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Fries, Christian P.: Stable Monte-Carlo Sensitivities for Bermudan Callable Products. 2009.

The gamma of a standard cancelable swap. The gamma is evaluated by finite differences applied to standard Monte-Carlo simulation using the standard backward algorithm (red), finite differences (!) applied to a C^{0} smoothed backward algorithm (green) and finite differences applied to a C^{1} smoothed backward algorithm. The Monte-Carlo error standard backward algorithm (red transparent corridor) explodes for small shift sizes. The Monte-Carlo error of the C^{1} smoothed backward algorithm (yellow transparent corridor) remains stable for small shift sizes.

In this paper we discuss the valuation and sensitivities of financial products with early exercise rights (e.g., Bermudan options) using a Monte-Carlo simulation. The usual way to value early exercise rights is the backward algorithm. As we will point out, the Monte-Carlo version of the backward algorithm is given by an unconditional expectation of a random variable whose paths are discontinuous functions of the initial data. This results in noisy sensitivities, when sensitivities are calculated from finite differences of valuations.

We present a simple localized smoothing of the Monte-Carlo backward algorithm which results in stable, variance reduced sensitivities. In contrast to other payoff smoothing methods, the smoothed backward algorithm will converge to the true Bermudan value in the Monte-Carlo limit. However, it looses the property of being a strict lower bound.

The method is easy to implement since it is a simple modification to the pricing algorithm and it is independent of the underlying model.