Informationen zum Lesekurs Stochastisches Maschinelles Lernen (WS 2019/2020)

Estimation of Risk with ML 

The estimation of risk is a highly important topic in Mathematical Finance. Surprisingly, little is known about the estimation of risk. Here we will revisit some approaches and (hopefully) crack the open question how to estimate risk with deep neural networks.

This project already has quite a big code. We know how to efficiently estimate in certain circumstances (normal distribution, VaR, ES) and also managed to approximate the best estimator in some cases with NN. However, the empirical performance is not always good (this shall be analysed) and further estimation problems should be targeted.

This is a topic with possibly lots of work to do (could be done by 2 persons)

Literature

* Pitera/Schmidt (2018): Unbiased estimation of risk
* McNeil/Frey/Embrechts: Chapter on Backtesting and estimation of risk
* Grundmach: Code ERMAI
* Skawran: Bachelorarbeit

Deep hedging in affine models

The deep hedging approach from Bühler e.a. uses reinforcement learning to find hedging methodologies. We are interested in applying this approach in affine models and, say, on classical interest rate markets. A further goal would be to learn hedges in the case where jumps exists.

Code is already available in Lecture 3 of Josef Teichman's lectures and we need to adapt this to the new settings which we want to consider.

Literature

* Bühler, Gonon, Teichman, Wood (2018): Deep Hedging (on Arxiv)
* Föllmer/Schied: Stochastic Finance (as reference for hedging, efficient hedging, etc.)
* Filipovic (2015): Term Structure Models (as reference for affine models, term structures, etc.)

Deep Calibration

Calibration is an essential step in Mathematical Finance: how to fit a given model to observed data. In Statistics, this is done from historical data, in MF this comes from fitting model prices to observed option prices. A variety of questions arise in this context, which can be attacked with neural networks. In this task we will revisit and analyse the two calibration notebooks from Josef Teichmann (Heston / local vol).

Literature

• see Calibration notebooks
• Horváth e.a. (2019): Deep Learning Volatility, SSRN

Deep Portfolio Optimization

Again, following Bühler e.a.(2018) we study the problem of portfolio optimization. The basis for us is the indifference price and that, using a simulation routine as above, the indifference price can be learned with neural networks.

Code is already available in Lecture 4 of Josef Teichmann's lectures and we will try to adapt this to affine models.

Literature

* Bühler, Gonon, Teichman, Wood (2018): Deep Hedging (on Arxiv)
* Föllmer/Schied: Stochastic Finance (as reference for hedging, efficient hedging, etc.)
* Filipovic (2015): Term Structure Models (as reference for affine models, term structures, etc.)

Simulation tool for affine models

As is apparant from the above examples, we need an efficient implementation of simulation for affine models. This could be done by converting already existing code from, say R.

Literature

* Filipovic (2015): Term Structure Models 
     (as reference for affine models, term structures, etc.)
* Alfonsi (2015): Affine Diffusions and Related Processes: Simulation, Theory and Applications
* Zhang e.a. (2015): Affine Point Processes: Approximation and Efficient Simulation

Statistical Arbitrage

Following the recent work Rein e.a. (2019) we will analyze statistical arbitrages in this project. These are strategies which are performed repeatedly and give a profit on average. While some strategies already exist, we are going to seek new strategies and analyze their performance on data.
 

Literature

* Rein e.a. (2019), Generalized statistical arbitrage concepts and related gain strategies 
                    (arxiv: 1907.09218)
* Cuturi e.a. (2015), Mean-reverting portfolios: tradeoffs between sparsity and volatility 
                    (arxiv: 1509.05954)

Deep Optimal Trading

Here we follow the article Han & E (2019) and formulate a stochastic control problem (optimal trading) such that it can be trained with a deep network. The key to this is to set up the strategy as a function of the current price (a Markovian strategey) and hence allow this to be trained by a DNN. Our goal will be to train the optimal strategy on an estimated, say affine, model and then apply this to real data to study the generalization of the approach. It could also be interesting, in a second step, to improve the achieved strategy on the real data.

Literature

* Han, J. and W. E (2019): Deep Learning Approximation for Stochastic Control Problems