Hedging exotic derivatives via stochastic optimization models: a focus on Asian and Barrier Options and Worst Performance derivatives

Stochastic optimization techniques are applied to the framework of hedging exotic options from the perspective of a bank which sells derivatives and thus is exposed to potential future liabilities. The objective is to formulate an optimal strategy that minimizes the impact of potential losses. In practice, hedging involves the construction of a portfolio, referred to as hedging portfolio, which will be periodically adjusted in response to market changes that have an impact on the considered derivative. The optimization process relies on scenario trees generated through stochastic models. Two methods for simulating underlying stock prices (Geometric Brownian Motion and Moment Matching) are presented. Several optimization problems are then developed and compared based on their hedging performance, associated costs and profit and loss. Self-financing and non-self-financing strategies are analyzed and compared, exploring also rebalancing frequencies, transaction costs and their influence on overall hedging performance. Additionally, the scenario trees structure is analyzed to explore the trade-off between accuracy of the results and computational time. The analysis focuses on hedging exotic derivatives, such as Asian and Barrier options and Worst Performance derivatives, whose structural features and valuation models are explained in detail. European vanilla options are also used and considered as benchmarks to validate the models and compare stochastic optimization with traditional delta hedging.