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Optimal exercise of russian options in the binomial model

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.

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Optimal exercise of russian options in the binomial model

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  1. Optimal exercise of russian options in the binomial model Robert Chen Burton Rosenberg University of Miami

  2. A Russian Option • Pays max price looking back. • “Interest” penalty Computational Finance 2006 Chen and Rosenberg

  3. 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

  4. Binomial Model Computational Finance 2006 Chen and Rosenberg

  5. Arbitrage Pricing • Case of new maximum price: Computational Finance 2006 Chen and Rosenberg

  6. 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

  7. Worked example • Stock prices and option values Computational Finance 2006 Chen and Rosenberg

  8. Worked example … • Backward induction (apply formula) Computational Finance 2006 Chen and Rosenberg

  9. Worked example … • Continue backwards: adapt pricing argument or use martingale measure Computational Finance 2006 Chen and Rosenberg

  10. The full model • Time value r • Martingale measure and expectation Computational Finance 2006 Chen and Rosenberg

  11. Option pricing formula • Liability at N: • Backward recurrence (=1/(1+r)): Computational Finance 2006 Chen and Rosenberg

  12. 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

  13. Induction Theorems • First Induction Theorem • Second Induction Theorem • Monotonicity properties: expectation increasing in j and k. Computational Finance 2006 Chen and Rosenberg

  14. 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

  15. Exercise boundary … • Least integer hnsuch 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

  16. 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

  17. Algorithm … Computational Finance 2006 Chen and Rosenberg

  18. The end  • Thank you for your attention. • Questions? Comments? Computational Finance 2006 Chen and Rosenberg

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