PT Unknown
AU Dickmann, F
TI Multilevel approach for Bermudan Option Pricing
PD 07
PY 2015
LA en
DE Multilevel; Monte Carlo; Bermudan Option; American Option; Variance Reduction
AB The Multilevel approach has been introduced into stochastics by Heinrich 2001 and Giles 2008. It is an idea about how to reduce the complexity of Monte Carlo simulations, if the precision and computational time of these simulations depend on a parameter.

In this work, the Multilevel approach will be applied to approximate the fair price of a Bermudan option. The latter is a financial option that gives the holder the right to get an amount of money depending on a stochastic process at one of finitely many exercise dates. The strategy when to exercise the option should therefore by optimized. The task is now to find stochastic methods that calculate a lower and an upper bound for the fair price of such an option.

If the performance of such methods is measured via its mean-squared error epsilon, the calculation of lower bounds has a complexity of epsilon^{-3} in the usual case. Here, the usual case is the combination of a good-natured problem and the use of a stochastic mesh method. By using the Multilevel approach, we can reduce the complexity down to epsilon^{-2.5}. In other cases, the reduction of complexity can be even of order epsilon^{-1} instead of epsilon^{-0.5}.

In order to find upper bounds, we use the dual method from Rogers 2002 and Haugh and Kogan 2004. It expresses the fair price of the option as a minimization problem over a set of martingales. Andersen and Broadie 2004 exploit this idea by approximating martingales with nested simulations. These nested simulations lead to a high computational complexity. Depending on the problem, this complexity can be of order epsilon^{-3} or even epsilon^{-4}. The Multilevel approach reduces this order in each case to epsilon^{-2} up to a logarithmic factor.
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