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?