Experiments with GANs for Simulating Returns (Guest post)

  • The tails of an empirical return distribution are always thick, indicating lucky gains and enormous losses are more probable than a Gaussian distribution would suggest.
  • Empirical distributions of assets show sharp peaks which traditional models are often not able to gauge.
  1. The generator is able to output the feature set which is identical in distribution to the real dataset on which both the networks were trained
  2. The discriminator is able to tell real data from the generated one
  • log(D(x)) + log(1 — D(G(z))) — Done by the discriminator — Increase the expected ( over many iterations ) log probability of the Discriminator D to identify between the real and fake samples x. Simultaneously, increase the expected log probability of discriminator D to correctly identify all samples generated by generator G using noise z.
  • log(D(G(z))) — Done by the generator — So, as observed empirically while training GANs, at the beginning of training G is an extremely poor “truth” generator while D quickly becomes good at identifying real data. Hence, the component log(1 — D(G(z))) saturates or remains low. It is the job of G to maximize log(1 — D(G(z))). What that means is G is doing a good job of creating real data that D isn’t able to “call out”. But because log(1 — D(G(z))) saturates, we train G to maximize log(D(G(z))) rather than minimize log(1 — D(G(z))).
Fig 1. Returns by simple feed-forward GAN
Fig 2. shows the empirical distributions for AAPL starting 1980s up till now.
Fig 3. shows the generated returns by Geometric Brownian motion on AAPL.

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