Quantum pricing with a smile: implementation of local volatility model on quantum computer

Quantum algorithms for the pricing of financial derivatives have been discussed in recent papers. However, the pricing model discussed in those papers is too simple for practical purposes. It motivates us to consider how to implement more complex models used in financial institutions. In this paper, we consider the local volatility (LV) model, in which the volatility of the underlying asset price depends on the price and time. As in previous studies, we use the quantum amplitude estimation (QAE) as the main source of quantum speedup and discuss the state preparation step of the QAE, or equivalently, the implementation of the asset price evolution. We compare two types of state preparation: One is the amplitude encoding (AE) type, where the probability distribution of the derivative's payoff is encoded to the probabilistic amplitude. The other is the pseudo-random number (PRN) type, where sequences of PRNs are used to simulate the asset price evolution as in classical Monte Carlo simulation. We present detailed circuit diagrams for implementing these preparation methods in fault-tolerant quantum computation and roughly estimate required resources such as the number of qubits and T-count.