Beating the curse of dimensionality in options pricing and optimal stopping
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The fundamental problems of pricing high-dimensional path-dependent options and optimal stopping are central to applied probability and financial engineering. Modern approaches, often relying on ADP, simulation, and/or duality, have limited rigorous guarantees, which may scale poorly and/or require previous knowledge of basis functions. A key difficulty with many approaches is that to yield stronger guarantees, they would necessitate the computation of deeply nested conditional expectations, with the depth scaling with the time horizon T. We overcome this fundamental obstacle by providing an algorithm which can trade-off between the guaranteed quality of approximation and the level of nesting required in a principled manner, without requiring a set of good basis functions. We develop a novel pure-dual approach, inspired by a connection to network flows. This leads to a representation for the optimal value as an infinite sum for which: 1. each term is the expectation of an elegant recursively defined infimum; 2. the first k terms only require k levels of nesting; and 3. truncating at the first k terms yields an error of 1/k. This enables us to devise a simple randomized algorithm whose runtime is effectively independent of the dimension, beyond the need to simulate sample paths of the underlying process. Indeed, our algorithm is completely data-driven in that it only needs the ability to simulate the original process, and requires no prior knowledge of the underlying distribution. Our method allows one to elegantly trade-off between accuracy and runtime through a parameter epsilon controlling the associated performance guarantee, with computational and sample complexity both polynomial in T (and effectively independent of the dimension) for any fixed epsilon, in contrast to past methods typically requiring a complexity scaling exponentially in these parameters.
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