160624.pdf

Abstract

In the field of financial mathematics, partial differential equations (PDEs) naturally arise in a variety of settings. However, developing algorithms for solving high-dimensional, nonlinear PDEs has been a difficult task given the classic “curse of dimensionality” and shortcomings of Monte Carlo techniques. This thesis applies a deep learning-based approach (referred to as the Deep BSDE solver) to handle a high-dimensional problem by introducing stochastic volatility in two settings: (1) option pricing and hedging and (2) portfolio optimization.

In the first half of this thesis, we apply the Deep BSDE solver to price European calls under the Black-Scholes, Heston and multiscale models. We exploit the architecture of the Deep BSDE solver in order to recover the delta of our approximated option price, a methodology we refer to as Deep Delta Hedging. As an application, we analyze the implied volatility surface of the option prices as produced by the Deep BSDE solver via principal component analysis.

In the second half of this thesis, we apply the Deep BSDE solver to determine the value functions and optimal strategies associated with the portfolio optimization problem under the Heston and multiscale models. We refer to the optimal strategy as determined by the Deep BSDE solver as the Deep Portfolio Strategy. As an application for the multiscale case, a simulation study is ran comparing the Deep Portfolio Strategy against the zeroth order approximated strategy, or Merton Strategy. The overall results suggest the Deep BSDE solver is suitable for all these scenarios compared to traditional benchmarks.