• Luyang Chen, Markus Pelger and Jason Zhu

Deep Learning in Asset Pricing

At AFTLab at Stanford, we have been developing new financial technologies that harness advances in big data, deep neural networks and high-performance computing and apply them to core problems in finance. In our recent paper “Deep Learning in Asset Pricing”, we show how to use deep learning to successfully predict stock prices and design optimal trading strategies.

The most fundamental question in asset pricing is to understand why different assets have different average returns. No-arbitrage pricing theory provides a clear answer - expected returns differ because assets have different exposure to systematic risk. All pricing information is summarized in the stochastic discount factor (SDF) or pricing kernel. Once, we know the SDF, we can predict which assets will have higher or lower future returns and trade accordingly. Therefore, our asset pricing model is directly relevant for optimal portfolio investment. Solving for the SDF is actually equivalent to obtaining the mean-variance efficient portfolio, which offers an optimal risk-return tradeoff.

The empirical quest in asset pricing for the last 40 years was to estimate a stochastic discount factor that can explain expected returns of all assets. There are four major challenges that the literature so far has struggled to overcome in a single model: (1) The SDF could by construction depend on all available information, which means that the SDF is a function of a potentially very large set of variables. (2) The functional form of the SDF is unknown and likely complex. (3) The SDF can have a complex dynamic structure and the risk exposure for individual assets can vary over time depending on economic conditions and changes in asset-specific attributes. (4) The risk premium of individual stocks has a low signal-to noise ratio, which complicates the estimation of an SDF that explains the expected returns of all stocks.

In this paper we estimate a general non-linear asset pricing model with deep neural networks for all U.S. equity data based on 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 that explains all stock returns from the conditional moment constraints implied by no-arbitrage. The use of machine learning techniques like deep neural networks is a natural idea to deal with the high dimensionality of the problem. One crucial insight of our work is that it is essential to incorporate economic conditions into the machine learning problem. Including the no-arbitrage constraint in the learning algorithm significantly improves the risk premium signal and makes it possible to explain individual stock returns. Empirically our general model outperforms out-of-sample the leading benchmark approaches and provides a clear insight into the structure of the pricing kernel and the sources of systematic risk.

Our estimation approach combines no-arbitrage pricing and three neural network structures in a novel way. It considers four key elements concurrently: First, we can explain the general functional form of the SDF as a function of the information set using a feedforward neural network. Second, we capture the time-variation of the SDF on macroeconomic conditions with a recurrent Long-Short-Term-Memory (LSTM) network that identifies a small set of macroeconomic state processes. Third, a generative adversarial network identifies the states and portfolios with the most unexplained pricing information which allows us to price all assets. Fourth, the no-arbitrage constraint helps to separate the risk premium signal from the noise and serves as a regularization to identify the relevant pricing information.

Our empirical analysis is based on a data set of 31,000 U.S. stocks with monthly returns from 1967 to 2016 combined with 46 time-varying firm-specific characteristics and 178 macroeconomic time series. It includes the most relevant pricing anomalies and forecasting variables for the equity risk premium. Our approach outperforms all other benchmark approaches, including linear models and deep neural networks that forecast risk premia. Our model has an annual out-of-sample Sharpe Ratio of 2.60 compared to 1.73 for the linear special case of our model, 1.52 for the deep learning forecasting approach and 0.76 for the Fama-French five factor model. At the same time, we can explain 8% of the variation of individual stock returns and explain 23% of the expected returns of individual stocks, which is substantially larger than the other benchmark models.

This post serves as a brief introduction of our model to illustrate that a successful use of machine learning methods in finance requires both subject specific domain knowledge as well as a state-of-the-art technical implementation. Our paper provides more details on model implementations and comprehensive empirical results.

The full paper can be downloaded here.

Luyang Chen, Stanford University.

Markus Pelger, Stanford University.

Jason Zhu, Stanford University.

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