Optimal exercise of russian options in the binomial model

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Optimal exercise of russian options in the
binomial model

• Robert Chen
• Burton Rosenberg
• University of Miami

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A Russian Option

• Pays max price looking back.
• Interest penalty

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Previous Work

• Introduced by Shepp Shiryaev, Ann. Applied Prob.,
  1993.
• Analyzed in the binomial model by Kramokov and
  Shiryaev, Theory Prob. Appl. 1994.

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Binomial Model
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Arbitrage Pricing

• Case of new maximum price

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The hedge

• Receive 2su/(u1) cash
• Buy u/(u1) shares stock at s
• If up
• Sell stock for su2/(u1)
• Plus su/(u1) cash gives su
• If down
• Sell stock for s/(u1)
• Plus su/(u1) cash gives s

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Worked example

• Stock prices and option values

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Worked example

• Backward induction (apply formula)

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Worked example

• Continue backwards adapt pricing argument or use
  martingale measure

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The full model

• Time value r
• Martingale measure and expectation

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Option pricing formula

• Liability at N
• Backward recurrence (?1/(1r))

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Dynamic ProgramingSolution

• Liability value at N, all j,k (actually k-j)
• Work backwards N-1, N-2, etc.

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Induction Theorems

• First Induction Theorem
• Second Induction Theorem
• Monotonicity properties expectation increasing
  in j and k.

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Exercise boundary

• Exercise decision depends only on delta between
  maximum and current prices
• If k-j?k-j then E(n,j,k)?nuk implies
  E(n,j,k)?nuk

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Exercise boundary

• Least integer hn such that E(n,k-hn,k) obtains
  liability value.
• If hn exists then hn exists for nnN, and hn
  is decreasing in n.
• In fact, 0hn-hn11.

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Algorithm

• Value of option depends essentially on delta
  between maximum and current prices
• O(n2) for all values, O(n) to trace
• exercise boundary only

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Algorithm

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The end ?

• Thank you for your attention.
• Questions? Comments?