We present a numerically efficient approach for learning minimal equivalent martingale measures for market simulators of tradable instruments, e.g. for a spot price and options written on the same underlying. In the presence of transaction cost and trading restrictions, we relax the results to learning minimal equivalent "near-martingale measures" under which expected returns remain within prevailing bid/ask spreads. Our approach to thus "removing the drift" in a high dimensional complex space is entirely model-free and can be applied to any market simulator which does not exhibit classic arbitrage. The resulting model can be used for risk neutral pricing, or, in the case of transaction costs or trading constraints, for "Deep Hedging". We demonstrate our approach by applying it to two market simulators, an auto-regressive discrete-time stochastic implied volatility model, and a Generative Adversarial Network (GAN) based simulator, both of which trained on historical data of option prices under the statistical measure to produce realistic samples of spot and option prices. We comment on robustness with respect to estimation error of the original market simulator.

翻译：我们提出了一种在数字上有效的方法,用于学习可交易工具市场模拟器的最低限度等效马丁格措施,例如现货价格和根据同一基本条件拟订的选择方案。在交易成本和贸易限制的情况下,我们放松结果,学习最低限度等效的“近期价格措施”,根据这些措施,预期回报仍然在现行投标/风险扩散的范围内。我们采用的方法是完全没有模型的,可以适用于没有典型仲裁的市场模拟器。由此产生的模型可用于风险中性定价,或者在交易成本或交易限制的情况下,用于“深置”交易。我们展示了我们的方法,将它应用到两种市场模拟器,一种自动递增的离心时间隐含的波动模型,以及一种基于“Generational Aversarial Net(GAN)”的模拟器,这两种模拟器都对根据统计措施得出的选择价格的历史数据进行了培训,以产生现实的现货和选择价格样本。我们评论了在估计原始市场模拟器错误方面的稳健性。