Optimal exercise of russian options in the binomial model Robert Chen Burton Rosenberg University of...
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Transcript of Optimal exercise of russian options in the binomial model Robert Chen Burton Rosenberg University of...
Optimal exercise of russian options in the
binomial modelRobert 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?