Solving Black Scholes PDE with a quantum computer

A fundamental task of quantitative finance is calculating the fair price for options, derivative contracts that give the holder the right to buy or sell an underlying asset on a specified date and strike price. The first successful approach to this problem was Black-Scholes model [1]. Although an analytical solution exists, numerical methods are extensively analyzed to solve this model, serving as the ground for more complex models. It has been tackled stochastically employing a quantum Monte Carlo method with quadratic speedup [2] implemented in the IBM computers [3] and also using an unary representation of the asset value [4].

We present a digital quantum algorithm with exponential speed-up to solve the partial differential equation of the model in the case of vanilla options, showing a feasible approach to price financial derivatives on a digital quantum computer by Hamiltonian simulation techniques. The protocol introduced can be easily extended to include additional degrees of freedom, e.g. time dependent volatility or coupled options.

References:
[1] F.Black and M.Scholes The Journal of Political Economy 81, 637(1973).
[2] P.Rebentrost et al. Phys. Rev. A 98, 022321 (2018).
[3] N.Stamatopoulos et al, arXiv:1905.02666 (2019).
[4] S.Ramos-Calderer et al, arXiv:1912.01618 (2019)