We estimate a general non-linear asset pricing model  with deep neural networks applied to all U.S. equity data combined with a substantial set of macroeconomic and firm-specific information. Our crucial innovation is the use of the no-arbitrage condition as part of the neural network algorithm. We estimate the stochastic discount factor (SDF or pricing kernel) that explains all asset prices from the conditional moment constraints implied by no-arbitrage. For this purpose, we combine three different deep neural network structures in a novel way: a feedforward network to capture non-linearities, a recurrent Long-Short-Term-Memory network to find a small set of  economic state processes and a generative adversarial network to identify the portfolio strategies with the least explained pricing information . Our model allows us to understand what are the key factors that drive asset prices to identify mispricing of stocks and to generate the mean-variance efficient portfolio. Empirically, our approach outperforms out-of-sample all other benchmark approaches: Our optimal portfolio has an annual Sharpe Ratio of 2.6, we explain 8\% of the variation in individual stock returns and explain over 90\% of average returns for all anomaly sorted portfolios .
We propose a novel approach to estimate asset pricing models for individual stock returns that takes advantage of the vast amount of conditioning information, while keeping a fully flexible form and accounting for time-variation. Our general non-linear asset pricing model is estimated with deep neural networks applied to all U.S. equity data combined with a substantial set of macroeconomic and firm-specific information. We estimate the stochastic discount factor that explains all asset returns from the conditional moment constraints implied by no-arbitrage. Our asset pricing model outperforms out-of-sample all other benchmark approaches in terms of Sharpe ratio, explained variation and pricing errors. We trace its superior performance to including the no-arbitrage constraint in the estimation and to accounting for macroeconomic conditions and non-linear interactions between firm-specific characteristics. Our generative adversarial network enforces no-arbitrage by identifying the portfolio strategies with the most pricing information. Our recurrent Long-Short-Term-Memory network finds a small set of hidden economic state processes . A feedforward network captures the non-linear effects of the conditioning variables . Our model allows us to identify the key factors that drive asset prices and generate profitable investment strategies .
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For this purpose, we combine three different deep neural network structures in a novel way: a feedforward network to capture non-linearities, a
Our asset pricing model outperforms out-of-sample all other benchmark approaches in terms of Sharpe ratio, explained variation and pricing errors. We trace its superior performance to including the no-arbitrage constraint in the estimation and to accounting for macroeconomic conditions and non-linear interactions between firm-specific characteristics. Our generative adversarial network enforces no-arbitrage by identifying the portfolio strategies with the most pricing information. Our
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to identify mispricing of stocks and to generate the mean-variance efficient portfolio. Empirically, our approach outperforms out-of-sample all other benchmark approaches: Our optimal portfolio has an annual Sharpe Ratio of 2.6, we explain 8\% of the variation in individual stock returns and explain over 90\% of average returns for all anomaly sorted portfolios
and generate profitable investment strategies