Deep Learning and Deep Hedging

Nowadays, if you want to be taken serious as a quant, you need to create an aura of being an expert in machine learning (or actually be one). With this keyword in your agenda it is currently easy to lock in a job, a consulting mandate, a paper or a conference talk. However, to really find out what is behind it at a deeper level, one needs to screen the scientific literature, which can take a really long time. We can help you to speed it up.

The foundation is the so-called universal approximation theorem, which essentially says that almost any function can be approximated by layers of affine and so-called activation functions. Many people have been working on this field, beginning with Kolmogorov in the 50s and one of the most-cited results stems from Kurt Hornik, Maxwell Stinchcombe and Halbert White from 1989.

If the number of layers is at least two, then the network is called a “deep” network. An activation function is a non-constant, bounded, and monotonically-increasing continuous function; one can think of it as a cumulative probability distribution function.

To get a flavor of the applications in finance, there are two papers that can help you getting up the curve, one about deep learning to improve predicting the value of a stock price at the next tick based on order book information, the other one on deep hedging a portfolio of derivatives. We will list both of them with abstracts.


  • Universal Features of Price Formation in Financial Markets: Perspectives from Deep Learning by Justin Sirignano and Rama Cont, March 16, 2018


Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific.

The universal model — trained on data from all stocks — outperforms, in terms of out-of-sample prediction accuracy, asset-specific linear and nonlinear models trained on time series of any given stock, showing that the universal nature of price formation weighs in favor of pooling together financial data from various stocks, rather than designing asset or sector-specific models as commonly done. Standard data normalizations based on volatility, price level or average spread, or partitioning the training data into sectors or categories such as large/small tick stocks, do not improve training results. On the other hand, inclusion of price and order flow history over many past observations is shown to improve forecasting performance, showing evidence of path-dependence in price dynamics.


  •  Deep Hedging by Hans Bühler, Lukas Gonon, Josef Teichmann, Ben Wood, Feb 2018


We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, market impact, liquidity constraints or risk limits using modern deep reinforcement machine learning methods.

We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show that the set of constrained trading strategies used by our algorithm is large enough to ϵ-approximate any optimal solution.

Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available.

We illustrate our approach by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard “complete market” solution.


Overall we observe that while neural networks are actually old hats, it is today’s computational power that allows more and more applications. Cont and Sirignano for example used 500 GPUs to compute their results. However, one may want to keep in mind is that the best papers are those never published, as the results will go straight into a trading application rather than an academic journal.

Thorsten Schmidt and Uwe Wystup, MathFinance


K. Hornik, M. Stinchcombe, H. White (1989). Multilayer Feedforward Networks are Universal Approximators, Neural Networks2(5), 359-366.

Justin Sirignano and Rama Cont. Universal Features of Price Formation in Financial Markets: Perspectives From Deep Learning, March 2018: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3141294

Hans Bühler, Lukas Gonon, Josef Teichmann, Ben Wood. Deep Hedging, Feb 2018: https://arxiv.org/abs/1802.03042


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