Predicting Conditional Expectations For Path-Dependent Events Using TDBP-Learning
116 Pages Posted: 25 Sep 2020 Last revised: 2 Nov 2020
Date Written: August 25, 2020
Conditional expectations (or probabilities) of events with multiple time steps into the future provide information about the likelihood of future events and, as such, are central to all decision making processes. The knowledge of such formulae is of fundamental importance, and has applications in all sectors of the economy. Hence, we are going to explore the problem of predicting conditional expectations for some path-dependent events driven by stochastic processes. That is, multi-step prediction problems where the correctness of the prediction is not revealed until more than one step after the prediction is made.
On one hand, parametric models for computing conditional expectations of path-dependent events are tied to their ability of capturing the dynamics of the underlying stochastic processes. As a result, their misspecification will lead to computation errors, since their formulae depend on the particular form of the dynamics of the stochastic processes chosen.
On the other hand, non-parametric models, such as neural networks (NNs), use data to estimate the implicit stochastic processes driving the dynamics of the future events and their relationship with conditional expectations. However, while these methods are good at solving single-step prediction problems, they cannot solve multi-step prediction problems. One solution is to consider methods specifically designed for solving multi-step prediction problems, such as temporal difference (TD) procedures, which are widely used in reinforcement learning (RL). We show that these methods are particularly well suited for capturing the conditional probability distribution function of stochastic processes. To infer the characteristics of conditional expectations for path-dependent events, we propose to combine the error backpropagation algorithm with the TD methods, getting the temporal difference backpropagation (TDBP) model.
We propose a framework for learning conditional expectations, both in discrete and continuous space, and perform a thorough analysis of our model by testing it on well known continuous stochastic processes, namely, the Black-Scholes-model, the Merton model, the Heston model, and the Bates model. We show that directly feeding the continuous stochastic processes into Deep Networks improve results, leading to smoother and more accurate conditional expectations. Further, since Deep Networks generalise learning across similar states, they can be used for learning more complex nonlinear functions. Another advantage of using Deep Networks with the TDBP model is that the continuous process can be a vector, allowing for the computation of multiple processes conditional expectations.
We will show that, in the continuous space, the TDBP-model recover nearly exactly the forward price, the call option price, and the digital option generated with Monte Carlo simulations. Further, we will extend our analysis to strong path-dependent payoffs such as forward starting contracts, barrier options, and American options. Finally, we will compute multiunderlying contingent claims such as basket options. In conclusion, we can approximately predict conditional expectations for path-dependent events with a single TDBP-model or an ensemble of such models.
Keywords: Multi-step Prediction Problems, Reinforcement Learning, Temporal Difference Backpropagation Learning, Predictive Representation Theory, Conditional Expectations, Conditional Probabilities
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