopotamia

• Mercantile forward contracts, written in
  cuneiform on clay tablets, circa 1700 BC

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China

• Forward contracts on rice, entered into prior to
  planting, circa 2000 BC

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Belgium and The Netherlands

• Antwerp and Amsterdam
• Grain
• Herring
• Tulips

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Tulip Bubble

• Mid-1630s
• Tulip demand exploded and prices skyrocketed
• Options and futures were used to ensure price and
  supply
• Bubble burst in 1637

9
America

• 19th century
• Privileges
• Non-standardized / over-the-counter
• Synthetic loans
• Financier Russell Sage
• Put-call parity
• Get around usury laws

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Chicago Board Options Exchange

• Began trading standardized options on April 26,
  1973
• 911 contracts traded on first day (options on 16
  different underlying companies)

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Optionsand theirCharacteristics
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A Type of Derivative

• A forward is the obligation to buy or sell
  something at a pre-specified time and at a
  pre-specified price
• An option is the right to buy or sell something
  at a pre-specified time (or during a
  pre-specified time-period) and at a pre-specified
  price

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Types of Options

• Call option the holder has a right to buy the
  underlying asset
• Put option the holder has a right to sell the
  underlying asset
• Counterparties parties to the option agreement
• One can buy (long) or sell (short) an option

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Parameters of Options

• Exercise price strike price price at which
  the holder of the option can exercise the option
  (and thus buy or sell the underlying asset)
• Premium amount paid for the option
• Expiration date
• Style
• American option can exercise any time up to and
  including expiration date
• European option can exercise only on expiration
  date

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Examples of Options --Theyre Everywhere

• Traded options
• On stocks, indices, FX, interest rates, futures,
  swaps, options,...
• Warrants
• Convertible bonds
• Call provisions on bonds
• On projects
• To expand, abandon, postpone
• Insurance

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Value Of Options At Expiration

• C Max ST - X, 0
• C Call option value (or payoff) at expiration
• ST Price of underlying asset at expiration
• X Exercise price
• P Max X - ST, 0
• P Put option value (or payoff) at expiration

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Example

• Amy sells Bob a January European call option on
  one share of ABC stock
• Suppose ABC stock is initially trading at 32.5
• Exercise price 35
• Premium 3
• In January, suppose ST30 ST40
• Total payoff profit/loss
• Amy 0 3 -5 -2
• Bob 0 -3 5 2

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Option Values

• Prior to expiration Call Put
• In-the-money St gt X St lt X
• At-the-money St X St X
• Out-of-the-money St lt X St gt X
• Intrinsic value profit that could be made if
  the option was immediately exercised
• Call stock price - exercise price
• Put exercise price - stock price
• Time value the difference between the option
  price and the intrinsic value

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Option Values Payoff Charts
Payoff

• Call -- long position
• Call -- short position
• Put -- long position
• Put -- short position

ST
X
X
ST
ST
X
X
ST
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Payoff vs. Profit/LossLong a Call Option
Payoff
Profit/Loss
ST
Call Premium
X
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Purposes of Derivatives

• Speculative
• Highly risky
• Highly leveraged
• Very volatile
• Hedging
• Combine with other securities
• Hedge (minimize) risk from other securities

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Hedging

• Hedge Take a position that offsets a
• risk
• Risk Uncertainty regarding the value of
• the underlying asset
• By hedging, one changes the risk inherent in
  owning the underlying asset
• The return distribution of the underlying asset
  is not changed

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Using Options to Hedge

• Combine the underlying asset with an option or
  options
• Can reduce or eliminate downside risk while
  retaining upside potential
• Can protect against falls in held asset values,
  or against increases in input prices

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Option Strategies

• Protective put
• Own stock (long position)
• Own put (long position)
• Covered call
• Own stock (long position)
• Sell call (short position)
• Straddle
• Spread

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Protective Put

• Investor owns asset
• Investor also buys (holds) a put on the asset
• Guarantees investment portfolio proceeds are at
  least equal to the exercise price of the put

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Protective Put Example

• Suppose you own a share of stock, and you
  purchase a put option with an exercise price of
  22.5 on that stock, for a premium of 0.75
• ST 30 25 20 15
• Premium -0.75 -0.75 -0.75 -0.75
• Put Payoff 0 0 2.50 7.50
• Overall 29.25 24.25 21.75 21.75

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Covered Call

• Investor purchases stock
• Investor also sells (writes) a call option on the
  stock
• Option position is covered by owning the
  underlying stock itself
• (vs. naked option)
• Provides additional (premium) income

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Covered Call Example

• Suppose you own a share of stock, and you write a
  call option with an exercise price of 35 on that
  stock, for a premium of 2.00
• ST 30 35 40 45
• Premium 2 2 2 2
• Call Payoff 0 0 - 5 -10
• 
  
• Overall 32 37 37 37

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Straddle

• (Long) Straddle buy both a call and a put on a
  stock
• Each option has the same exercise price and
  expiration date
• Believe stock will be relatively volatile
• Worst-case no movement in stock price

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Spread

• Combination of options
• Two or more calls, or
• Two or more puts
• Horizontal spread sale and purchase of options
  with different expiration dates
• Vertical spread simultaneous sale and purchase
  of options with different exercise prices -- e.g.,

X2


X1
X1
X2
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Exotic Options

• Certain characteristics of plain vanilla
  options are adjusted to produce exotic options
• Some characteristics of plain vanilla options
• American or European
• Linear payoff
• Does not disappear
• Value of underlying at exercise

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Put-Call ParityGeneral Concept

• Arbitrage implies a certain relationship between
  put, call, and underlying asset prices
• Assuming same exercise prices and expiration
  dates, and non-dividend-paying stock, two
  portfolios have, at payoff, identical values
• One European call option cash of PV(X)
• One European put option one share of stock
• C PV(X) P S

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Put-Call ParitySpecific Relationships

• McDonald text terminology
• Call put PV(forward price strike price)
• For non-dividend-paying stock

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Put-Call ParitySpecific Relationships (cont.)

• For dividend-paying stock
• For index options

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Put-Call Parity Example

• Find the value of the one-year (T 1) put with
  exercise price (X) of 110 when
• C(110,1) 10.16
• S0 100 d 0
• r 0.10
• P(110,1) 10.16 110 e -.10 1 - 100 9.69

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Put-Call Parity Example (cont.)

• Find the value of the one-year (T 1) put with
  exercise price (X) of 110 when
• C(110,1) 10.16
• S0 100 d 0.02 (2 div yield)
• r 0.10
• P(110,1) 10.16 110 e -.10 1 - 100 e -.02
  1
• 11.67