Abstract
We present the Forest of Stochastic Trees (FOST) method for pricing multiple exercise options by simulation. The proposed method uses stochastic trees in place of binomial trees in the Forest of Trees algorithm originally proposed to value swing options, hence extending that method to allow for a multi-dimensional underlying process. The FOST can also be viewed as extending the stochastic tree method for valuing (single exercise) American-style options to multiple exercise options. The proposed valuation method results in positively- and negatively-biased estimators for the true option value. We prove the sign of the estimator bias and show that these estimators are consistent for the true option value. This method is of particular use in cases where there is potentially a large number of assets underlying the contract and/or the underlying price process depends on multiple risk factors. Numerical results are presented to illustrate the method.
Highlights
The broad class of stochastic optimal control problems includes many important applications in management sciences and quantitative finance such as development of natural resources, project initiation or abandonment, maintenance scheduling, land use decisions, and valuation and hedging of complex contracts
We consider the valuation of multiple exercise options as a stochastic optimal control problem with three relevant state variables—the underlying variable (S), number of exercise rights remaining (N ), and usage level (U) assuming some volume control
We work with the time-discretized problem and use dynamic programming to solve for the optimal exercise policy and the corresponding optimal value
Summary
The broad class of stochastic optimal control problems includes many important applications in management sciences and quantitative finance such as development of natural resources, project initiation or abandonment, maintenance scheduling, land use decisions, and valuation and hedging of complex contracts. For example the Black–Scholes–Merton formula for pricing European-style options does not have an American-style option analog Approaches such as binomial lattice methods, partial differential equation (PDE) methods, variational inequalities and integral equations have been adopted for pricing these types of derivatives. Multiple exercise options (MEOs) are generalizations of American-style options as they provide the holder more than one exercise right and sometimes control over one or more other variables, such as the amount exercised. Similar to pricing American-style options, MEO valuation is a problem in stochastic optimal control. For American-style options, the solution provides both a value and optimal exercise rule—a stopping time. Multiple exercise option valuation algorithms are generalizations of those used for pricing American-style options. We discuss MEO valuation in the context of swing options and note that the FOST method and the FOST price estimators’ properties apply to general MEOs
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