Deep learning exotic derivatives
Abstract: Monte Carlo methods in derivative pricing are computationally expensive, in particular for evaluating models partial derivatives with regard to inputs. This research proposes the use of deep learning to approximate such valuation models for highly exotic derivatives, using automatic differentiation to evaluate input sensitivities. Deep learning models are trained to approximate Phoenix Autocall valuation using a proprietary model used by Svenska Handelsbanken AB. Models are trained on large datasets of low-accuracy (10^4 simulations) Monte Carlo data, successfully learning the true model with an average error of 0.1% on validation data generated by 10^8 simulations. A specific model parametrisation is proposed for 2-day valuation only, to be recalibrated interday using transfer learning. Automatic differentiation approximates sensitivity to (normalised) underlying asset prices with a mean relative error generally below 1.6%. Overall error when predicting sensitivity to implied volatililty is found to lie within 10%-40%. Near identical results are found by finite difference as automatic differentiation in both cases. Automatic differentiation is not successful at capturing sensitivity to interday contract change in value, though errors of 8%-25% are achieved by finite difference. Model recalibration by transfer learning proves to converge over 15 times faster and with up to 14% lower relative error than training using random initialisation. The results show that deep learning models can efficiently learn Monte Carlo valuation, and that these can be quickly recalibrated by transfer learning. The deep learning model gradient computed by automatic differentiation proves a good approximation of the true model sensitivities. Future research proposals include studying optimised recalibration schedules, using training data generated by single Monte Carlo price paths, and studying additional parameters and contracts.
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