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 most unexplained pricing information. Our model allows us to understand what are the key factors that drive asset prices , identify mis-pricing of stocks and  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 2.1 , we explain 8\% of the variation in individual stock returns and explain over 90\% of average returns for all anomaly sorted portfolios.
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.
R
A
a
D
,
R
most unexplained
least explained
R
, identify mis-pricing
to identify mispricing
A
to
R
2.1
2.6