Quantum Pricing with a Smile: Implementation of Local Volatility Model on Quantum Computer

Applications of the quantum algorithm for Monte Carlo simulation to pricing of financial derivatives have been discussed in previous papers. However, up to now, the pricing model discussed in such papers is Black-Scholes model, which is important but simple. Therefore, it is motivating to consider how to implement more complex models used in practice in financial institutions. In this paper, we then consider the local volatility (LV) model, in which the volatility of the underlying asset price depends on the price and time. We present two types of implementation. One is the register-per-RN way, which is adopted in most of previous papers. In this way, each of random numbers (RNs) required to generate a path of the asset price is generated on a separated register, so the required qubit number increases in proportion to the number of RNs. The other is the PRN-on-a-register way, which is proposed in the author's previous work. In this way, a sequence of pseudo-random numbers (PRNs) generated on a register is used to generate paths of the asset price, so the required qubit number is reduced with a trade-off against circuit depth. We present circuit diagrams for these two implementations in detail and estimate required resources: qubit number and T-count.