Dynamic Hedging of exotic derivatives via Stochastic Optimization: development and benchmarking with Delta Hedging and Deep Hedging

In the context of financial markets, a constant element is the presence of risks, which can make the management of financial instruments challenging. The complexity may increase in the case of exotic derivatives, whose value may depend on a variety of underlying variables and risk factors. In this framework, the present work develops a strategy for the dynamic hedging of exotic derivatives through stochastic optimization. The main idea behind a hedging problem is to optimize the management of a hedging portfolio designed to offset potential future liabilities arising from a position in the exotic derivative. The term 'dynamic', instead, refers to a multi-stage decision process where multiple decisions can be made over time. To deal with a dynamic hedging problem, the implemented strategy lies within the stochastic optimization framework, formulating, at each decision stage, an optimization problem which reflects the underlying idea of an Asset-Liability-Management problem. The optimization model accounts for a set of stochastic variables and leverages scenario trees for a discrete representation of possible future outcomes. To accurately approximate the space of future scenarios, two methods to simulate financial instrument dynamics are employed: the Geometric Brownian Motion simulation and the Moment Matching method. After an initial development phase, the proposed approach is evaluated in terms of effectiveness and efficiency: Monte Carlo simulation is exploited to test the implemented strategy over a set of simulated paths for the financial market dynamics. Lastly, a comparative analysis is performed to assess the effectiveness of the proposed strategy compared to traditional methods, such as Delta Hedging, and more recent approaches, such as Deep Hedging.