In the first part of the seminar, we review the historical development of classical computing from its origin until nowadays and introduce a short but general overview of quantum computing, with special emphasis on the state-of-the-art of gate-based quantum computers and applications in finance. In the second part, we present a novel digital quantum algorithm to solve the pricing problem using a Black-Scholes model by mapping it to the Schrödinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding the dynamics into an enlarged Hilbert space, which makes use of only one additional ancillary qubit. More generally, this algorithm shows a feasible approach for solving forward and backward Kolmogorov equations on a digital quantum computer based on Hamiltonian simulation. Progress and extensions of this algorithm are discussed. For some cases, the algorithm remarkably provides a quantum advantage since the terms in the Hamiltonian can be truncated by a polynomial number of interactions while keeping its error bounded. We report expected accuracy levels comparable to classical numerical algorithms by using 10 qubits and 94 entangling gates on a fault-tolerant quantum computer.