Dual Pricing of Multiple Exercise Options

Christian Bender
Universität des Saarlandes

In this talk we discuss simulation methods for pricing of options with multiple early exercise opportunities. Options of this type are popular in interest rate markets (chooser flex caps) and, in particular, in electricity markets (swing options). The pricing problem for multiple exercise options can be reformulated in terms of a multiple optimal stopping problem and, hence, any close-to-optimal family of stopping rules leads to good lower bounds for the option price. We will therefore focus on how to calculate tight upper bounds for the option price. For American options (i.e. the case of a single early exercise
right) a duality approach for computing upper bounds was suggested independently by Rogers and Haugh & Kogan, who reformulated the American option pricing problem as a minimization problem over a set of martingales. It was generalized, in different ways, by Meinshausen & Hambly and Schoenmakers to multiple exercise options in discrete time with at most one right to be exercised per time step. After an introduction to these duality approaches we will present two generalizations. The first one admits to incorporate general volume constraints on the maximum number of rights to be exercised at the different days. The second one is a dual formulation for multiple exercise options in continuous time.


Haugh, M., Kogan, L.: Pricing American options: a duality approach.
Operations Research 52, 258-270 (2004)
Meinshausen, N., Hambly, B. M.: Monte Carlo methods for the valuation of multiple-exercise options. Math. Finance 14,
557-583 (2004)
Rogers, L.C.G.: Monte Carlo valuation of American options. Math.
Finance 12, 271-286 (2002)
Schoenmakers, J.: The real multiple dual. Preprint (2009).

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