Economics 134a

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Economics 134a

• Lecture Notes

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• Exam prefer non-cumulative?

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Midterm 3
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Unit V -- Derivatives

• Derivatives securities with payoffs that depend
  on (are derived from) the payoffs on other
  securities
• Most important types futures and options

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Options

• Options traded on several exchanges (Chicago
  Board Options Exchange).
• Also traded over-the-counter (customized hedging
  devices)
• Financial engineering customized options
• Options on stocks, bonds, foreign currencies,
  commodities, metals
• Uses gambling and insurance (speculating and
  hedging)

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• The company on the stock of which options are
  written may or may not play any role in trading
  the options.
• of options on the long side of options on
  short side.
• Think of options as a specialized kind of bet on
  the price of the stock.
• option to buy is a call option to sell is a put.
• Call long side has the right, but not the
  obligation, to buy the stock from the short side
  for a fixed price (the exercise price), on the
  exercise date
• Because the long side has a right, not an
  obligation, the price (premium) of any option
  will be positive

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Option terms

• Either kind of option has a long side and a short
  side.
• long side has the option
• short side grants the option
• Taking a short position in an option writing an
  option.
• Exercise (strike) price exercise date
• Purchase price premium
• profit on a long position in an option payoff
  minus premium

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Option payoffs

• The holder (long side) of an option pays the
  premium the writer (short side) receives it.
• The long position has the option to buy the stock
  from the writer for the exercise price
• European option on the exercise date
• American option on or before.
• What is the value of the payoff? For a call,
  its Max(stock price minus exercise price, 0)

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Call
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For a put

• Its
• Max(exercise price stock price, 0)

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Put
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options lingo

• An option is in-the-money if its immediate
  exercise would yield a positive payoff
• (at the money, out of the money)
• Thus a call is in-the-money if the current stock
  price is greater than the exercise price.

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Profit

• The profit on a long position in an option is its
  payoff minus the premium, so if the option
  expires out of the money, the long side has a
  loss.
• The profit for the short side is the premium
  minus the payoff.

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Options trading

• Again IBM is not directly involved in trading
  options on IBM stock
• Prices determined by supply and demand. Markets
  are organized by specialists, just as with
  stocks.
• If demand gt supply, the price will rise ...
• Option clearing corporation organizes the
  transfer to shares of stock from writer to buyer.

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Option strategies

• Straddle simultaneous purchase of a put with a
  low exercise price and a call with a high
  exercise price
• a bet on volatility

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Straddle
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Portfolio Insurance

• Buying a put with exercise price below the
  current value of the portfolio.
• Then the investor is protected from a big loss,
  because if that happens the put will be in the
  money.
• The following diagram shows the payoff of a
  portfolio plus the put (doesnt show the premium,
  which would lower the line).

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Portfolio insurance

• Buying a put with an exercise price below the
  current price

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Put-Call Parity

• Consider the payoff on a portfolio consisting of
  a call plus the (present value of the) exercise
  price
• At the exercise date of the call, the payoff
  equals the exercise price of the call or the
  price of the stock, whichever is greater.
• Were assuming European options here (no early
  exercise)

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Payoff of call plus risk-free investment
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• Now consider a portfolio consisting of a holding
  of the stock, plus a put (with the same exercise
  price and exercise date as the call).

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Payoff of Stock plus Put
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• The payoffs are the same!
• Since the payoffs on the two portfolios are the
  same, so must be their current values.
• (Otherwise there would exist an arbitrage be
  sure you understand how you would set up the
  arbitrage to exploit a discrepancy).

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• Put-call parity, applied to current values
• Call premium PV of the exercise price
• Put premium current stock price.
• Implication if you can value a call, you can
  value a put via put-call parity.

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Option Valuation

• value (price) intrinsic value time value
• intrinsic value value if exercised now
• time value whatever is left over ...
• Time value cant be negative, since otherwise you
  would throw away an out of the money option, and
  exercise an in the money option.

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Bounds on option value

• obvious the value of a call is (1) always
  positive (2) always less than value of stock (3)
  always greater than or equal to its intrinsic
  value
• Less obvious call value gt stock price - present
  value of exercise price.

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A lower bound on call value
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Derivation via Put-Call Parity

• You can derive this bound via put-call parity.
• Claim C gt S - PVE,
• or C - S PVE gt 0
• Put-call parity C - S PVE P,
• so the claim reduces to P gt 0.
• The current price of a put is always positive.

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Direct derivation

• If C gt S - PVE werent satisfied, there would be
  an arbitrage
• Suppose C lt S - PVE.
• (1) buy the call, and (2) invest the present
  value of the exercise price, and (3) short the
  stock.
• This generates a positive amount of cash now.

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• If the call matures in-the-money, exercise it and
  cover the short position. Zero cash at the
  exercise date.
• If the call matures out-of-the-money, throw it
  away. Use cash to close out the short position.
  Positive cash at the exercise date.
• So its an arbitrage profitable no matter what
  happens to the stock price!

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Major determinants of option value

• exercise price -- the higher the exercise price,
  the lower (higher) the value of a call (put)
• exercise date -- the farther in the future is the
  exercise date, the greater is the value of either
  a call or a put
• stock price volatility -- the greater the stock
  price volatility, the greater the value of either
  a call or a put