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
Optimal exercise of russian options in the
binomial model
Robert ChenBurton RosenbergUniversity of Miami
Computational Finance 2006 Chen and Rosenberg
A Russian Option Pays max price looking back. “Interest” penalty
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.
Computational Finance 2006 Chen and Rosenberg
Binomial Model
Computational Finance 2006 Chen and Rosenberg
Arbitrage Pricing Case of new maximum price:
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
Computational Finance 2006 Chen and Rosenberg
Worked example Stock prices and option values
Computational Finance 2006 Chen and Rosenberg
Worked example … Backward induction (apply formula)
Computational Finance 2006 Chen and Rosenberg
Worked example … Continue backwards: adapt pricing argument or use martingale measure
Computational Finance 2006 Chen and Rosenberg
The full model Time value r Martingale measure and expectation
Computational Finance 2006 Chen and Rosenberg
Option pricing formula Liability at N:
Backward recurrence (=1/(1+r)):
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.
Computational Finance 2006 Chen and Rosenberg
Induction Theorems First Induction Theorem
Second Induction Theorem
Monotonicity properties: expectation increasing in j and k.
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’
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.
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
Computational Finance 2006 Chen and Rosenberg
Algorithm …
Computational Finance 2006 Chen and Rosenberg
The end
Thank you for your attention.
Questions? Comments?