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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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 Presentation

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Page 1: Optimal exercise of russian options in the binomial model

Optimal exercise of russian options in the

binomial model

Robert ChenBurton RosenbergUniversity of Miami

Page 2: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

A Russian Option Pays max price looking back. “Interest” penalty

Page 3: Optimal exercise of russian options in the binomial model

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.

Page 4: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Binomial Model

Page 5: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Arbitrage Pricing Case of new maximum price:

Page 6: Optimal exercise of russian options in the binomial model

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

Page 7: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Worked example Stock prices and option values

Page 8: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Worked example … Backward induction (apply formula)

Page 9: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Worked example … Continue backwards: adapt pricing argument or use martingale measure

Page 10: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

The full model Time value r Martingale measure and expectation

Page 11: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Option pricing formula Liability at N:

Backward recurrence (=1/(1+r)):

Page 12: Optimal exercise of russian options in the binomial model

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.

Page 13: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Induction Theorems First Induction Theorem

Second Induction Theorem

Monotonicity properties: expectation increasing in j and k.

Page 14: Optimal exercise of russian options in the binomial model

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’

Page 15: Optimal exercise of russian options in the binomial model

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.

Page 16: Optimal exercise of russian options in the binomial model

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

Page 17: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

Algorithm …

Page 18: Optimal exercise of russian options in the binomial model

Computational Finance 2006 Chen and Rosenberg

The end

Thank you for your attention.

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