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. A Russian Option. Pays max price looking back. “Interest” penalty. Previous Work. Introduced by Shepp Shiryaev, Ann. Applied Prob., 1993. - PowerPoint PPT PresentationTRANSCRIPT
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Optimal exercise of russian options in the
binomial model
Robert ChenBurton RosenbergUniversity of Miami
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Computational Finance 2006 Chen and Rosenberg
A Russian Option Pays max price looking back. “Interest” penalty
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Computational Finance 2006 Chen and Rosenberg
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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Computational Finance 2006 Chen and Rosenberg
Binomial Model
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Computational Finance 2006 Chen and Rosenberg
Arbitrage Pricing Case of new maximum price:
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Computational Finance 2006 Chen and Rosenberg
The hedge Receive 2su/(u+1) cash Buy u/(u+1) shares stock at s If up:
Sell stock for su2/(u+1) Plus su/(u+1) cash gives su
If down: Sell stock for s/(u+1) Plus su/(u+1) cash gives s
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Computational Finance 2006 Chen and Rosenberg
Worked example Stock prices and option values
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Computational Finance 2006 Chen and Rosenberg
Worked example … Backward induction (apply formula)
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Computational Finance 2006 Chen and Rosenberg
Worked example … Continue backwards: adapt pricing argument or use martingale measure
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Computational Finance 2006 Chen and Rosenberg
The full model Time value r Martingale measure and expectation
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Computational Finance 2006 Chen and Rosenberg
Option pricing formula Liability at N:
Backward recurrence (=1/(1+r)):
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Computational Finance 2006 Chen and Rosenberg
Dynamic ProgramingSolution Liability value at N, all j,k (actually k-j)
Work backwards N-1, N-2, etc.
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Computational Finance 2006 Chen and Rosenberg
Induction Theorems First Induction Theorem
Second Induction Theorem
Monotonicity properties: expectation increasing in j and k.
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Computational Finance 2006 Chen and Rosenberg
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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Computational Finance 2006 Chen and Rosenberg
Exercise boundary …
Least integer hn such that
E(n,k-hn,k) obtains liability value.
If hn exists then hn’ exists for n≤n’≤N, and hn is decreasing in n.
In fact, 0≤hn-hn+1≤1.
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Computational Finance 2006 Chen and Rosenberg
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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Computational Finance 2006 Chen and Rosenberg
Algorithm …
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Computational Finance 2006 Chen and Rosenberg
The end
Thank you for your attention.
Questions? Comments?