Deep-PDE-Solvers

Code of numerical experiments in https://arxiv.org/abs/1810.05094.

   @misc{vidales2019unbiased,
          title={Unbiased deep solvers for parametric PDEs}, 
          author={Marc Sabate Vidales and David Siska and Lukasz Szpruch},
          year={2019},
          eprint={1810.05094},
          archivePrefix={arXiv},
          primaryClass={q-fin.CP}
    }

We use Feed Forward networks to price European options at any time t. We use two different learning algorithms:

• The BSDE method --> we also learn the hedging strategy
• Directly learn the price as the conditional expectation E[X_T | F_t], which is the orthogonal projection of X_T onto the sigma-algebra F_t.

In addition, we de-bias the approximation of the solution of the PDE by leveraging Monte Carlo and the Deep Learning solution (see our paper). In that sense, the deep learning approximation of the stochastic integral arising from the Martingale representation of g(X_T) (where g is the payoff at time T) can be seen as a control variate. Under this perspective, we have another learning algorithm:

• Minimising the variance of g(X_T)-cv, where cv is the approximation of the stochastic integral.

Running the code

Pricing exchange options in the Black-Scholes world by solving the 2-dimensional PDE:

python pde_BlackScholes_exchange.py --use_cuda --device 0 --d 2

and visualizing the results

python pde_BlackScholes_exchange.py --d 2 --visualize

Solving the 2-dimensional Heat equation with time reversed

python pde_Brownian.py --use_cuda --device 0 --d 2

and visualizing the results

python pde_Brownian.py --d 2 --visualize