Exotic Options: I

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Chapter 14 Exotic Options I
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Exotic (nonstandard) options

• Exotic options solve particular business problems
  that an ordinary option cannot
• They are constructed by tweaking ordinary options
  in minor ways
• Relevant questions
• How does the exotic payoff compare to ordinary
  option payoff?
• Can the exotic option be approximated by a
  portfolio of other options?
• Is the exotic option cheap or expensive relative
  to standard options?
• What is the rationale for the use of the exotic
  option?
• How easily can the exotic option be hedged?

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Asian options

• The payoff of an Asian option is based on the
  average price over some period of time
  path-dependent
• Situations when Asian options are useful
• When a business cares about the average exchange
  rate over time
• When a single price at a point in time might be
  subject to manipulation
• When price swings are frequent due to thin
  markets
• Example
• The exercise of the conversion option in
  convertible bonds is based on the stock price
  over a 20-day period at the end of the bonds
  life
• Asian options are less valuable than otherwise
  identical ordinary options

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Asian options (cont.)

• There are eight (23) basic kinds of Asian
  options
• Put or call
• Geometric or arithmetic average
• Average asset price is used in place of
  underlying price or strike
• Arithmetic versus geometric average
• Suppose we record the stock price every h periods
  from t0 to tT
• Arithmetic average Geometric average

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Asian options (cont.)

• Average used as the asset price Average price
  option
• Geometric average price call max 0, G(T) K
• Geometric average price put max 0, K G(T)
• Average used as the strike price Average strike
  option
• Geometric average strike call max 0, ST
  G(T)
• Geometric average strike put max 0, G(T) ST
• All four options above could also be computed
  using arithmetic average instead of geometric
  average
• Simple pricing formulas exist for geometric
  average options but not for arithmetic average
  options

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Asian options (cont.)

• Example of use of average price option
• Receiving funds in a foreign currency every month
  for a year, with translation of the cash flow
  into US Dollars taking place every month. At the
  end of the year, the firm is concerned with the
  average exchange rate obtained, i.e. does not
  want to have received funds at a rate below a
  specific exchange rate.
• Example of use of average strike option
• Acquiring shares of a security S gradually by
  purchasing a fixed amount every month. Wanting to
  unload the full position at the end of the year,
  the firm needs an option that yields a payoff
  proportional to the difference between average
  security value and terminal value.

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Asian options (cont.)

• Comparing Asian options

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Asian options (cont.)

• For Asian options that average the stock price,
  the averaging reduces the volatility of the value
  of the stock entered in the payoff.
• Hence the price of the option is less than an
  otherwise similar traditional option.
• For Asian options that average the strike price,
  more averaging reduces the correlation between
  the terminal value of the stock and the strike
  price (computed as average of past stock prices),
  hence increasing the value of the option.

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Asian options (cont.)

• XYZs hedging problem
• XYZ has monthly revenue of 100m, and costs in
  dollars
• x is the dollar price of a euro
• In one year, the converted amount in dollars is
• Ignoring interest what needs to be hedged is
• A solution for XYZ
• An Asian put option that puts a floor K, on the
  average rate received

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Asian options (cont.)

• Alternative solutions for XYZs hedging problem

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Barrier options

• The payoff depends on whether over the option
  life the underlying price reaches the barrier
  path dependent
• Barrier puts and calls
• Knock-out options go out of existence if the
  asset price
• down-and-out has to fall to reach the barrier
• up-and-out has to rise to reach the barrier
• Knock-in options come into existence if the
  asset price
• down-and-in has to fall to reach the barrier
• up-and-in has to rise to reach the barrier
• Rebate options make a fixed payment if the asset
  price
• down rebates has to fall to reach the barrier
• up rebates has to rise to reach the barrier
• Barrier options are less valuable than otherwise
  identical ordinary options

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Barrier options (cont.)
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Barrier options (cont.)

• Knock-In option Knock-out option Ordinary
  option

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Barrier options (cont.)

• Overall use of Barrier options
• They come into or go out of existence depending
  on whether a barrier has been reached, hence they
  are cheaper than standard options. One could
  prefer to pay less and have an option that is
  alive only if needed, or have the option
  disappear if not needed after all.

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Compound options

• An option to buy an option

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Gap options

• A gap call option pays S K1 when S gt K2
• The value of a gap call is
• where
• and

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Gap Call, paying S-90 when Sgt100
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Gap Put, paying 90-S when Slt100
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Gap options (cont.)
Advantage of Gap options the payout can be
anything, and is not tied to the exercise price
the way traditional options are. From an
insurance point of view, one can see this as the
assurance that one will be 100 covered if the
loss exceed a certain amount. Or can create a
cheaper Gap option by choosing a higher
deductible.
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Exchange options

• Pays off only if the underlying asset outperforms
  some other asset (benchmark) outperformance
  option
• The value of a European exchange call is
• where
• 
  ,
• and