Pricing and hedging American-style options with deep learning

Abstract

This paper describes a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies yielding small hedging errors.

A deep learning approach for computations of exposure profiles for high-dimensional Bermudan options

“…This makes the phenomenon difficult to analyze since the effect may lead underestimated exposure, unchanged exposure (from offsetting effects), or overestimated exposure. In addition to the classical regression methods, with e.g., polynomial basis functions, which are cited above there are several papers in which a neural network take the roll as a basis functions, see e.g., [21], [22], [23] and [24].…”

“…The minus sign in the loss-function transforms the problem from a maximization to minimization, which is the standard formulation in the machine learning community. Note the straightforward relationship between the loss function and the average cashflows in (22). In practice, the data is often divided into mini-batches, for which the loss-function is minimized consecutively.…”

“…The SGBM and LSM only compute the exercise strategy implicitly, i.e., by comparing the approximate option value and the immediate pay-off. Therefore small errors of the approximate option value close to the exercise boundary can lead to significant errors in the exercise strategy 22 . We therefore start by presenting a comparison of the exercise boundaries.…”

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

“…The main difference is that the optimization problem in (8), which is used to determine the evaluation points in (9), only depends on the random variable W whereas the optimization problem (see ( 4)) to determine the evaluation points in (6) in the LRV strategy depends on the random variable W and the function φ.…”