Implied volatility is critical in financial markets, especially for option pricing. Traditional methods for its calculation sometimes are not well suited to some scenarios. Recent developments in neural networks have provided more efficient alternatives.

Leveraging advances in quantum computing, our research introduces quantum neural networks for computing implied volatility, assessing the feasibility and characteristics of this novel approach. We focus on two quantum neural network architectures: Dissipative Quantum Neural Networks (DQNN) and Parameterized Variational Quantum Circuits (PVQNN). DQNN, similar to classical neural networks in structure and training ease, faces challenges with quantum state outputs and data decoding, impacting performance negatively. Besides, limited by the reliance on network output states at each layer, DQNN faces challenges in implementation with the current state of quantum hardware.

In contrast, PVQNN offers a more promising solution. Compared to DQNN, PVQNN requires fewer qubits, can apply traditional optimizers to train the model, and can run on NISQ devices. This research thoroughly examines various aspects influencing PVQNN's performance, including training data characteristics, data re-uploading technology, network size, data encoding methods, and quantum circuit design. The selected PVQNN model can achieve high accuracy in implied volatility computation with $R^2$ of approximately 0.999. In addition, we find that the PVQNN can obtain satisfactory results even with limited training data, setting it apart from traditional neural networks.

This thesis not only adopts a new model to compute implied volatility but also deepens the understanding of quantum neural networks in financial modeling. However, due to resource constraints, our experiments are conducted in simulations on traditional computers, and thus our study focuses mainly on the expressive power of QNNs rather than their operational efficiency.